lauri tamminen pricing of constant maturity spread...
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Lauri Tamminen
Pricing of Constant Maturity Spread Options in theDeterministic Libor Market Model FrameworkMaster of Science Thesis
Examiners:Samuli Siltanen, Juho
Kanniainen
Topic accepted by the Faculty
Council on 3.9.2008
ABSTRACT
TAMPERE UNIVERSITY OF TECHNOLOGYMaster of Science Degree Programme in Science and EngineeringTamminen Lauri: Pricing of Constant Maturity Spread Options in the Determin-istic Libor Market Model FrameworkMaster of Science Thesis, 63 pages, 4 Appendix pagesTopic accepted by the Faculty Council in September 2008Major: Technical mathematicsExaminers: professor Samuli Siltanen, researcher Juho KanniainenKeywords: CMS spread option, Range accrual note, Libor Market Model, BGM, Semi-parametric correlation, Swaption, Cap, Caplet, Geometric Brownian motion, Brownianbridge
In this work we have investigated the pricing of exotic CMS spread options and CMSrange accrual notes in the general framework Libor Market Model (LMM) framework.The aim has been to put the treatment of the general n-factor market model on a firmmathematical basis, details which have not been discussed in literature before.
The pricing has been done via calibrating (fitting) the Libor market n-factor modelto the market quoted swaption volatilities. In the forward rate framework swaptionvolatilities are dependent on the forward rate correlations and the search of the optimalmodel parameters has been done via semi-parametric approach. The use of semi-parametric approach has the advantage of having automatically empirically appealingfeatures embedded in the forward rate correlations. The optimization routines, becauseof their relatively low dimensionality, remain also stable. The fitting quality of themethod for swaption volatility quotes turned out to be accurate and better comparedto the classical angles form parametrization usually used in the literature. The methodby construction recovers the caplet prices exactly. It was observed that the number ofmodel factors had a slight impact for the model implied term structure of volatility.It turned out to be more time homogenous when the number of model factors wereincreased.
The CMS spread option prices were found out to be highly dependent on the numberof used model factors. Using principal component analysis, we extracted the forwardrate responses for different factors and investigated how different factors change theyield curve. From the factor loadings we found out that the first factor with the mostexplanative power was responsible for the parallel shifts in the yield curve. Higherorder factors were responsible for bending the yield with respect to the different axisby changing the slope and curvature of the yield curve. At least 3 factors were necessaryto produce realistic spread option prices, while 5 factors were found out to be sufficientin 99% accuracy in contrast to the full-factor case. The need of extra fourth and fifthfactor were emphasized when pricing very long expiring options with a long simulationstep procedure. Our proposed method to price CMS range accruals, whose payoff isbased to the daily reference index performance, was based on mixing the LMM tothe simple geometric Brownian motion like dynamics for the swap rates ”in between”dates by using Brownian bridging techniques.
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TIIVISTELMA
TAMPEREEN TEKNILLINEN YLIOPISTOTeknis-luonnontieteellinen koulutusohjelmaTamminen Lauri: Vakiomaturiteettisten Korkoero Optioiden Hinnoittelu Deter-ministisella Libor MarkkinakorkomallillaDiplomityo, 63 sivua, 4 liitesivuaAihe hyvaksytty tiedekuntaneuvoston kokouksessa syyskuussa 2008Paaaine: Teknillinen matematiikkaTarkastajat: professori Samuli Siltanen, tutkija Juho KanniainenAvainsanat: CMS korkoero optio, Range accrual optio, Libor Markkinakorkomalli, BGM,Semiparametrisoitu korrelaatio, Swap-optio, Korkokatto, Geometrinen Brownin liike,Brownin silta
CMS spread optio eli vakiomaturiteettinen korkoero-optio on eksoottinen korko-optio,jossa allaoleva kohde-etuus on kahden eri maturiteettisen kiintean koron valinenkorkoero. Korkoero-optioiden hinnoittelu korkomallilla on haasteellista, silla niidenperformanssi on hyvin riippuvainen korkokayran muodonmuutoksista. Siten niidenhinnoittelussa kaytetty korkomalli joutuu todelliseen testiin sen suhteen, kuinka hyvinse kykenee muuttamaan korkokayran muotoja. Klassisille 1 faktori spot korkoma-lleille, jossa spot korko on vain yhden Brownin liikkeen ajama prosessi (ja jossa havait-tavia markkinakorkoja mallinnetaan vain epasuorasti), ovat realistiset kayramuutoksetolleet tunnettu ongelma. Ongelma spot-korkomalleille on niiden implikoimat lahesyhdensuuntaiset kayraliikkeet ja tasta eparealistisuudesta johtuen ne hinnoittelevatspread optiot yleensa liian halvoiksi
Olemme tarkastelleet CMS spread optioiden ja kertyvien korkokaytava spreadoptioiden hinnoittelua Libor markkinakorkomallilla (LMM). Kun plain vanillakoronvaihtosopimus optioit (swap optiot) ovat riippuviisia markkinoiden nakemyk-sesta allaolevan kiintean koron implisiittisesta volatiliteetista, ovat CMS spread optiotlisaksi riippuvaisia kiinteiden korkojen valisesta implisiittisesta korrelaatiosta. CMSspread optioille ei ole olemassa porssimarkkinaa kuten plain vanilla swap optioille,joiden markkinat ovat hyvin likvideja ja vaihdoltaan suuria. Epalikvidista spread optiomarkkinasta johtuen ei ole olemassa markkinaodotuksia kiinteiden korkojen korre-laatioille. Tavoitteea on siten hinnoitella CMS spread optiot konsistentisti havaitunlikvidin swap-optio markkinan kanssa.
Modernimmassa Libor markkinakorko lahestymistavassa on hyvana puolena,etta markkinoilla havaittavia forward korkoja mallinnetaan suorasti. Yksi tyonkeskeisimpia tavoitteita on ollut saada yleisen ns. n-faktorimallin teoria tasmallisellematemaattiselle perustalle ja perustella kuinka eksoottisten korko-optioiden hinnoit-telu Monte Carlo simulaatiolla kaytannossa tapahtuu. Tata aihetta ei ole yksityisko-htaisesti kirjallisuudessa ennen kasitelty.
Hinnoittelu on toteutettu kalibroimalla (sovittamalla) yleisen Libor n-faktorimallintuntemattomat parametrit markkinoilta havaittaviin swap optioiden implisiittisiinvolatiliteeteihin, siten etta mallin hinnoitteluvirhe markkinoihin on mahdollisimman
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pieni. Osoittautuu, etta Libor mallin ymparistossa koronvaihtosopimus optioidenvolatiliteetit ovat riippuvaisia mallinnettavien forward korkojen valisista korre-laatioista. Forward korrelaatioiden sovittaminen swap optioiden volatiliteet-teihin tuo siten linkin mallin optimaalisten parametrien hakemiselle. Korrelaa-tioiden sovitus on tehty ns. semi-parametrisella lahestymistavalla. Idea onparametrisoida forward korkojen valinen korrelaatio vain muutamalla parametrilla.Etuna kaytetyssa parametrisoinnissa on, etta se sisaltaa sisaanrakennettuna empiiris-esti havaitut korrelaatiopinnan mielekkaat ominaisuudet, seka minimointiongelmiensuhteellisen pienen ulottuvuuden johdosta ne ovat numeerisesti stabiileja.
Tyossa havaittiin, etta menetelmalla saatu sovitteen laatu swap optioiden volatili-teetteihin osoittautui olevan todella tarkka, parempi kuin yleisesti monissa kirjal-lisuuslahteissa kaytetylla klassisella kulmaparametrisointi menetelmalla. Kalibroin-timenetelman konstruktion johdosta korkokatto optioiden (cap optioiden) implisiittisetvolatiliteetit saadaan mallilla eksaktisti sovitettua markkinoihin. Yleisesti korkoma-llin stabiilisuuden testaukseen kaytetty volatiliteetin aikarakenteen muuntautuminenajansuhteen osoittautui olevan riippuvainen kaytettyjen faktoreiden lukumaarasta.Aikarakenne osoittautui ajansuhteen sita pysyvammaksi, mita enemman faktoreitamallissa oli kaytossa. Ajan suhteen pysyvaa volatiliteetin aikarakennetta voidaan yleis-esti pitaa mallille hyvana ominaisuutena, silla markkinoiden volatiliteetin aikarakenneon havaittu markkinoilla olevan ajan suhteen pysyva.
CMS spread optioiden hinnoittelussa havaittiin hintojen olevan vahvasti riipu-vaisia kaytettyjen faktoreiden lukumaarasta. Paakomponenttianalyysia kayttaenselvitimme forward korkojen vasteet eri faktoreille ja analysoimme kuinka eri faktoritmuuttavat korkokayraa ja vaikuttavat siten optioiden hintoihin. Havaitsimme,etta ensimmaisella faktorilla oli suurin selittava vaikutus, mutta forward-korkojenvasteiden tasaisuudesta johtuen ensimmainen faktori aiheutti lahes yhdensuuntaisiakayraliikkeita. Muut faktorit kaytannossa aiheuttivat korkokayran taipumisen erikohtien suhteen. Vahintaan kolme faktoria oli valttamattomyys tuottamaan realistisiaspread optioiden hintoja. Toisaalta 5 faktoria havaittiin olevan riittava maara 99%tarkkuudella tuottamaan taysfaktorimallin mukaisia hintoja. Neljannen ja viidennenfaktorin vaikutus havaittiin olevan erityisesti valttamatonta pitkamaturiteettistenoptioiden hinnoittelussa.
Libor mallin ymparistossa spot markkinakorot voidaan kaytannossa simu-loida vain forward korkojen maturiteeteissa. Kertyvien korkokaytava spreadoptioiden performanssi riippuu toisaalta paivittaisista spot markkinakorkojenfiksauksista, joten tarvitaan menetelma jolla korot voidaan simuloida paivittain.Kaytetyssa menetelmassa rakensimme spot markkinakoroille geometrisen Brownin liik-keen mukaisen dynamiikan siten, etta se konvergoitui realisaatioittain Libor mallinimplikoimiin markkinakorkoihin forward korkojen maturiteeteissa. Tama saatiinkaytannssa aikaan kayttaen Brownin siltoja.
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PREFACE
This thesis was written while I was working in derivatives sales at Pohjola Bank. Istarted my job as a summer trainee on the summer 2007. The subject for the thesiswas found from the need to widen the class of traded products in the derivatives salesat Pohjola Markets. I started to write the thesis on the following summer 2008, butthe preliminary research and investigation was done during the winter 2007-2008.
I would like to express my gratitude for the senior researcher Juho Kanniainen fromwhom I found the job from Pohjola Bank and who has given me feedback while readingthis thesis. Also I would like to thank the whole derivatives sales group for fundingthis project and giving me 100% focus on doing this work. Special thanks goes toPetri Kangas, who has given me a great amount of fixed income market knowledgeand who has answered patiently for my questions. Matti Hanninen has been a greathelp in technical LATEX issues while I was writing this manuscript. Also thanks goesto Professor Samuli Siltanen for examining this thesis. Finally I am grateful for myfamily and girlfriend Martina for supporting me in my studies.
Helsinki 26th August 2008
Lauri [email protected]
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Contents
1 Introduction 1
2 Basic Definitions and Facts from Probability Theory 4
2.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Ito’s Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Properties of Ito Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Girsanov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Basic Building Blocks 18
3.1 Zero Coupon Bonds and Libor Rates . . . . . . . . . . . . . . . . . . . 18
3.2 Swap Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Caplets and Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Assumptions under Libor Market Model 26
4.1 Libor Dynamics under Forward Measure . . . . . . . . . . . . . . . . . 26
4.2 Joint Dynamics of the Libor Rates . . . . . . . . . . . . . . . . . . . . 29
4.3 Parametrization of instantaneous correlation . . . . . . . . . . . . . . . 33
5 Calibration of the Model 36
5.1 Black Swaption Volatilities under LMM . . . . . . . . . . . . . . . . . 36
5.2 Term Structure of Volatility . . . . . . . . . . . . . . . . . . . . . . . . 39
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5.3 Optimization Based Calibration Algorithm . . . . . . . . . . . . . . . . 39
6 Drift Approximation Methods and Simulation 43
6.1 Evolution of the Stochastic Part . . . . . . . . . . . . . . . . . . . . . . 45
6.2 Evolution of the Deterministic Part . . . . . . . . . . . . . . . . . . . . 46
7 Numerical Results 48
7.1 Model Calibration with Different Factors . . . . . . . . . . . . . . . . . 49
7.2 Pricing CMS Spread Options . . . . . . . . . . . . . . . . . . . . . . . 56
7.3 Pricing CMS Range Accrual Notes . . . . . . . . . . . . . . . . . . . . 60
8 Conclusion 63
A Derivation of Swaption Volatility Coefficients 66
B Used Model Input Data 68
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Chapter 1
Introduction
The purpose of this work is to develop a pricing engine, that can be used to priceso called constant maturity spread options (shortly CMS spread options), where theunderlying instrument is the spread between two market swap rates.
In the Black & Scholes framework, where the underlying instruments S1 and S2 aredriven by simple geometric Brownian motions with constant drift and volatility, it ispossible to compute simple Black & Scholes approximations for spread option prices(see for example [5, pp. 604–606] for the derivation of approximate formula). Theproblem is, however, that to arrive at a unique price, we need in addition to the impliedswap rate volatilities the quantity called implied correlation, because the net volatilityof the underlying S1 − S2 is dependent on the interrelated correlation between indexrates S1 and S2. As spread options are illiquid options, where there does not existsan efficient market in high volumes, there is no market access for implied correlations.Currently market makers in the spread option market have quite large disagreementsabout spread option implied correlations, which can be probably explained that thereis not a generally accepted way to price market rate spread options. The philosophygoes that implied correlation can be just considered a way to quote the right price,not the other way around!
Therefore we need to take a different route, and develop a model that prices the CMSspread options consistently with the observed market. The classical approach wouldbe to model the infinitesimally short spot rate. These so called spot rate models,however, have known problems in interest rate modeling. The biggest problem is thatthey lack in realism, because by construction the whole yield curve is characterizedby only one quantity which is an infinitesimally short spot rate. The infinitesimallyshort spot rate does not even exists in practise! The modern approach in the interestrate modeling is to model the observed market rates. The drawback of this approachis, however, that the problem dimension explodes dramatically. If the approach isto model forward interest rates, among academics this approach is known as LiborMarket Model (LMM), while the practioners in industry usually know it as the BGMmodel by the original authors Brace, Gatarek and Musiela. We will refer to it as LiborMarket model or shortly as LMM.
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In this thesis we are discussing deterministic Libor market model, which means thatthe volatility processes of the underlying rates are deterministic functions of timein contrast to the stochastic Libor Market model, which assumes the forward ratevolatility being a certain stochastic process. The assumption for a deterministicvolatility process has the implication that the model implies always flat volatilityfor plain vanilla options. This means that for the same underlying, the model impliedvolatility is independent of the option strike price. This is the worst drawback indeterministic LMM, because the smile in plain vanilla option market can be observed.Volatility smile in swaptions, however, makes modeling much more complicated as theswaption smile surface would be 3 dimensional. It would be dependent on the optionexpiry and the underlying swap tenor with additional dependence on option strikes.Stochastic Libor market model imply volatility smile but it is quite a new model stillunder much research. If a reader is interested in stochastic market models, then astarting point in this area would be the paper by Zhang [23] who develops the Hestontype stochastic volatility equity model in market model framework. We do not pursuestochastic LMM any further however.
The structure of this thesis starts from the basic mathematical definitions and toolsfrom stochastic calculus which are presented in Chapter 2. The stochastic calculus is atechnical branch of mathematical analysis as its fundaments are based on very difficultresults from abstract measure theory. We have, however, tried to keep the approach inthis work in a most practical level, focusing the discussion in the application not in theunderlying mathematical machinery. Where we have thought it would be appropriate,an interested reader is advised to look for mathematical details in the references.
In Chapter 3 the necessary financial instruments that are needed in the constructionof the Libor market model are presented. These instruments are basic zero couponbonds or discount bonds and forward and swap rates with basic instruments based onthese rates called forward rate agreements and swaps. Finally the plain vanilla capletand swap option instruments are presented where the underlying can be considered tobe a swap rate in the latter while a forward rate in the former.
Chapter 4 presents the basic assumptions in the Libor market model. The constructionis motivated to base on the consistency between the market way to price or moreprecisely quote the price for caplets using the Black & Scholes type model. It turnsout that the Libor Market Model can be considered as a series of Black & Scholestype models used simultaneously. The no arbitrage joint dynamics for the rates arespecified using martingale pricing techniques.
From our application perspective, Chapter 5 shows how the swap rate based financialinstruments can be handled in the forward rate framework. Further a method to solvethe unknown model parameters consistent with the observed market prices, usuallyknown as the model calibration is presented. This calibration is based on the impliedcorrelation approach. In the implied correlation approach, the forward correlationsare considered as the fitting parameter for the model thus being outputs from themodel rather than inputs as considered in some implementations. We also investigatehow using the principal component analysis it is possible to reduce the number of
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random noises that drive the yield curve and thus find the explaining factors that areconsidered most important.
As the underlying stochastic differential equations turn out to be high dimensionalcontaining complex expressions, it is not possible to solve them analytically. Thereforethe relevant option pricing is done via simulation. Chapter 6 explains how this is donein practise after the model parameters are known. Finally in Chapter 7, we use realmarket data to calibrate the model and use it to price the CMS spread options andCMS range accrual notes. The work ends in the conclusion in Chapter 8.
It should be noted, however, that the discussion will be kept on a more general levelin each chapter. The Libor market model is a very generic model enabling it to beapplied in many different exotic option pricing problems. This is due to the fact thatpricing is done by modeling the real market observables by Monte Carlo simulation.Our aim is to explain how the practical pricing is done and CMS spread option pricingis just one application of the model use.
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Chapter 2
Basic Definitions and Facts fromProbability Theory
This chapter assumes the reader has a basic knowledge about measure theoreticconcepts (sigma algebras, measurable functions, filtrations etc.) and basic measuretheoretic probability. The motivation of this chapter is to present mathematical resultswhich are in one way or another used in the forward construction of the Libor MarketModel. They are fundamental in their nature for mathematical finance, as they arecontinuously assumed as facts and used in more advanced papers. We want to presentand discuss their ideas here, so as not to leave the reader in confusion when we usethem later on.
We will start with the very definition of the underlying probability space, present theconcept of the Brownian motion which is the usual model of uncertainty in the riskyassets. Quadratic variation and covariation, which are some of the most importantconcepts in stochastic analysis are defined. We will present the basic construction ofthe Ito integral, which is a stochastic integral with respect to the Brownian motion anddiscuss why it is a financially appealing concept for modeling the financial portfoliodynamics. The theory and basic facts of conditional expectations are presented witha brief introduction to the theory of martingales. Further a more general class ofstochastic processes based on Ito integrals called Ito processes are defined. We willpresent the most important relation in stochastic analysis known as Ito’s lemma whichcan be seen as an stochastic analogue for the fundamental theorem of calculus. Finallywe close the chapter in Girsanov’s theorem, which is a deep measure theoretic resultshowing how the Brownian motion changes under the change of probability measure.Girsanov’s theorem is the tool which explains why Ito processes change drift whenchanging the probability measure. This indeed is the key idea in martingale pricingespecially in no arbitrage drift computations in Chapter 4.
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2.1 Brownian Motion
We will basically follow the discussion and definitions from [15] and [16].
Let(Ω,F , P, Ftt≥0
)be a filtered probability space, where Ω is the space of events,
F is the collection of measurable events (i.e. events which can be assigned probability)and Ftt≥0 forms an increasing sequence of σ-algebras, i.e. Fs ⊆ Ft when s ≤ t. Fora technical condition we suppose the filtration Ftt≥0 to satisfies the so called usualassumptions, that is it is right continuous i.e. Ft = Ft+ = limstFs := ∩s>tFs andFt contains the measure zero sets of F for each t. P is a probability measure whichassigns a probability for each measurable event in F .
A stochastic process X(t, ω) is a measurable function defined in the product space[0,∞)× Ω. That means in particular that
• for each t ∈ [0,∞), X(t, ·) defines a random variable, that is a measurablefunction with X−1 ([0,∞)× B) ∈ F , where B is the collection of Borel sets inR. This random variable is sometimes written in a shorthand notation Xt.
• for each ω ∈ Ω, X(·, ω) is a realization of the process, called a sample path. Thesample path is thus a function of time which differs on each realization of theprocess.
We give now a definition for The Brownian motion.
Definition 2.1.1. A stochastic process W (t, ω) is called a Brownian motion if itsatisfies the following properties
• The process starts from zero, that is W (0, ω) = 0 a.s.
• for each s < t, W (t) −W (s) is normally distributed with mean 0 and variancet− s
• The process has independent increments, that is for t0 ≤ t1 ≤ · · · ≤ tnW (t0), W (t1)−W (t2), . . . , W (tn)−W (tn−1) are independent
• The process has continuous sample paths, that is W (t, ·) is continuous a.s.
That a Brownian motion truly exists in mathematical sense, see the construction ofthe associated canonical probability space from [15, Chapter 3].
From the option pricing point of view it is important that we can construct Brownianmotions whose increments are correlated. This is because a Brownian motion and itsincrements is the usual model that is used to move the underlying market observables.The construction of the correlated Brownian motions is done as follows:
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Let 0 ≤ ρ ≤ 1 be given and define
X(t) = ρW1(t) +√
1− ρ2W2(t),
where W1(t) and W2(t) are independent Brownian motions.
We show next that the stochastic process X is honest Brownian motion and its incre-ments are correlated with W1 by ρ. To show that X is a Brownian motion, it mustsatisfy the assumptions in its definition.
1. Clearly the process starts from zero, because W1 and W2 start from zero.
2. We have for each s < t that
X(t)−X(s) = ρ (W1(t)−W1(s)) +√
1− ρ2 (W2(t)−W2(s))
W1(t)−W1(s) and W2(t)−W2(s) are normal variables with mean zero, and variancet− s. Thus X(t)−X(s) being a linear combination of two random variables is normalwith the mean sum of the means (0), and the variance (because of independency)
ρ2 (t− s) +(1− ρ2
)(t− s) = t− s
Thus X satisfies the second condition in the definition.
3. It is also clear that increments of X are independent because they are continuous(Borel measurable) functions of different increments between W1 and W2, which byassumption are independent.
4. It is obvious that the sample paths are a.s. continuous because sum of two continuousfunctions is continuous (almost surely sense).
Further the increments of X are correlated with the ones for W1 because for any s ≤ t
E ((X(t)−X(s)) (W1(t)−W1(s)))
= ρE((X(t)−X(s))2) +
(1− ρ2
)E ((W1(t)−W1(s)) (W2(t)−W2(s)))︸ ︷︷ ︸
independent increments, mean zero
= ρ (t− s)
As Var (W1(t)−W1(s)) = E((X(t)−X(s))2) = t− s, the correlation is by definition
ρ
The generalization for higher dimensions is straightforward. Given a positive semidef-inite correlation matrix C(t) with Cholesky decomposition C(t) = A(t)A(t)T withaij = A(t)ij, set
Xi =N∑
k=1
aikWk,
where Wk’s are independent Brownian motions. Xi defines a Brownian motion withcorrelation structure
Corr ((Xi(t)−Xi(s)) (Xj(t)−Xj(s))) = ρij(t)
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Even though the sample paths of a Brownian motion are continuous, they are highlyirregular. It actually happens to be the case that
P (ω| W (·, ω) is differentiable) = 0, (2.1)
that is the sample paths of a Brownian motion are nowhere differentiable almost surely(a.s.). This drawback is a consequence of a non vanishing quadratic variation for themap (t, ω) 7→ W (t, ω). We will give the precise definition as we will need to computequadratic variations for stochastic processes later on.
Definition 2.1.2 (Quadratic variation and Quadratic Covariation process). Let Ut
and Vt be stochastic processes defined on finite interval [a, b]. Let a = t0 < t1 < · · · <tn = b be a partition of the interval with a property that ‖∆n‖ = max1≤i≤n(ti − ti−1)tends to zero. If for each sequence of partitions tkn≥0 in [a, b] the sequence
Qn =n∑
k=1
(U (tk)− U (tk−1))2 (2.2)
is convergent in probability, then the limit limn→∞ Qn is called the quadratic variationof U.
If for each sequence of partitions tkn≥0 in [a, b] the sequence
Qn =∑
k=1
(U (tk)− U (tk−1)) (V (tk)− V (tk−1)) (2.3)
is convergent in probability, then the limit limn→∞ Qn is called the quadratic covariationbetween U and V .
We denote a quadratic variation process for U in finite interval [a, b] as 〈U〉ba and thequadratic covariation process between U and V as 〈U , V 〉ba. If one keeps t as a variable,the quadratic variation and covariation of U and V on [0, t] define a stochastic processdenoted 〈U〉t and 〈U, V 〉t.From elementary analysis it is quite easy to see using the mean value theorem thata differentiable function always implies a vanishing quadratic variation on any finiteinterval. Unfortunately this is not a case for a Brownian motion.
Theorem 1 (Non vanishing quadratic variation for the Brownian motion). Let [a, b]be any finite interval. The quadratic variation for the Brownian motion W (t) is
limn→∞
n∑
k=1
(W (tk)−W (tk−1))2 = b− a (2.4)
Further, if W1(t) and W2(t) are two independent Brownian motions, then theirquadratic covariation is
limn→∞
n∑
k=1
(W1 (tk)−W1 (tk−1)) (W2 (tk)−W2 (tk−1)) = 0 (2.5)
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Proof. We show the convergence in equations (2.4) and (2.5) in L2 norm sense as itautomatically implies the convergence in probability. Note that b−a =
∑nk=1(ti−ti−1)
and thus define
χn =n∑
i=1
[(W (ti)−W (ti−1))
2 − (ti − ti−1)]
=n∑
i=1
Xi,
whereXi = (W (ti)−W (ti−1))
2 − (ti − ti−1)
It is thus required to show that E (χ2n) = E (
∑ni=1
∑ni=1 XiXj) → 0, as n → ∞ .
Because of independent increments for the Brownian motion we have for i 6= j thatE(XiXj) = 0 and it requires to work out E(X2
i ). This is
E(X2i ) = E
((W (ti)−W (ti−1))
4 − 2(ti − ti−1) (W (ti)−W (ti−1))2 + (ti − ti−1)
2)
= 3(ti − ti−1)2 − 2(ti − ti−1)
2 + (ti − ti−1)2 = 2(ti − ti−1)
2,
where we have used the fact that W (ti) − W (ti−1) is normal with mean zero andvariance (ti − ti−1). Using standard tables or moment generating functions it is easyto evaluate higher order moments of normal variables as was done. We thus have that
E(χ2n) =
n∑
k=1
2(ti − ti−1)2 ≤ 2‖∆n‖
n∑
k=1
(ti − ti−1) → 0, as n →∞ (2.6)
The proof for the vanishing quadratic covariation is much easier. We show the conver-gence in L2 norm. Convergence implies straight from the independence of the twoprocesses. Define now
Xi = (W1(ti)−W1(ti−1)) (W2(ti)−W2(ti−1))
and show that E(∑n
i=1
∑nj=1 XiXj
)= 0. By independence of the processes and their
increments E(XiXj) = 0 for i 6= j and E(X2i ) = (ti − ti−1)
2. The limiting case followsusing same reasoning as in (2.6).
As an evident corollary we have the following
Corollary 2.1.1 (Quadratic covariation between correlated Brownian motions). LetW1(t) and W2(t) be two correlated Brownian motions with the correlation coefficientρ. The quadratic covariation for W1(t) and W2(t) in [a, b] is
〈W1 ,W2〉ba = ρ(b− a) (2.7)
and the quadratic covariation between the Brownian motion W (t) and t is
〈W , t〉ba = 0 (2.8)
Proof. For latter (2.7) write W2(t) = ρW1(t) +√
1− ρ2W3(t) for W3 independent ofW1 and use the earlier result. The former is trivial. Note that quadratic covariationis bilinear.
8
We write in the differential form d 〈W (t)〉 = dt and d 〈W1(t) ,W2(t)〉 = ρdt becauseformally
〈W 〉t0 =
∫ t
0
d 〈W (s)〉 =
∫ t
0
ds = t
and
〈W1 ,W2〉t0 =
∫ t
0
d 〈W1(s) ,W2(s)〉 =
∫ t
0
ρ ds = ρt
Also d 〈W, t〉 = 0 has the same interpretation.
2.2 Conditional Expectation
Conditional expectation is one central concept in option pricing theory. It is alsonecessary to present it in order to understand the concept of martingales defined inthe next section. We explain first the intuition and motivation behind it before givingthe rigorous mathematical definition. Note that each atom ω in the sample space forstochastic process maps to a trajectory in R+. When we observe the process trajectory(stock price, rate, ...) at a fixed time t we are only observing the trajectory up to timet. This is because we have only access to the information that is recorded in the subσ-field Ft. This collection defines events that can be described by process values up totime t. This means that if A is an arbitrary event in Ft and the sample path (atom) ωis in A, then any other sample path ω with ω(s) = ω(s), for s ≤ t is in A. Although theprice process realization is technically drawn at once (by the God of time?), we needto make decisions (trade) up to the information we have observed. This information iscontained in Ft. This motivates to define a stochastic object that is the best estimateof the future process value. This object should be Ft measurable, because we want toknow its value at time t. This is indeed the idea in conditional expectation.
To make ideas more precise denote the space Lp(G) as the set of random variableswhich are G measurable having finite p’th moment, that is
Lp(F) =
X : Ω → R
∣∣ X−1 (B) ∈ G,
∫|X|p dP < ∞
(2.9)
We will give the classical definition following [15].
Definition 2.2.1. Let X ∈ L1(F) be a random variable, and let G ⊆ F be a sub σ-algebra of F . The conditional expectation of X given the information G is a functionY satisfying the following 2 properties
1. Y is a G measurable function
2.∫
AY dP =
∫A
X dP , for all A ∈ GRemark 2.2.1. The conditional expectation Y is usually written as E(X|G). Theintegral taken in the definition is an abstract Lebesgue integral in the probability space(Ω,F , P ).
9
Using Hilbert space theory the result which captures the intuition behind the definitioncan be shown to be true.
Proposition 2.2.1 (Conditional expectation as a orthogonal projection). Let(Ω,F , P ) be a probability space, and let G ⊆ F be a sub-σ-algebra of F . The condi-tional expectatation of X ∈ L2(F) with respect to G is the orthogonal projection ontothe subspace L2(G)
E(·|G) : L2(F) → L2(G), X 7→ E(X |G ), (2.10)
with E(X|G) being the unique minimizer in L2(G) such that
‖X − E(X|G)‖2 = infg∈L2(G)
‖X − g‖2 (2.11)
Proof. See Theorem 22.4 and Theorem 23.9 from [20, pp. 251–252, 263–264]
If one wants to find more about the structure of conditional expectations and otherpreliminary measure theory the book by Schilling [20] is a good and clear reference.
2.3 Martingales
In the mathematical sense, martingales are stochastic processes satisfying certainuseful convergence properties. They can be understood (in discrete time) as a general-ization of random partial sums and they pop out in many applications of probability.The wide range of their properties are not discussed here, as it is practically irrelevantfor this thesis. Deep and rigorous theory for martingales can, however, be found atfirst chapter in Medvegyev [16].
To give a definition for martingales we need to know what is an adapted process.
Definition 2.3.1 (Adapted process). Let Ftt≥0 be a fixed filtration. A stochasticprocess X(t, ω) is said to be adapted to the filtration Ftt≥0, if Xt is for each t ≥ 0Ft measurable.
The definition for adapted process just means that the process cannot see into thefuture. It is a natural assumption that all market observables are adapted to thefiltration that contains all the information up today. We define now martingales.
Definition 2.3.2 (Martingale property). Let X(t, ω) be a stochastic process adaptedto the filtration Ftt≥0. A stochastic process is a martingale if Xt for each 0 ≤ s ≤ tsatisfied the condition
E (Xt |Fs ) = Xs
A martingale property translates to the fact that the best possible future estimatefor process value is its current value. The martingale is important concept in option
10
pricing because martingales effectively define fair gambles. Namely if Xt is a wealthprocess for some gambling strategy, the martingale property guarantees that mostlikely gambler’s wealth in the future is his current wealth. We need the martingaleconcept when we are arguing fair option values through martingale pricing.
2.4 Ito’s Integral
This section gives the construction for the stochastic integral
∫ b
a
X dY,
where we restrict the attention for the integrator being a standard Brownian motion,and the integral is a stochastic process satisfying certain technical conditions and forour purposes 0 ≤ a ≤ b < ∞. In practical applications in finance these technicalitiesare not usually interfered as they naturally always hold. From now on we fix the filtra-tion Ftt≥0 in our filtered probability space
(Ω,F , P, Ftt≥0
)to be the augmented
filtration generated by Brownian motion (dee definition below).
Definition 2.4.1 (Ito integral). Let a = t0 < t1 < · · · < tn = b be a partition of theinterval [a, b] with a property that ‖∆n‖ = max1≤i≤n(ti − ti−1) tends to zero. Let X(t)be an stochastic process that is adapted to the augmented filtration generated by thepast of a Brownian motion up o time t, that is X(t) is Ft = σσW (s), s ≤ t ∪N measurable, where N contains the measure zero sets of F . Suppose further that theprocess X(t) is uniformly bounded, that is |X(t, ω)| ≤ K for some K ∈ R+ and theprocess X(t) has left- and righthand limits. An Ito integral of a stochastic process X(t)with respect to the Brownian motion is defined as
∫ b
a
X(t) dW (t) = limn→∞
n∑
k=1
X(tk−1) (W (tk)−W (tk−1)) (2.12)
Remark 2.4.1. The augmentation is a technical condition that will make the filtrationused to satisfy the usual assumptions mentioned at the beginning of this chapter. It isactually the case that filtration generated by the Brownian motion does not necessarycontain measure zero sets of F (see [16, p. 9]). As the interval where the stochasticintegral is defined is compact, the boundness of the integrand is a very natural assump-tion. The interpretation for the stochastic integral is that it is a net gain of somegambling process. More precisely if X(t) represents the net gambler’s investment forthe gamble at time t, then the amount X(tk−1) (W (tk)−W (tk−1)) represents the netgain from the gamble in small time interval [ti, ti−1]. It is thus reasonable to assume theleft evaluation point for integrand as the size of a gambling portfolio should be decidedat the start of the interval [ti, ti−1] before a new price is announced. The adaptnes ofthe integrand just means that the size of the investment should be known at the timethe gamble is being entered.
11
It has been shown in [16, pp. 126–127] that the assumptions in the definition implythe (2.12) to converge in L1 mean.
If we choose t to be a changing variable and take the integral of a process X(t) over [a, t]by Eq. (2.12), then the Ito integral defines an integral process. From ordinary calculusit is a well known fact that the value of a Riemann-Stieltjes integral is independent ofthe choice of the evaluation point of the integrand in the subinterval [ti−1, ti]. This isdue to the fact that the integrators have finite variation over any finite interval. Quitea surprising result is that with probability one, the Brownian motion has unboundedvariation over any finite interval. Sample paths are thus continuous but the netabsolute change in value during any interval is infinite. This is a reason why newrules of calculus need to be developed for stochastic integrals. Before presenting themwe give a definition for a more general type of stochastic process called Ito process.Motivation is that modeled forward rates in Libor market model obey dynamics thatare certain type of Ito processes.
Definition 2.4.2. An Ito process is a stochastic process of the form
Xt = Xa +
∫ t
a
f(s)dW (s) +
∫ t
a
g(s)ds, a ≤ t ≤ b (2.13)
Here Xa is Fa measurable, and we assume stochastic processes f(s) and g(s) contin-uous with f(s) adapted to the augmented filtration generated by Brownian motion.
Remark 2.4.2. The second integral is just the ordinary Riemann integral which isdefined pathwise, in other words
(∫ t
a
g(s)ds
)(ω) =
∫ t
a
g(s, ω)ds
Remark 2.4.3. The above integral representation is usually written in a shorthandnotation using a ”stochastic differential form”
dXt = f(t)dW (t) + g(t)dt
f(t) is called the drift and g(t) the diffusion part of Xt. It should be noted that thestochastic differential is not any mathematical object, it is just a shorthand notationused for the integral representation in (2.13).
2.5 Properties of Ito Integral
This section presents some important properties for Ito integrals that will be referredto in later chapters. We will again refer the reader to [15] or [16] for mathematicalrigor. The motivation goes as follows: later on, we will need to compute expectationsand higher order moments of Ito integrals and thus we show how this can be donein practise. Further in modern option pricing based on the martingale approach weneed to identify price processes that are martingales. There actually exists a necessaryand sufficient condition for identifying the martingales that have interest in this work.
12
Finally we move to the stochastic counterpart of the classical fundamental theorem ofcalculus which is a celebrated result by Ito. This result is important because it letsus calculate stochastic returns from stochastic quantities driven by Brownian motions.We start with a theorem usually referred as Ito’s isometry.
Theorem 2 (Ito’s isometry). Suppose f satisfies the conditions in definition 2.4.1.Then the Ito integral I(f) =:
∫ t
af(s)dW (s) is a random variable for each a ≤ t ≤ b
with E(I(f)) = 0 and
E(I(f)2) =
∫ t
a
E(f(s)2) ds
If g is a second function, where I(g) denotes its Ito integral with respect to the sameBrownian motion, then
E(I(f)I(g)) =
∫ t
a
E(f(s)g(s)) ds
Proof. See Lemma 4.3.2 in [15, pp. 44–45] and Theorem 4.3.5 with Corollary4.3.6 [15, p. 48]. For a proof in much more general case see Proposition 2.64 in [16,pp. 156–157].
If the integrand is assumed to be a deterministic process, we can tell more and actuallyknow the distributional properties of Ito integrals.
Corollary 2.5.1. Suppose f is a deterministic function satisfying the conditions indefinition 2.4.1. Then the Ito integral I(f) =
∫ t
af(s)dW (s) is a Gaussian random
variable. Additionally for each a ≤ t ≤ b
∫ t
a
f(s)2 dW (t) ∼ N(
0,
∫ b
a
f(s)2 ds
),
where N (µ, σ2) denotes normal random variable with mean µ and variance σ2.
Proof. See Lemma 2.3.1 and Theorem 2.3.4 in [15, pp. 10–11].
It turns out that the choice of the left evaluation point for the integrand is the keythat makes every Ito integral with respect to a Brownian motion a honest martingale.
Theorem 3. Suppose f satisfies the conditions in definition 2.4.1. Then the stochasticprocess ∫ t
a
f(s)dW (s), a ≤ t ≤ b
is a martingale with respect to the filtration generated by the Brownian motion W (t).
Proof. See Theorem 4.6.1 in [15, pp. 53–55].
13
Even the converse of the above theorem is true for the martingales which are adapted tothe filtration generated by the sample paths of Brownian motion, and satisfying someother technical conditions. In literature this theorem is referred as the MartingaleRepresentation Theorem. We will not present the theorem here because the assump-tions are rather technical. But the idea is that all martingales discussed in this thesiscan be characterized by Ito integrals, that is Ito processes with zero drift.
Before closing this section we will write down a stochastic counterpart for the funda-mental theorem of calculus. If f(t) and X(t) are differentiable functions, then thecomposition f(X(t)) has the representation
f(X(t))− f(X(a)) =∫ t
af ′(X(s)) dX(s) =
∫ t
af ′(X(s))X ′(s) ds (2.14)
by the fundamental theorem of calculus. Because of the non vanishing quadraticvariation for Brownian motion it is nowhere differentiable. The representation in(2.14) thus cannot make sense for a Brownian motion, but a stochastic analogue for(2.14) exists and is known as Ito’s lemma. We present it in its most general form. Therigorous proof for this theorem is rather technical and involved, but we shall presenthere a symbolic derivation of the result, which turns out to be very handy in practise asa rule of thumb and in making calculations in practise. For a mathematically correctproof for Ito’s lemma see [15, pp. 96–101].
Theorem 4 (1-dimensional Ito’s lemma). Let Xt be a Ito process of the form
Xt = Xa +∫ t
af(s) dW (s) +
∫ t
ag(s) ds, for a ≤ t ≤ b .
Let θ(t, x) be a continuous function with continuous partial derivatives ∂θ∂t
, ∂θ∂x
, and ∂2θ∂x2 .
Then θ(t,Xt) is also an Ito process and it is given by
θ(t,Xt) = θ(a,Xa)
+∫ t
a
∂θ
∂x(s,Xs)f(s) dW (s) +
∫ t
a
(∂θ
∂t(s,Xs) +
∂θ
∂x(s,Xs)g(s) +
12
∂2θ
∂x2(s,Xs)f(s)2
)ds
In the stochastic differential form, this reads as
dθ(t,Xt) =∂θ
∂xf(t)dW (t) +
(∂θ
∂t+
∂θ
∂xg(t) +
12
∂2θ
∂x2f(t)2
)dt
Proof. The derivation presented here is not rigorous, rather a rule of thumb how itcan be quickly remembered. From the Taylor’s approximation
dθ(t,Xt) =∂θ
∂tdt +
∂θ
∂xdX +
12
∂2θ
∂x2(dX)2 +
∂2θ
∂x∂t(dt)(dX) +
12
∂2θ
∂t2(dt)2
+ higher order terms.
Writing explicitly the process dX and carrying out the multiplications based on thefollowing formal product table the result follows using elementary algebra.
14
dt dW1
dt 0 0dW2 0 ρ1,2dt
Table 2.1: Ito product table
Remark 2.5.1. The symbolic identities in Table 2.1 are based on the rigorous factsin Eqs. (2.7) and (2.8).
The applications in finance, especially the problems in fixed income are usually highdimensional. This means that the rate of returns are characterized by more than onestate variable driven by Ito processes. Therefore we need a multidimensional versionfor Ito’s lemma.
Theorem 5. Let W(t) = [W1(t), . . . ,WN(t)] be a multidimensional Brownian motion,where the components W (t)j are correlated Brownian motions with correlation coeffi-cient ρj,k between Wj(t) and Wk(t). Let Xj(t), j ∈ 1, 2, . . . , N be multidimensionalIto processes of the form
dXj(t) = µj(t, Xj)dt + σTj (t,Xj)dW(t)
= µj(t, Xj)dt + σj,1(t,Xj)dW1(t) + · · ·+ σj,N (t,Xj)dWN (t)(2.15)
and let θ(t, x1, . . . , xN) be a smooth function of t and xj, j ∈ 1, . . . , N with contin-
uous partial derivatives ∂θ∂xj
and ∂2θ∂xixj
.
Then θ(t,X1, . . . , XN) is also a Ito process with dynamics
dθ(t,X1, . . . , XN ) =∂θ
∂t(t, X1, . . . , XN ) dt +
N∑
i=1
∂θ
∂xi(t,X1, . . . , XN ) dXi
+12
N∑
i=1
N∑
j=1
∂2θ
∂xixj(t,X1, . . . , XN )d 〈Xi , Xj〉 ,
(2.16)
where d 〈Xi , Xj〉 is a quadratic covariation between Xi and Xj.
Proof. See Proposition 6.1 in [16, pp. 353–356] and Corollary 6.4 in [16, pp. 356–358].
Loosely speaking for processes discussed in this thesis, the quadratic covariationprocess between Ito processes Xi and Xj is the formal product dXidXj using therules in 2.1.
2.6 Girsanov Theorem
This section shows how the Brownian motion changes under changing from a proba-bility measure to another. Motivation is that in the option pricing based on martingale
15
approach, we need to perform certain measure changes for normalized price processes.As the price processes are certain Ito diffusions driven by Brownian motion(s), itturns out that the price dynamics change by a changed drift. This drift change comesfrom the Brownian motion acquiring an extra drift. Girsanov’s theorem shows how itchanges precisely.
Before presenting the result, we begin with the definition of equivalent measures.
Definition 2.6.1 (Equivalent measures). Let (Ω,F , P ) and (Ω,F , Q) be two measurespaces with measures P and Q respectively. We say that measures P and Q are equiv-alent if for each A ∈ F
P (A) = 0 ⇐⇒ Q(A) = 0 (2.17)
Remark 2.6.1. The definition implies directly that for each A ∈ FP (A) = 1 ⇐⇒ Q(A) = 1 (2.18)
If measures P and Q are equivalent probability measures, they can generally assignvery different probabilities for measurable events in F . Equivalency guarantees onlythat events which are impossible under P will be impossible under Q, and vice versa.
If P is a probability measure, we say that Q is an equivalent martingale measure if Pand Q are equivalent and all tradeable asses price processes X are martingales underQ.
It is a fundamental theorem of martingale pricing that a market M in(Ω,F , P, Ftt≥0) with a collection of tradeable asset prices Xi, i = 1, . . . , N is arbi-
trage free if and only if there exists an equivalent martingale measure Q and a marketM in (Ω,F , Q, Ftt≥0) which is obtained from M by transforming the asset prices in
units of tradeable numeraire asset. If Xj is the numeraire, the market M consist ofthe normalized prices processes Xi/Xj, i = 1, . . . , N . For the fundamental theorem,see Harrison&Kreps [9] and Harrison&Pliska [10], [11]. The practical use of this toprice options becomes clear later on.
When searching for an equivalent martingale measure, it is important to know howthe Brownian paths change when performing equivalent measure changes. The resultwhich relates all equivalent measures for a given measure P is a measure theoreticresult called
Theorem 6 (Radon-Nikodym theorem). Let (Ω,F) be a measurable space. MeasuresP and Q are equivalent ⇐⇒ there exists a F measurable function F : Ω → R+ suchthat
Q(A) =
∫
A
FdP, for all A ∈ F . (2.19)
The integral in consideration is the abstract Lebesgue integral in the measure space(Ω,F , P ). If the measures are probability measures, plugging the event Ω in the abovecondition means, that
1 = Q(Ω) =
∫
Ω
F dP, that is EP (F ) = 1. (2.20)
16
Remark 2.6.2. The function F : Ω → R+ in the above theorem is usually called theRadon-Nikodym derivative or the density of the measure Q and it is formally writtenas
F =dQ
dP. (2.21)
We need the following definition.
Definition 2.6.2 (Exponential process). Suppose h satisfies the properties required todefine Ito integral in 2.4.1. A stochastic process defined by
Eh(t) := exp
[∫ t
0
h(s)dW (s)− 1
2
∫ t
0
h(s)2ds
], 0 ≤ t ≤ T (2.22)
is called an exponential process with kernel h(t).
An exponential process is closely related to the equivalent measure changes. The maintheorem of this section makes it precise.
Theorem 7 (Girsanov theorem). Assume that the exponential process Eh(t) satisfiesthe condition Ep(Eh(t)) = 1 for all t ∈ [0, T ] and that W (t) is a Brownian motion withrespect to P . Then the stochastic process
W (t) = W (t)−∫ t
0
h(s)ds, 0 ≤ t ≤ T (2.23)
is a Brownian motion with respect to the probability measure
Q(A) =
∫
A
Eh(t)dP, A ∈ F . (2.24)
Proof. See Theorem 8.9.4 in [15, pp. 143–144] or proposition 6.31 in [16, pp. 381–382] in more general case and Theorem 6.32 for this particula.
In practical option pricing when using the fundamental martingale pricing theorem,one is searching for the dynamics for normalized price process such that the dynamicsbecome drift free. The extra drift from Brownian motion when performing anequivalent measure change makes it possible to search for the kernel h(t) such thatQ(·) =
∫· Eh(t) dP is an equivalent martingale measure. In this measure the practical
option pricing is done.
17
Chapter 3
Basic Building Blocks
This chapter assumes that the reader has some knowledge about mathematical formu-lation of financial markets, portfolios, portfolio dynamics and options (contingentclaims). For a good mathematical reference see the text by Bjork [1], for less mathe-matical but more practical treatment see Hull [12]. In the definitions this work followsbasically Bjork.
We present the basic plain vanilla instruments that are needed for the Libor marketmodel and its calibration. These instruments are basic zero coupon bonds (discountbonds) and LIBOR-forward rates, which shall be derived from zero coupon bonds usingno arbitrage arguments. Further we show that forming portfolios of zero coupon bondsit is possible to introduce interest rate swaps. Finally the chapter presents the marketconvention for quoting prices for plain vanilla options based on these instruments.
3.1 Zero Coupon Bonds and Libor Rates
Definition 3.1.1 (Zero Coupon Bond). A zero coupon bond is a financial contractwhich guarantees the holder 1 unit of currency paid on time T . We denote the timet ≤ T value of this contract by P (t; T ).
The date T is said to be the maturity of the bond. One unit of currency is called theface value of the contract. It should be noted that the face value can be anything elsetoo. The unit 1 is used for normalizing purposes. Note that the price must satisfyP (T, T ) = 1 to avoid arbitrage.
Now suppose the current time is t and two times T0 and T1 are fixed in the future.The purpose is to write a contract at current time t, which allows us to make 1 unitof investment in future at time T0 and to have a known rate of return paid at T1 foran invested unit of currency. To achieve this, it is required to make a self financingportfolio, which is zero cost today, costs us 1 unit of currency at time T0 and gives usa deterministic amount of cash at time T1. Self financing portfolio means that one is
18
not required to make any further investments or take any money out of the portfolioafter it has been initiated.
• At time t we sell one T0 bond and receive P (t; T0) amount of money.
• We use this income to buy P (t;T0)P (t;T1)
T1 bonds.
• Our net investment today is overall P (t; T0)− P (t;T0)P (t;T1)
P (t; T1) = 0
• At time T0 we are required to pay the maturing T0 bond, that is unit amount ofcash.
• At time T1 we receive the amount P (t;T0)P (t;T1)
The overall net investment of 1 unit of currency at T0 has produced a known amountof money
P (t; T0)P (t; T1)
, paid at time T1.
Calculating the interest using simple compounding, it is possible to define the interestF1(t) for this investment during the time span [T0, T1] by
1 + τ1F1(t) =:P (t; T0)P (t; T1)
.
τ1 is called the year fraction or day count convention between the dates T1 and T0.τ1 reflects the way the interest is accumulated. There are many market conventionshow year fractions are calculated. For example it can be actual days between [T0, T1]divided by 360. There are, however, many other conventions used. From now on, theyear fraction is not specified, because it varies in different markets and also in differentfinancial instruments by the needs of the clients. The assumed day count conventionis also irrelevant for the theory.
Definition 3.1.2 (LIBOR forward rates). The simple forward rate or LIBOR-ratebetween dates T0 and T1 determined at t is the simply compounded interest rate
F1(t) =P (t;T0)− P (t; T1)
τ1P (t;T1)(3.1)
If t = T0, the interest is called the simple LIBOR spot rate, because the investment ismade today.
T0 is the expiry and T1 the maturity of the forward rate. T0 is called the expiry,because as time approaches T0 the forward rate must converge to the observed spotrate in the market. At time T1 the interest is paid.
Remark 3.1.1. It should be emphasized that the market does not quote forward rates,they are only computational based on arbitrage free arguments. The market quotesonly daily deposit rates (in Euro market EURIBORs).
19
3.2 Swap Rates
Consider the set of expiry maturity pairs T0, . . . , TN for the spanning forward rates.Let τ1, . . . , τN be the corresponding set of year fractions, which shows how forwardrates accumulate interest. For the ease of notation let T−1 correspond the currentspot date. Forward rates need to be identified with two time indices and the notationFi(t; Ti−1, Ti) denotes a forward rate at time t for a spanning period from Ti−1 to Ti. Inthis thesis, an ease of notation Fi(t) is made to denote the forward rate Fi(t; Ti−1, Ti).It should not raise any confusion. We denote the spot Libor rate at time t as L(t).To avoid arbitrage forward rate values must converge to the local spot rate at forwardreset times, that is there must hold L(Ti−1) = Fi(Ti−1) for each i ∈ 1, 2, . . . , N.An interest rate swap is a financial contract between two market participants in whichone party exchanges a stream of fixed rate interest payments for another party’s streamof variable floating spot rate interest payments. More precisely at times T1, . . . , TNthe other party pays a fixed predetermined interest R to the other and receives thefloating variable spot Libor rate L(Ti−1) from the other. The other counter party isin the opposite situation. It receives the fixed rate and pays the floating. The interestis paid for a predetermined principal value K.
From the point of view of the one that pays the fixed and receives the floating, thetotal cash flows at times T1, . . . , TN can be expressed as
Kτi(L(Ti−1)−R) = Kτi
(1− P (Ti−1;Ti)τiP (Ti−1; Ti)
−R
)= K
1P (Ti−1; Ti)
−K − τiKR,
which is paid at time Ti, for all i ∈ 1, 2, . . . , NThe time t value of K is of course KP (t; Ti) and for τiKR it is τiKRP (t; Ti). ForK 1
P (Ti−1;Ti)time t value can be determined by using the following self financing port-
folio:
• At time t, buy one Ti−1 bond. This will cost amount P (t; Ti−1).
• At time Ti−1 invest the received 1 to buy 1P (Ti−1;Ti)
pieces of Ti maturing bonds.
• At time Ti receive 1P (Ti−1;Ti)
Thus the time t value of this interest rate swap is given by
KN∑
i=1
[P (t; Ti−1)− (1 + τiR)P (t; Ti)] = K
(P (t; T0)− P (t; TN )−R
N∑
i=1
τiP (t;Ti)
).
Definition 3.2.1 (Swap rates). The market swap rate denoted Sα,β is such a swaprate at time t such that the mark to market value of the swap is zero at t, that is
Sα,β =P (t;Talpha)− P (t;Tβ)∑β
i=α+1 τiP (t; Ti)(3.2)
20
From the perspective of the counterparty that pays the fixed rate we say that the swapis a payer swap. We say that the swap is a receiver swap if the counterparty receivesthe fixed rate (and pays floating).
Remark 3.2.1. From now on, the notation Sα,β is used to denote a market swap ratefor an interest rate swap starting at Tα, and having payments at times Tα+1, . . . , Tβ.Remark 3.2.2. The process PVBPα,β(t) =
∑βi=α+1 τiP (t, Ti) for α < β is called the
present value of a basis point. In the swap market, only par swap rates are quoted fordifferent maturities and the present value PVα,β(t; R) of a swap with swap rate R canbe calculated from the formula
PVα,β(t;R) = (Sα,β(t)−R)PVBPα,β(t) (3.3)
It is a crucial observation first observed by Rebonato that swap rates can be expressedas a weighted average of forward rates.
Sα,β(t) =β∑
α+1
wi(t)Fi(t) (3.4)
wi(Fα+1(t), . . . , Fβ(t)) =τi
∏ij=α+1
11+τjFj(t)∑β
k=α+1 τk∏k
j=α+11
1+τjFj(t)
(3.5)
This simple observation can be seen as follows: Write the denominator in (3.2) as
P (t; T0)− P (t; TN ) =N∑
i=1
P (t; Ti−1)− P (t; Ti) =N∑
i=1
τiP (t; Ti)Fi(t).
Now write down the zero coupon fraction P (t;Ti)P (t;T0)
by expanding the bond fraction, thatis
P (t; Ti)P (t;T0)
=P (t;T1)P (t;T0)
P (t;T2)P (t;T1)
. . .P (t; Ti)
P (t; Ti−1)=
i∏
k=1
11 + τkFk(t)
(3.6)
Finally write the P (t; Ti)’s in (3.2) using (3.6).
For another terminological issue we say that the swap is forward starting swap if thestart T0 of the swap spanning period [T0, TN ] is in the future. If we keep the swaptenor fixed we call the corresponding swap rate a constant maturity rate.
3.3 Caplets and Swaptions
There exists a huge market for options with an underlying being market par swaprate. These financial contracts are called swaptions. A more precise definition is givenbelow.
Definition 3.3.1 (Swaption). A Tα×(Tβ−Tα) payer swaption with swaption strike Ris a financial contract, which at time Tα gives the holder the right but not the obligationto enter into a [Tα, Tβ] payer swap with the swap rate R.
21
As market swaprates reflect the market expectations for future interest rates, swaptionsare contracts which allow investors to bet for the future market expectations for interestrates. They are used for hedging high interest rates in the future but at the same timebeing able to benefit from low rates.
Denote SWT(t; Tα, Tβ, R) time t value of a payer swaption which gives the right toenter at time Tα into a [Tα, Tβ] swap paying fixed rate of interest R. Time Tα valueSWT(Tα; Tα, Tβ, R) of a swaption can be calculated using (3.3) by
SWT(Tα; Tα, Tβ, R) = max [PVα,β(Tα; R), 0] = max [Sα,β(Tα)−R, 0]PVBPα,β(Tα)
Because PVBPα,β(t) is a tradeable asset (portfolio of zero coupon bonds), there existsa martingale measure Qα,β under which all PVBPα,β(t) normalized traded assetsbecome martingales. Thus the time t value of a swaption can be calculated from themartingale condition
SWT(t; Tα, Tβ, R)PVBPα,β(t)
= EQα,β
(SWT(Tα; Tα, Tβ, R)
PVBPα,β(Tα)
∣∣∣∣Ft
)= EQα,β
(max [Sα,β(Tα)−R, 0] | Ft)
Particularly at time zero the value of a swaption is
SWT(0;Tα, Tβ, R) = EQα,β(max [Sα,β(Tα)−R, 0] | F0)PVBPα,β(0) (3.7)
From (3.7) it can be seen that under Qα,β, the swaption can be just considered a calloption for market swap rate Sα,β.
A convention in the market is to price the swaptions with a known Black 76 formulaused originally to price commodity options. To achieve the Black 76 pricing convention,the market assumes the market swap rate Sα,β to be a lognormal driftless martingaleunder Qα,β, with Qα,β dynamics given by
dSα,β(t) = Sα,β(t)σα,β.dWα,β
σα,β is the (implied) volatility of Sα,β and W α,β is a 1-dimensional Brownian motionunder Qα,β. Dynamics like above is called Geometric Brownian motion (shortly GBM)and using Ito’s lemma for co-ordinate transformation ln(Sα,β), it is possible to solve theabove stochastic differential equation and to find out that the solution is lognormallydistributed. Lognormal distribution for rates is much more appealing than normaldistribution, because in theory normally distributed rates can have negative realiza-tions which are not financially very appealing. The solution for stochastic process attime Tα, conditional to the time t ≤ Tα value is
Sα,β(Tα) = Sα,β(t) exp(∫ Tα
tσα,β dWα,β(s)− 1
2
∫ Tα
tσ2
α,β ds
)(3.8)
Using the theorems from chapter two, the exponent is normally distributed with mean
−12
∫ Tα
tσ2
α,βds = −12
(Tα − t) σ2α,β
and variance ∫ Tα
tσ2
α,βds = (Tα − t)σ2α,β,
22
thus by definition the swap rate is lognormal. As knowing the distribution of Sα,β(Tα)under the martingale measure Qα,β, the expectation in (3.7) is routine integration (seefor example Hull [12, pp. 310–312]).
Definition 3.3.2 (The Black 76 Formula for Swaptions). The Black 76 time t pricefor a payer Tα × (Tβ − Tα) swaption with swaption strike R can be computed from
SWT(0;Tα, Tβ, R) = PVBPα,β(0) [Sα,β(t)N (d1)−RN (d2)] ,
where
d1 =1
σα,β
√Tα − t
(ln
(Sα,β(t)
R
)+
12σ2
α,β (Tα − t))
d2 = d1 − σα,β
√Tα − t,
and N(t) is the CDF of a standard normal variable.
N(t) =1√2π
∫ t
−∞e−t2/2 dt
In the above formula σα,β is called the implied (Black) swaption volatility. In themarket, a convention is to only quote the implied Black volatilities, because at time tit is the only unknown quantity, to get the unique swaption price from market usingBlack 76 formula.
Finally for calibrating the Libor Market model, we present another very natural classof plain vanilla options called interest rate caps.
Definition 3.3.3 (Cap & Caplets). Consider the date structure T0, . . . , TN and thecorresponding forward rates Fi(t), for i ∈ 1, . . . , N.An interest rate cap with the cap rate R is an option paying at times Ti for eachi ∈ 1, . . . , N the amount
τi max [Fi(Ti−1)−R, 0] = τi max [L(Ti−1)−R, 0] (3.9)
Caplet is an individual Ti−1 claim.
Denote the time t price of a caplet paying time Ti−1 spot Libor rate at time Ti byCPL(t; Ti−1, R) for unit notional. Time Ti value of i’th caplet is then of course givenby Eq. (3.9), that is
CPL(Ti; Ti−1, R) = τi max [Fi(Ti−1)−R, 0] ,
Remark 3.3.1. An interest rate cap is thus a simple portfolio of corresponding caplets,and it is enough to value a single caplet for each simple timespan [Ti−1, Ti].
It should be noted that the amount of interest paid is already determined at Ti−1 for eachi. Caplets can also be considered as a special case of swaptions, where the swaptiontenor structure is the simplest possible, namely spanning only [Ti−1, Ti] for fixed i.
23
The market standard pricing equation for caplets can be derived analogously with theone for swaptions. As P (t; Ti) is a tradeable asset, there exists a martingale measureQTi (usually called the forward measure )under which all tradeable assets normalizedby the numeraire P (t; Ti) become martingales. Because caplet payoff is a tradeable
asset, in particular CPL(t;Ti−1,R)P (t,Ti)
is a martingale under QTi . Its time t value can becomputed from the martingale condition
CPL(t;Ti−1, R)P (t; Ti)
= EQTi
(CPL(Ti;Ti−1, R)
P (Ti; Ti)
∣∣∣∣Ft
)= EQTi (τi max [Fi(Ti−1)−R, 0] |Ft) .
(3.10)
Note from section 3.1 that it is possible to write the forward rate Fi(t) as
τiFi(t) =P (t; Ti−1)− P (t; Ti)
P (t; Ti).
As P (t; Ti−1)− P (t; Ti) is a portfolio of tradeable asset (long P (t; Ti−1), short P (t; Ti)bond ) it is itself a tradeable asset. Further P (t; Ti) is a tradeable asset. Thus takingP (t; Ti) a numeraire, under the equivalent martingale measure QTi , all P (t; Ti) normal-ized tradeable assets become martingales. It must be the case that τiFi(t) is a martin-gale under QTi and further (because τi is a constant) Fi(t) is necessarily a martingaleunder QTi . To quote the price for caplets, the market standard is to assume Libor-forwards Fi(t) to be lognormal martingales under the forward measure QTi , that iswith the dynamics
dFi(t) = Fi(t)σidW Ti ,
Using the same co-ordinate transformations as in swap-rate dynamics, the solution forthe stochastic differential equation is the same as in (3.8), that is
Fi(Ti−1) = Fi(t) exp(∫ Ti−1
tσidW Ti(s)− 1
2
∫ Ti−1
tσ2
i ds
)(3.11)
Given that the Libor forwards are lognormal under measure QTi , it is possible tocompute the expectation 3.10 and to arrive at an analogous formula for market capletprices
Definition 3.3.4 (Black 76 Formula for Caplets).
CPL(t; Ti−1, R) = P (t; Ti)τi (Fi(t)N [d1]−RN [d2]) ,
where
d1 =1
σi
√Ti − t
(ln
(Fi(t)R
)+
12σ2
i (Ti − t))
d2 = d1 − σi
√Ti − t
In the above formula σi is called the implied (Black) caplet volatility. The only require-ment is to know the implied caplet volatility to arrive at a unique caplet price.
Remark 3.3.2. The convention in the market is to think of the whole portfolio ofcaplets as a whole and to quote only flat cap volatilities. Flat volatilities mean that thesame volatility is used to price all individual caplets in the whole cap contract. If one
24
knows all the cap volatilities for each maturity Ti, i ∈ 0, . . . , N, the individual capletvolatilities can be solved by iteratively bootstrapping the volatilities. The discussionabout this can be found in detail in [12, pp. 620–624 and Problem 26.20]. The ideais that if we know the 1 and 2 year flat volatilities we know the 1 year and 2 yearcap prices. Their price difference is by no arbitrage arguments the second caplet in 2year cap contract. It is thus required to solve a volatility that implies this caplet price.Same procedure is continued further.
25
Chapter 4
Assumptions under Libor MarketModel
The motivation of this chapter is to present the basic assumptions under Libor Marketmodel. The discussion is largely based on the original paper given by Brace, Garatekand Musiela [2].
For the model to be for a practical use, it needs to be able to price certain liquid marketinstruments consistently with the market quoted prices. This is because when pricingfor some exotic interest rate derivative with the used model, liquid plain vanilla marketinstruments are the basis, which are used to hedge the risk from the exotic contract.That is, when producing the hedge parameters, the underlying market instrumentprices need to be consistent with the model.
As noted in the previous chapter the market convention for quoting swaption andcaplet prices is to use implied volatility to quote the price, which plugged into theBlack 76 formula for caplets or swaptions produce the right market price. However, itshould be noted that the option prices are determined by normal supply and demandwhich again reflect the market expectations of future interest rates that are governedby future monetary politics, inflation and other macro factors that drive the interestrates in the market.
4.1 Libor Dynamics under Forward Measure
As was pointed out in the previous chapter, the forward rate Fi(t) is a martingale underits own payoff measure QTi . Further assuming that the forward rate is lognormal, itis possible to compute the expectation explicitly in 3.10 and to arrive at the Black 76formula for caplets. It should, however, be pointed out that when assuming forwardrates to follow lognormal dynamics under their payoff measure, the swaprates cannotbe simultaneously made lognormal under their own annuity measure. The converse alsoholds true. That is assuming swaprates lognormal under their own annuity measure,
26
forward rates cannot be lognormal simultaneously in their own forward measure. Theproof for this mismatch can be found for example in Brigo and Mercurio [5].
Definition 4.1.1 (Isolated Libor Market Model). Let T0, . . . , TN be the setof expiry maturity pairs and the corresponding set of zero coupon bond pricesP (t; T0), . . . , P (t; TN) be given.
For each i ∈ 1, . . . , N, the Libor Market model assumes the following dynamics forthe Libor forward rates
dFi(t) = Fi(t)σi(t)dW Ti(t), for 0 ≤ t ≤ Ti−1 (4.1)
in the forward measure QTi. Here σi(t) is a deterministic time dependent instantaneousvolatility, and W Ti is a Brownian motion under the martingale measure QTi with thequadratic covariation process defined by
d〈W Ti ,W Tj 〉(t) = ρi,j(t)dt (4.2)
Exogenous correlation structure ρ ∈ RN×N between forward rates can thus be recoveredrequiring that
ρ dt = dW dW T
Despite the time dependency of the instantaneous volatility, one can solve the GBMEq. (4.1) with the same procedure as in Chapter 3. The solution is given by
Fi(t) = Fi(0) exp(∫ t
0σi(t) dW Ti(s)− 1
2
∫ t
0σi(t)2 ds
), for 0 ≤ t ≤ Ti−1.
The dynamics given above gives perfect description for the modeled forward rates,but there exists as many driving uncertainties as there are modeled forward rates.There exists no clear way how to reduce the dimension (that is the driving Brownianmotions) of the problem in any clear way. That is why we will present another morecommonly used description for the Libor dynamics.
Definition 4.1.2 (n-factor Libor Market Model). Assume that the number of modelfactors, that is the number n of driving Brownian motions is fixed. Further assume theBrownian drivers are independent, that is d 〈Wi,Wj〉 = δi,jdt, where δi,j is the usualKronecker delta
The n-factor Libor Market model assumes the following GBM dynamics for forwardrate Fi under its payoff measure QTi
dFi = Fi(t)γi(t)T dWTi(t), (4.3)
where W Ti(t) is a standard n-dimensional Brownian motion under QTi , and γi(t) isa deterministic vector process given by γi(t)
T = [σi,1(t), . . . , σi,n(t)]T . The interpreta-tion for γi(t) is that it contains the responsiveness of the i’th forward for n differentindependent random shocks.
Using the Ito’s lemma, one can again solve the GBM equation. Further using Ito’sisometry with the fact that the variance of sum of independent normal variables is
27
the sum of their component variances one can find the distributional properties to belognormal.
Fi(t) = Fi(0) exp(
∼N(0,∫ t0 ‖γi(s)‖2 ds)︷ ︸︸ ︷
∼N(0,∫ t0 σ2
i,1(s) ds)︷ ︸︸ ︷∫ t
0σi,1(s) dW1(s)+ · · ·+
∼N(0,∫ t0 σ2
i,n(s) ds)︷ ︸︸ ︷∫ t
0σi,n(s) dWn(s)+
∫ t
0‖γi(s)‖2 ds)
The two different Libor formulations are thus equivalent in the sense that they willproduce same distributional properties and price caplets consistently with the market(for factors n > 1 in infinitely many ways) if one requires that
∫ Ti−1
0σ2
i (t)Inst =
∫ Ti−1
0‖γi(t)‖2 dt =
∫ Ti−1
0
(σi,1(t)2 + · · ·+ σi,n(t)2
)dt = Ti−1σ
Blacki ,
(4.4)where σBlack
i is the Black implied caplet volatility extracted from the market. Theabove equation is yet highly undetermined, as there are infinitely many possible choicesfor instantaneous volatility structure satisfying the condition in Eq. (4.4). The traderthus needs to specify some functional form for instantaneous volatility. This concern,however, will be tackled in the next sections after the no arbitrage joint dynamics havebeen determined.
The dynamics in Eq. (4.3) does not yet distinguish the correlation and volatility infor-mation of the forward rates. Writing the equation in the form
dFi(t) = Fi(t)d∑
k=1
‖γi(t)‖dWk(t) = Fi(t)‖γi(t)‖d∑
k=1
σi,k(t)‖γi(t)‖dWk(t)
and defining the quantities
bi,k(t) =σi,k(t)‖γi(t)‖
one can write the dynamics in a compact way
dFi(t) = Fi(t)‖γi(t)‖d∑
k=1
bi,kdWk(t) (4.5)
If one requires that∑d
k=1 b2i,k = 1 then the caplets will be automatically priced
correctly.
The formulation in Eq. (4.5) will be useful and will be the basis of a n-factor LMM.It distinguishes the volatility dependence of the forward rates and the correlationstructure ρ, which can be equivalently defined by its pseudo square root b by
ρ = bbT
Remark 4.1.1. The difference between the two LMM formulations (4.1) and (4.3) orits equivalent form (4.5) is that in (4.1) one places the emphasis on the total volatilityσi(t) of forward rate and the Brownian shock dW Ti is forward rate specific. On theother hand in the formulation (4.3) one places emphasis on risk factor specific inde-pendent shocks dWi(t) and their corresponding responsiveness σi,k. The total volatilityin this case is given by ‖γi(t)‖. From now on we denote the total volatility, regardlessof the formulation by σInst
i (t) or shortly σi(t).
28
4.2 Joint Dynamics of the Libor Rates
The goal of this section is to specify the dynamics of the forward rates under a singleprobability measure. The construction of the model was motivated by the fact thatwith the assumptions as defined in the previous section it was possible to arrive at themarket consistent caplet prices. However, the Libor dynamics needed to be specified indifferent martingale measures! For the sake of simulation arbitrage free joint forwarddynamics needs to be specified, that is the Libor processes need to be returned under asingle measure where the simulation is carried out. It is thus necessary to make equiva-lent probability transformations, where Girsanov’s theorem from Chapter 2 come intoplay. This thesis follows the convention to choose the terminal measure QTN asthe measure under which the joint dynamics of forwards are specified and interest rateoptions priced. Of course the option prices are not dependent on the chosen numeraire.
Theorem 8. Suppose the forward rates Fi are driven by the usual n-factor dynamicsin their own martingale measure QTi given by
dFi = Fi(t)γi(t)T dWTi(t)
or its equivalent formdFi(t) = Fi(t)‖γi(t)‖bT
i (t)dWTi(t)
= Fi(t)‖γi(t)‖d∑
k=1
bikdWk(t)
Then under the terminal measure QTN dynamics of the forward rates Fi(t), i ∈1, . . . , N are given by
dFi(t) = Fi(t)µFi(t)dt + γi(t)T dW TN (t) , for 0 ≤ t ≤ Ti−1, (4.6)
where the drift term can be computed to be
µFi(t) = −‖γi(t)‖N∑
k=i+1
‖γk(t)‖ρi,k(t)Fk(t)τk
1 + Fk(t)τk(4.7)
Remark 4.2.1. If i = N , that is the final spanning forward, the sum in (4.7) collapsesto the empty sum, which is defined to be zero and final forward is driftless under QTN
as it should be.
Before the proof for drift corrections, we give the following useful lemma, which isvery useful in the calculations, such like the ones for drift corrections for Libor forwarddynamics.
Lemma 4.2.1. Let processes X, and Yi, i ∈ 1, 2, . . . , N satisfy GBM dynamics
dX(t) = X(t)µXdt + X(t)σX(t)bTX(t)dW(t),
anddYi(t) = Yi(t)µi(t)dt + Yi(t)σi(t)bT
i (t)dW(t),
29
where W is a n-dimensional Brownian motion with dWdWT = In×ndt , and bX , bi
are n-dimensional vector processes satisfying the exogenous correlation requirement
bi,1(t)bX,1(t) + · · ·+ bi,n(t)bX,1(t) = ρX,Yi(t)
and the quadratic covariation between X and Yi thus being
d 〈X(t), Yi(t)〉 = X(t)Yi(t)σX(t)σi(t)ρX,i(t)dt, for i ∈ 1, 2, . . . , N
Then the quadratic covariation process between X(t), and∏N
i=1 Yi(t) is given by
d
⟨X(t),
N∏
i=1
Yi(t)
⟩=
N∑
i=1
d 〈X(t), Yi(t)〉
N∏
j=1,j 6=i
Yi(t)
Proof. We give the proof by induction with respect to the amount of GBM processesin the product.
The case N = 1 is trivial from the definition of the covariation process from Chapter 2.For case N = 2, we get the dynamics for Y1(t)Y2(t) using the Ito product formula.
d(Y1Y2(t)) = Y1(t)dY2(t) + Y2(t)dY1(t) + d 〈Y1(t), Y2(t)〉= Y1(t)Y2(t) (µ1(t) + µ2(t) + σ1(t)σ2(t)ρ1,2(t)) dt
+ Y1(t)Y2(t)σ2(t)bT2 (t)dW(t) + Y2(t)Y1(t)σ1(t)bT
1 (t)dW(t)
Now multiplying the dW terms, and remembering that dtdt and dWdt terms vanishwe get what is wanted:
d 〈X(t), (Y1Y2)(t)〉= Y2(t) (X(t)Y1(t)σX(t)σY1(t)ρX,Y1(t)) dt + Y1(t) (X(t)Y2(t)σX(t)σY2(t)ρX,Y2(t)) dt
= Y2d 〈X(t), Y1(t)〉+ Y1d 〈X(t), Y2(t)〉(4.8)
It is straightforward to iterate (4.8) to get the general claim of the lemma.
Now we have available tools to derive the no arbitrage drifts.
Proof. During the whole proof, choose the discount bond P (t; TN) as a numeraire.
There is a slight problem with the forward as they are themselves not tradeable assetsand thus they cannot be normalized with P (t; TN) and to argue the martingale condi-
tion for a normalized process Fi(t)P (t;TN )
.
However this problem can be tackled by identifying that P (t; Ti) is a tradeable asset,
and thus under QTN , fraction P (t;Ti)P (t;TN )
should be a martingale to avoid arbitrage.
Further Fi(t)P (t; Ti) is a tradeable asset, and to avoid arbitrage opportunities, the
fraction Fi(t)P (t;Ti)P (t;TN )
needs to be a martingale. Martingale Representation Theoremfrom Chapter 2, says that it can be represented by a Ito process with zero drift.
30
Supposing that the Fi(t) dynamics under QTN are of the form
dFi(t) = Fi(t)µFi(t)dt + Fi(t)γi(t)T dWTN (t)
= Fi(t)µFi(t)dt + Fi(t)‖γi(t)‖bTi dWTN (t).
1
we need to find a drift µFi(t) that will close arbitrage opportunities and force the price
ratio Fi(t)P (t;Ti)P (t;TN )
driftless. Remembering that the bond fraction can be expressed as aproduct of forwards, it is possible to compute the covariation term explicitly using theearlier proved lemma. Note also that the dynamics of 1 + τkFk(t) are (by Ito’s lemmaagain)
d(1 + τkFk(t)) = τkFk(t)µFk(t)dt + τk‖γk(t)‖bk(t)dWTN (t)
We get
d
(Fi(t)
P (t; Ti)P (t; TN )
)=
P (t;Ti)P (t; TN )
dFi(t) + Fi(t)d(
P (t;Ti)P (t; TN )
)+ d
⟨Fi(t),
P (t; Ti)P (t; TN )
⟩
Compute the covariation term, as it contains drift components
d
⟨Fi(t),
P (t;Ti)P (t; TN )
⟩= d
⟨Fi(t),
N∏
k=i+1
(1 + τkFk(t))
⟩
=N∑
k=i+1
d 〈Fi(t), (1 + τkFk(t))〉N∏
j=i+1,j 6=k
(1 + τjFj(t))
=N∑
k=i+1
‖γi(t)‖Fi(t)τkFk(t)‖γk(t)‖ρi,k(t)N∏
j=i+1,j 6=k
(1 + τjFj(t)) dt
= ‖γi(t)‖Fi(t)N∑
k=i+1
τkFk(t)‖γk(t)‖ρi,k(t)N∏
j=i+1,j 6=k
(1 + τjFj(t)) dt
As the term P (t;Ti)P (t;TN )
dFi(t) contains also a drift component, namely
µFi(t)Fi(t)
∏Nj=i+1 (1 + τjFj(t)), then for the overall drift to be zero, the following
condition must satisfy
µFi(t)Fi(t)N∏
j=i+1
(1 + τjFj(t))
+ ‖γi(t)‖Fi(t)N∑
k=i+1
τkFk(t)‖γk(t)‖ρi,k(t)N∏
j=i+1,j 6=k
(1 + τjFj(t)) = 0 ⇐⇒
µFi(t) = −‖γi(t)‖N∑
k=i+1
τkFk(t)‖γk(t)‖ρi,k(t)
∏Nj=i+1,j 6=k (1 + τjFj(t))∏N
j=i+1 (1 + τjFj(t))⇐⇒
µFi = −‖γi(t)‖N∑
k=i+1
τkFk(t)‖γk(t)‖ρi,k(t)1 + τkFk(t)
This shows what was needed to prove.
1Actually according to the Girsanov’s theorem, the dynamics must be of this form, when we aredoing equivalent measure change. All that can be done is to change the drift!
31
According to the theorem, the forward rates are not lognormal martingales in QTN ,except for the final one. Even though with an additional drift, the (4.6) can beformally solved by the same coordinate transformations. With a state dependent driftthe problem is, however, that the solution of (4.6) is no longer lognormal. It is thusnot possible to compute expectations analytically. Although the drift component isquite complex and not very traceable it forms the basis for simulation procedures. Itis actually the case that covariance elements for forward rates are all that matters tospecify the dynamics uniquely. This will become very clear in Chapter 6.
Now as the joint dynamics of Libor rates are finally specified it it worth reviewingwhat is needed to specify the Libor Market model completely.
• The forward rate values today
• The correlation structure between modeled forward rates ρi,j, and
• The total instantaneous volatility of the forward rates σInsti (t) = ‖γi(t)‖ and its
functional form.
The first requirement is easy and straightforward, as forward rates can be observed orat least stripped straight from liquid market instruments (depos, futures, FRA’s andswaps).
The second requirement is not obvious, because the correlation between forward ratesis not an observable quantity. There are also no liquid market instruments from wherethe instantaneous correlation structure could be determined. However, it is possible toestimate it from historical data. One way to do it would be to obtain historical marketinstrument quotes used to construct the market yield and strip historical forward ratesfrom the curve. Using a huge amount of historical information the correlation matrixcould be determined by time series analysis. There lies, unfortunately, a drawback inthis approach: According to Rebonato in [18] pricing very slop dependent exotic prod-ucts like spread options, whose payoff is highly dependent of the flattening, steepeningor inverting the yield curve, the correlation structure is important. The historicaldata has a risk of containing outliers thus affecting the overall correlation rendering itunrealistic, therefore useless. Also history is not a good estimate in turbulent marketconditions when the yield curve is observed to be under heavy shape changes. It ishighly feasible that a historical time series analysis will produce very biased correlationestimates.
Fortunately there is another approach where correlation can also be seen as a fittingparameter, such that the model gives the market consistent prices for calibrated instru-ments. It is easy to see that the given correlation structure does not affect the capletprices (see the pricing equation for caplets) in any way. If, however, swaptions arechosen in the set of calibrated instruments in LMM, it turns out that swaption volatil-ities are dependent on the correlations of the forwards which are linked to swap rateSα,β(t) through Eq. (3.4). Thus by taking forward correlations as free parameters, itis possible to search an optimal fit for correlations so that the model fits swaptionvolatilities in the market. This kind of procedure will be presented later on. However,
32
by extracting forward rate correlations from swaption prices implies great faith in theefficiency between these two sister markets. Some discussion about the possibly incon-sistency between these markets can be found in this Fan [7]. Further discussion aboutthe correlations being inputs or outputs for the model can be found from the lecturenotes by Brigo [6].
For the third requirement that is the functional dependence of time dependent instan-taneous volatility, there exists few more or less standard choices for the functional formof the time dependent volatility. Rebonato in [19, pp. 667–680] and [18, pp. 141–171]discusses in great detail the different parametric forms for forward rate instantaneousvolatilities and their implications for the term structure of volatility. See also [3] fora review of different parametric forms usually chosen for the instantaneous volatilityand numerical studies for the implications for term structure of volatilities. In thiswork we follow the ”humped shape” exponentially decaying parametrization by [22].It is presented in Chapter 5.
4.3 Parametrization of instantaneous correlation
The parameterized correlation for forward rates should capture at least two empiricallyobserved features. The first feature is that the correlation between forward rates shoulddecrease when the forward rates are farther apart. This makes intuitively sense becausethe farther the forward rates are in time, the less they should move in stage with eachother. The other feature is that the correlation between same distance forward ratesshould be smaller for shorter expiries than for larger ones. This is because the volatilityin the short end is higher than in the long end. In mathematical language the firstfeature translates to the rows being monotonically decreasing from the diagonal. Thesecond feature requires the sub diagonals of the correlation matrix to be monotonicallyincreasing.
When calibrating the model for swaption instruments, the usual way is to use the anglesform proposed by Rebonato (see for example [19, pp. 639–642]). When using the n-factor model (if n is number of forwards the factor model is full) one parameterizesthe pseudo root b for correlation by
bi,k(t) = cos θi,k(t)k−1∏
j=1
sin θi,j(t), k = 1, . . . , n− 1
bi,k(t) =k−1∏
j=1
sin θi,j(t), k = n.
This will automatically solve the caplet pricing condition b2i,1(t) + · · · + b2
i,n(t) = 1for any combination for angles θi,j, because the coefficients will lie on the surface ofa n-dimensional unit sphere. This requires, however, O(N2) parameters to be solvedand the optimization routines tend to become high dimensional. Also the modelimplied correlation structures can be unstable and erratic in some cases, displayingthe empirically unappealing features. See for example Rebonato [19, pp. 649–658] for
33
different case studies with the angles form. We will not pursue the angles approachany further here.
A good aim for the model is to try achieving a correlation structure which will produceempirically (i.e. historically) appealing features. Schoenmakers and Coffie in [21] havedeveloped a method which they call semi-parametric formulation and which suitswell the implied correlation LMM calibration. The method is called semi-parametricbecause it requires only identifying O(N) parameters and producing O(N2) correla-tion numbers. The low parametric form will achieve the calibration procedure morestable and computationally more feasible than with the angles form.
The basic idea is as follows (see their paper for full details). Let b = (b1, . . . , bN−1)be a strictly increasing sequence of numbers such that the sequence bi/bi+1 is strictlyincreasing. Let further b1 = 1. Then the instantaneous correlation structure definedby
ρi,j =min(bi, bj)max(bi, bj)
is always a mathematically true correlation structure (i.e. positive definite symmetricmatrix with |ρi,j| ≤ 1 and ρi,i = 1 for each i) implying the features earlier set for desiredcorrelation. At the same time it is possible to show that the above parametrization isequivalently characterized by a sequence of non negative numbers ∆i, ∆i ≥ 0 wherei = 2, . . . , N − 1 and bi’s are given by
bi = exp
(N−1∑
l=2
min(l − 1, i− 1)∆l
)
and the implied correlation structure by
ρi,j = exp
(−
N∑
l=i+1
min(l − i, j − i)∆l
).
Now Shoenmakers and Coffey derive different low parametric forms from the deltasequences. For example the classical correlation structure characterized by only oneparameter β and suggested by Rebonato in [18, p. 191]
ρi,j = exp (−β|Ti − Tj |)
can be fitted into this framework defining ∆′s appropriately. For the details see [21].This correlation structure, however, does not show the empirically appealing featurethat the subdiagonals should be increasing.
Schoenmakers has suggested in [22] that the Schoenmakers and Coffie’s 3 para-metric form defined by
ρi,j(ρ∞, η1, η2)
= exp[−|j − i|
N − 1
(− ln ρ∞ +
(η1
i2 + j2 + ij − 3Ni− 3Nj + 3i + 3j + 2N2 −N − 4(N − 2)(N − 3)
)
−η2
(i2 + j2 + ij −Ni−Mj − 3i− 3j + 3N + 2
(N − 2)(N − 3)
))]
34
works well in practise in implied correlation calibration. This correlation structure canbe fitted in their semi-parametric approach by defining
α1 =6η1 − 2η2
(N − 1)(N − 2), α2 =
4η1
(N − 1)(N − 2)
and using the sequence of deltas ∆2 = α1, ∆N−1 = α2, ∆N = β and linear interpolationbetween ∆2 and ∆N−1 by
∆i = α1m− i− 1
m− 3+ α2
i− 2m− 3
.
35
Chapter 5
Calibration of the Model
The convention in EURO market for swaptions is to assume the underlying swapcontracts having annual fixed coupons while the floating leg is semi-annual 6 monthEURIBOR rate. If one wants to model the market swap rates through Libor marketmodel, it is thus natural to model 6 month forward rates in the model framework.
We show first how forward rate volatilities and correlations can be linked to the swap-tion volatilities in the Libor market model framework. In addition we discuss brieflythe term structure of volatility plot and finally present in detail the calibration methodthat has been used in this work.
5.1 Black Swaption Volatilities under LMM
Analogous to the Libor market model, another market model has been developedto model the underlying swap rates under its own annuity measure. This model issometimes called Swap Market Model (SMM) or Jamishidian model by the originalauthor. Lately there has also been research in the general market model frameworkand its calibration (see paper by Galluchio et al. [8]). This thesis does not handlethe Swap Market model in more detail than is necessary to understand the swaptionscalibration in the forward rate framework. The basic ideas in SMM are the same as inLMM. The difference is that in SMM the drift computations for swap rates are moreinvolved and do not have much traceability, when bringing the swap rate dynamicsunder a single pricing measure. For more details of SMM construction see the originalpaper from Jamishidian [13].
Under the martingale measure Qα,β Jamishidian assumes swap rates Sα,β(t) to followmarket consistent lognormal dynamics
dSα,β(t) = Sα,β(t)σα,β(t)dW α,β(t) (5.1)
36
To price swaptions consistently with the market, the following consistency conditionas in caplets must hold (
σBlackα,β
)2Tα =
∫ Tα
0σ2
α,β(t) dt (5.2)
The problem in the above is that under LMM deterministic swaption volatilities σα,β(t)are not possible!
However noting that swap rates can be expressed as a weighted average of the forwards
Sα,β(t) =
β∑i=α+1
wi(t)Fi(t), (5.3)
it is possible to write the consistency condition (5.2) using the quadratic variation
(σBlack
α,β
)2Tα =
∫ Tα
0
σ2α,β(t) dt =
∫ Tα
0
d 〈ln Sα,β(t), ln Sα,β(t)〉 ,
because by Ito’s lemma applied to (5.1) one has
d ln Sα,β(t) = σα,β(t)dWα,β(t)− 1
2σ2
α,β(t)dt.
By applying Ito’s lemma for ln Sα,β(t) and using the product formula for (5.3) one canwrite
d ln Sα,β(t) =1
Sα,β(t)
β∑i=α+1
wi(t)dFi(t) + Fi(t)dwi(t) + dwi(t)dFi(t) + dt terms
dwi(t)dFi(t) contains also only dt terms and thus does not affect the quadratic variationd 〈ln Sα,β(t), ln Sα,β(t)〉 in any way.
The result which was originally proposed by Rebonato and Jackel was to freeze the wi
terms assuming dwi = 0 and thus arrive at the approximation
Theorem 9 (Rebonato’s formula for swaption volatilities). Assuming the weights wi
constant the swaption volatility can be computed from the approximation
σ2α,β(t) ≈ 1
Sα,β(0)
β∑i=α+1
β∑j=α+1
Fi(0)Fj(0)wi(0)wj(0)σi(t)σj(t)ρi,j(t),
if the coefficients wi are frozen, that is assumed to be constant with their current valuewi(0).
Hull and White have made the approximation more accurate without more computa-tional complexity by computing the dwi(t) terms explicitly with Ito’s formula.
37
Lemma 5.1.1 (Hull and White formula for swaption volatilities). The instantaneousswap rate volatilities in Eq. (5.2) can be linked for forward rate’s instantaneous volatil-ities by the formula
σ2α,β(t) ≈ 1
Sα,β(0)
β∑i=α+1
β∑j=α+1
Fi(0)Fj(0)ζi(0)ζj(0)σi(t)σj(t)ρi,j(t),
if the weights ζi(t) are frozen, that is assumed to be constant with their current valueζi(0). The coefficients ζi(0) are known from today’s market data and computed fromthe formulas A.1 and A.2 given in Appendix A.
Proof. See the Appendix A for the derivation of the version of the Hull and Whiteformula for semi-annual floating and annual fixed legs. The involved coefficients forζi(t) will be listed in there too, and will later be used in the calibration schemes.
The above derived result by Rebonato is an industry standard formula for approxi-mating swaption volatilities within the forward rate framework. For the initially flatyield curve Rebonato proves in [17] that the correction by Hull and White vanishes.This is only the case for swaps, where payment frequencies for fixed and loating legsare equal. Overall the correction is neglible and the Rebonato’s formula suits well inmost cases. Schoenmakers however suggests in [22] that even for a flat yield curve,the Hull and White correction term is significant and should always been taken intoaccount when the different payment frequency for fixed and floating legs have beentaken into account. There have been lots of empirical studies of the accuracy of theRebonato’s approximation formula and its comparison with the more complex Hulland White version. The empirical results for Rebonato’s approximation have beenquite positive. For studies of the accuracy of the approximation see for example [17].
Suppose that the market given implied volatility quote for the swaption with theunderlying swap Sα,β is σBlack
α,β . Trusting for the approximation formula, the LiborMarket Model prices swaption type instrument consistently with the market if thefollowing consistency condition holds
(σBlack
α,β
)2Tα =
1
Sα,β(0)2
β∑i=α+1
β∑j=α+1
Fi(0)Fj(0)ζi(0)ζj(0)
∫ Tα
0
σi(t)σj(t)ρi,j(t) dt (5.4)
The condition in Eq. (5.4) is the fundamental result, in which all the Libor Marketmodel swaption calibration implementations are based on. When the volatilities andcorrelations have been assumed to have some functional form with unknown parame-ters, there is well defined inversion problem which can be solved.
Before moving to the calibration details we present one benchmark that is usuallydone for the calibration fit after the Libor model has been calibrated.
38
5.2 Term Structure of Volatility
Denote the set of modeled forward rates as usual F1, F2, . . . , FN with expiry maturitypairs T0, T1, . . . , TN−1, TN defining the time structure, and T−1 the spot date. Assumeas here is done that the forward rates have 6 month tenors. The model implied capletvolatility σModel
i with underlying expiring in 6 months is given by the root mean of theinstantaneous volatility
σModel6M (T−1) =
√1
T0 − T−1
∫ T0
T−1
σ1(t)Inst
After six months have elapsed the current spot is T0, and the original 6 month forwardexpiring in 1 year is 6 month forward expiring in 6 months. Now after 6 months themodel ”implies” at future time T0 the caplet volatility
σModel6M (T0) =
√1
T1 − T0
∫ T1
T0
σ2(t)Inst
This motivates the following definition for the evolution of the term structure ofvolatility
Definition 5.2.1 (Term structure of volatility). At time t = Tj, the volatility termstructure is a graph of points
(Tj+1, σ
Modelj+1 (Tj)
),(Tj+2, σ
Modelj+2 (Tj)
), . . . ,
(TN−1, σ
ModelN (Tj)
),
where
σModelh (Tj) =
√1
Th − Tj
∫ Th
Tj
σInsth (t)
and h ≥ j + 1
At time t = T−1 the term structure of volatility is just the plot of market caplet volatil-ities. Different assumptions for the instantaneous volatility imply different evolutionsfor the term structure of volatility. A time homogenous future term structure ofvolatility is a good property for a model to have because empirically the market termstructure of volatility observed in normal situations does not change its shape. It isusually observed to be a humped shape form.
5.3 Optimization Based Calibration Algorithm
We first implemented the so called cascade calibration proposed by Brigo and Moriniin [4]. The idea in their approach is to parametrize the volatility structure withpiecewise constant functions. With several simplifications it is possible to bootstrapthe unknown piecewise volatilities iteratively without the need to use minimizationalgorithms and the model fit for swaptions is by construction exact. We suggest the
39
reader to familiarize for their paper. The method is simple to implement but thereexists several drawbacks which we figured out. The fitted piecewise volatility droppedvery low almost to zero in the long maturities and the evolution of the yield curveon the long end was almost stationary. This cannot be the case in practise. Evenworse is possible as proposed by Brigo and Morini [4]. Because piecewise volatilitiesare extracted from quadratic equation it is possible that we may extract negative oreven complex volatilities. This is of course financially implausible. The second dangeris the overfitting which means that the model tries to ”explain too much”. This isreflected in the instability of model parameters for small changes in the input. This isbad because in addition to the correct option pricing, the equally important functionfor the exotic interest rate model is to produce sensible risk numbers for the exotics.These are the sensitivities of the option prices for movements in the underlying market(deltas, gammas, vegas) and for a trader these are crucial to know if he wants to hedgethe risks from the sold exotic option. We thus abandoned the cascade calibration.
The model calibration proposed by Schoenmakers was found out to work well in prac-tise and this thesis follows basically the ideas from his paper [22]. Schoenmakers takesthe usual tenor structure and assumes that the Libor rates are governed by
dFi(t) = −Fi(t)‖γi(t)‖N∑
j=i+1
τjFj(t)‖γj(t)‖1 + τjFj(t)
dt + Fi(t)γidWN (t).
Here N is the number of modeled forward rates and W N is a standard N -dimensionalBrownian motion under the terminal measure. Later on we explain how Schoenmakersreduces the dimension of the Brownian motion, that is the number of model factors.
The tenor of the modeled forward rates is semi-annual, because underlying marketswaps are composed of semi-annual forwards.
The major difference now compared to the Eqs. (3.4) and (3.5) is to take into accountthe different payment frequencies and day count conventions in fixed and floating legpayments in underlying swaps. This means that the swap rate equations in (3.4)and (3.5) need to be adjusted appropriately. The adjustment is quite obvious but canbe derived using the portfolio arguments as was done to derive the par swap ratesformula in Eqs. (3.4) and (3.5). One finds that the swap rates are now given at timet < Tα
Sα,β(t) =P (t; Tα)− P (t; Tβ)∑β−α
i=1 /2τ ′2i+αP (t; T2i+α)=
β∑
i=α+1
wi(t)Fi(t), (5.5)
where τ ′i is the day count fraction for fixed legs, namely τ ′i = Ti − Ti−2 and weightswi(t) are given
wi(Fα+1, . . . , Fβ, t)
=τiP (t; Ti)∑(β−α)/2
i=1 τ ′2i+αP (t; T2i+α)=
τi∏i
k=α+11
1+τkFk(t)∑(β−α)/2i=1 τ ′2i+α
∏2i+αk=α+1
11+τkFk(t)
(5.6)
40
We parametrize the instantaneous volatility (the volatility norm) for the forward rateFi by σinst
i (t) = ci(Ti−1)g(Ti−1 − t), where
g(t) = ga,b,g∞(t) = g∞ + (1− g∞ + at) e−bt, a, b, g∞ > 0,
and the instantaneous correlation ρi,j by the Schoenmakers and Coffie’s 3 parametricform denoting it now ρi,j(η1, η2, ρ∞).
The aim is to try to fit the model for market swaption quotes. More precisely given anarbitrary set of market implied swaption quotes σBlack
α1,β1, . . . , σBlack
αk,βkwe try to fit the set
of model parameters a, b, g∞, η1, η2, ρ∞ so that they imply the pricing error (soondefined explicitly) as small as possible for this set of instruments. It should be notedthat the calibration does not need the whole swaption matrix. It is possible to givearbitrary set of calibrated swaption instruments which fits in the set of expiry maturitypairs. The chosen set of calibrated instruments should naturally be those that are usedto hedge the exotic option. To test the model performance for fitting, we choose thewhole swaption matrix as the target set of calibrated instruments.
Caplet volatilities σBlacki will be fully recovered when
(σBlacki )2 (Ti−1 − T−1) =
∫ Ti−1
T−1
σinsti (t)2dt =
ci(Ti−1)2
Ti−1 − T−1
∫ Ti−1
T−1
g2a,b,g∞(Ti−1 − t)dt, (5.7)
Note that there is no correlation dependence in (5.7). The swaption volatility willbe approximately recovered by a version of Hull and White formula (introduced inappendix A) when
(σBlackα,β )2 (Tα − T−1) =β∑
i=α+1
β∑
j=α+1
ζi(0)ζj(0)Fi(0)Fj(0)Sα,β(0)2
ci(Ti−1)cj(Tj−1)∫ Tα
T−1
ga,b,g∞(Ti−1 − t)ga,b,g∞(Tj−1 − t)ρi,jdt
(5.8)The functional form for coefficients ζi are shown in the appendix. It is possible tomake a substitution getting away from the pure maturity dependent coefficients ci in(5.8) by defining
ωi,j,α :=√
Ti−1 − T−1√
Ti−1 − T−1
Tα − T−1
∫ Tα
T−1ga,b,g∞(Ti−1 − t)ga,b,g∞(Tj−1 − t) dt
√∫ Ti−1
T−1g2a,b,g∞(Ti−1 − t) dt
√∫ Tj−1
0 g2a,b,g∞(Tj−1 − t) dt
,
and at the same time using the caplet pricing condition (5.7) thus rewriting the swap-tion price condition
(σBlackα,β )2 =β∑
i=α+1
β∑
j=α+1
ζα,βi (0)ζα,β
j (0)Fi(0)Fj(0)Sα,β(0)2
σBlackj σBlack
j ωi,j,α ρi,j(η1, η2, ρ∞) =: (σModelα,β )2
Given a set of parameters a, b and g∞, the integrals in ωi,j,α can be calculated inclosed form. We have used the symbolic toolbox from Matlab to compute the definite
41
integrals and wrote an m-file subroutine being called from the calibration algorithm.The expressions are long, look quite involved and are thus not represented here. Acomputer can however carry the task much faster than by calling some numericalintegration scheme.
The idea in the calibration is to minimize the mean square error for the set of calibrated
instruments. That is the mean of the squares for the relative errorsσBlack
α,β −σModelα,β
σBlackα,β
. If n
is the number of calibrated instruments and the set of Black volatilities are indexedby i = 1, . . . , n one is faced with the minimization problem
mina,b,g∞
η1,η2,ρ∞
1n
n∑
i=1
(σBlack
α,β − σModelα,β
σBlackα,β
)2
(5.9)
After the model has been calibrated with full amount of factors and optimization hasfound the optimal set of parameters a, b, g∞, η1, η2, ρ∞ it is possible to reduce thenumber of model factors. This is done via principal component analysis.
The correlation structure ρ(η1, η2, ρ∞) is symmetric and of full rank. This means thatit has N real and positive eigenvalues. From elementary matrix algebra it can thus bediagonalized by
ρ = QλQT =
:=B︷ ︸︸ ︷(Q√
λ)(
Q√
λ)T
where λ is a diagonal matrix containing the positive eigenvalues λi in descendingorder λ1 ≥ λ2 ≥ · · · ≥ λN and
√λ = [
√λ1, . . . ,
√λN ] with Q = [q1,q2, . . . ,qN]
being an unitary matrix containing the corresponding eigenvectors of λ. The principalcomponent interpretation is that orthogonal directions qi explain the total variance byproportion λi∑N
k=1 λk. The matrix B given by B = [
√λ1q1, . . . ,
√λNqN] is the pseudo
root of ρ. Now for n < N define
B(n) = [√
λ1q1, . . . ,√
λnqn] ∈ RN×n,
which is by construction of rank n. B(n) B(n) Tis unfortunately not a honest correlation
matrix, because the diagonal elements are not generally 1. To define rank n correlationmatrix ρ(n) diagonal elements need to be adjusted to unity. Define
ρ(n)i,j =
B(n)i,j B(n)
i,j√B(n)
i,i
√B(n)
j,j
,
which is by construction a honest correlation matrix. Schoenmakers proposes in [22]to fix the correlation matrix ρ(n) and find the new optimal set of parameters a, b, g∞which minimizes (5.9).
42
Chapter 6
Drift Approximation Methods andSimulation
Up to now this thesis has discussed the methods which determines the unknown param-eters of model by ”marking the model onto market”. If the main goal would had beento price plain vanilla swaptions or caps/caplets, the method to do their pricing wouldhad been to use the Black 76 formula adjusted with appropriate volatility. The mainreason to develop the market model is however to price exotic interest rate options,whose complex payoff can be expressed in terms of market observable Libor forwardrates. However as noted in Chapter 4 the joint dynamics of forward rates underthe terminal measure was quite complex and the corresponding stochastic differentialequations were high dimensional and contained state dependent drift terms. It waspointed out in Chapter 4 that it was formally possible to compute the solution for GBMequation for forward rates , but the integrals were complex containing state dependentexpressions. Thus the joint distributional properties of forwards in terminal pricingmeasure were impossible to compute explicitly. This leaves two natural alternatives:
1. The naive and straightforward alternative is to discretize the correspondingstochastic differential equation using some discretization scheme (Euler orMilstein, for example). After that, using a very small time step it is possible toevolve the forward rates (slowly) for the price sensitive events.
2. The second alternative is to try to approximate the state dependent stochasticintegrals. If some efficient and accurate method can be obtained in time which issuperior to the discretization, then it is possible to solve the stochastic differentialequation approximately analytically. This makes it possible to make a longintegration step evolving the interest rates in price sensitive events by a singlesimulation step.
The first alternative is straightforward. It is, however, extremely time consumingbecause for each realization for the forward rate, the time step used must be small forthe discretization to be accurate. The second alternative is more appealing as for eachrealization of the forwards, only as many steps are needed as there are price sensitive
43
events. Therefore the second alternative is much more efficient. It has turned out thatthere exists very accurate integral approximation methods and some of them will bereviewed here. Note again that the arbitrage free dynamics of n-factor model in QTN
is given by
dFi(t) = Fi(t)µi(t) + Fi(t)‖γi(t)‖bi,1(t)dW1(t) + · · ·+ Fi(t)‖γi(t)‖bi,n(t)dWn(t), t ≤ Ti−1
(6.1)with the initial condition Fi(t0), and the no arbitrage drift given by (4.7). Further the
loadings bi,j = B(n)i,j with B(n) being the pseudo root of rank n, as introduced in the
previous chapter. The rank n correlation matrix ρ(n) is given by BBT = ρ(n). Notethat
bi,1bj,1 + · · ·+ bi,nbj,n = ρ(n)i,j .
It is possible to solve time T value of forward rates Fi(T ) in (6.1) with the initialcondition Fi(t0) and to arrive at the solution
Fi(T ) = Fi(t0) exp
−
N∑
j=i+1
∫ T
t0
‖γi(t)‖τkFk(t)‖γj(t)‖1 + τkFk(t)
dt− 12
∫ T
t0
‖γi(t)‖2 dt
+∫ T
t0
‖γi(t)‖bi,1 dW1(t) + · · ·+∫ T
t0
‖γi(t)‖bi,n dWn(t)) (6.2)
We call ∫ T
t0
µi(t) dt− 1
2
∫ T
t0
‖γi(t)‖2 dt (6.3)
the deterministic part of the evolution in (6.2) and denote
Xi(T ; t0) =
∫ T
t0
‖γi(t)‖bi,1dW1(t) + · · ·+∫ T
t0
‖γi(t)‖bi,ndWn(t). (6.4)
and call it the stochastic part of the evolution in (6.2).
By Ito’s isometry Xi(T ; t0) is a Gaussian random variable with mean 0. Further
because Brownian motions Wj(t)’s are independent, so are∫ T
t0‖γi(t)‖bi,1dWj(t)’s. This
means that the variance of Xi(T ; t0) is given by the sum of the component variances,which again using Ito’s isometry can be computed as follows
Var (Xi(T ; t0)) =d∑
k=1
Var(∫ T
t0
‖γi(t)‖bi,kdWk(t))
=d∑
k=1
∫ T
t0
‖γi(t)‖2b2i,k dt =
∫ T
t0
‖γi(t)‖2d∑
k=1
b2i,k =
∫ T
t0
‖γi(t)‖2 dt
44
The joint distribution of the vector [X1(t0, T ), . . . , XN(t0, T )]T ∈ R1×N is thus Gaus-sian with mean 0 and characterized by the marginal covariance matrix. The elementsof this can be computed using similar calculations
Cov (Xi(T ; t0), Xi(T ; t0))
= EQTN[(
Xi(T ; t0)− EQTN (Xi(T ; t0)))(
Xj(T ; t0)− EQTN (Xj(T ; t0)))]
= EQTN [Xi(T ; t0)Xj(T ; t0)] = EQTN
[d∑
k=1
d∑
l=1
∫ T
t0
‖γi(t)‖bi,k dWl(t)∫ T
t0
‖γj(t)‖bj,l dWk(t)
]
=d∑
k=1
d∑
l=1
EQTN
[∫ T
t0
‖γi(t)‖bi,k dWk(t)∫ T
t0
‖γj(t)‖bj,l dWl(t)]
Because the integrands are deterministic and the Brownian drivers Wk(t) and Wl(t)are independent for k 6= l, then the integral processes are also independent. Thus fork 6= l we have
EQTN
[∫ T
t0
‖γi(t)‖bi,k dWk(t)∫ T
t0
‖γj(t)‖bj,l dWl(t)]
= EQTN
[∫ T
t0
‖γi(t)‖bi,k dWk(t)]EQTN
[∫ T
t0
‖γj(t)‖bj,l dWl(t)]
= 0
For k = l, use the second part of Ito’s isometry to get
EQTN
[∫ T
t0
‖γi(t)‖bi,l dWl(t)∫ T
t0
‖γj(t)‖bj,l dWl(t)]
=∫ T
t0
‖γi(t)‖‖γj(t)‖bi,lbj,l dt
and finally
Cov (Xi(T ; t0), Xi(T ; t0))
=∫ T
t0
‖γi(t)‖‖γj(t)‖d∑
l=1
bi,lbj,l dt =∫ T
t0
‖γi(t)‖‖γj(t)‖ρi,j dt
Thus the marginal covariance matrix is given by
Cov(t0, T ) :=
∫ T
t0‖γ1(t)‖2 dt
∫ T
t0‖γ1(t)‖‖γ2(t)‖ρ1,2 dt . . .
∫ T
t0‖γ1(t)‖‖γN(t)‖ρ1,N dt∫ T
t0‖γ2(t)‖‖γ1(t)‖ρ1,2 dt
∫ T
t0‖γ2(t)‖2 dt . . .
∫ T
t0‖γ2(t)‖‖γN(t)‖ρ2,N dt
......
. . ....∫ T
t0‖γN(t)‖‖γ1(t)‖ρN,1 dt
∫ T
t0‖γN(t)‖‖γ2(t)‖ρN,2 dt . . .
∫ T
t0‖γN(t)‖2
(6.5)We present now how the simulation is being carried out.
6.1 Evolution of the Stochastic Part
Suppose time S value of forward rates Fi(S) are known and one wants to simulate tothe timepoint T > S. One should note that the marginal covariances given by (6.5)
45
become degenerate as forward rates start to reset. The simulation is carried out asfollows if one wants to use only n random factors that drive the yield curve.
1. Carry out the integration to find Cov(S, T ).
2. Make a random draw Z from standard n-dimensional normal distributionN (0n×1, In×n).
3. Find a rank n pseudo root C ∈ RN×n of Cov(S, T ).
4. Compute CZ.
By construction CZ is a sample from the joint vector [X1(S, T ), . . . , XN(S, T )]T .
6.2 Evolution of the Deterministic Part
Suppose time S value of forward rates Fi(S) are known and one wants to simulate tothe time point T > S. The second integral in the deterministic part is easy. If weknow the Cov(t0, T ), it is possible to read the second integral values from the diagonal.Therefore we focus on computing the drift integral. As suggested by Rebonato in [18,pp. 121–131], it is possible to make some simplifying assumptions for the drift integralin (6.3). The naive idea would be to freeze the drift of the forwards, that is to assumethem deterministic - more precisely their constant value determined at time S. Fori’th forward the Eq. (6.3) could then be written
−N∑
j=i+1
τjFj(0)1 + τjFj(0)
∫ T
S‖γi(t)‖‖γj(t)‖ρi,j(s)− 1
2
∫ T
S‖γi(t)‖ (6.6)
Unfortunately Rebonato reports and explains in [18, pp. 121–128] that such an approx-imation would be too crude especially when making a very long simulation step, andthus using the approximated Eq. (6.6) to predict long term forward distribution usingfrozen drift. Joshi et al. in [14] have, however, made empirical studies for approxi-mating the integral ∫ T
S
‖γi(t)‖‖γj(t)‖ρi,jFj(t)
1 + τjFj(t)ds
All the sophisticated approximations therein are some variants of the predictorcorrector methods. We present a few methods that we used to test their mutualaccuracy. The discussion is based on the paper by Joshi [14].
Denote the drift integral approximation for i:th forward by µi. To estimate µi thenaive method, as earlier proposed, would be to freeze the forward rates Fj, j > i fortheir current value Fj(S). This is precisely done and we call this estimate the Eulerapproximation and it is presented below:
46
Euler approximation: Freeze the Fj for j > i for their current value S and compute
µi = −N∑
j=i+1
Fj(S)τj
1 + Fj(S)τj
∫ T
S‖γi(t)‖‖γj(t)‖ρi,j dt
After the stochastic part has been sampled, Euler method then computes the samplefrom Fi(T ) using Eq. (6.2).
Euler method used on its own is, however, too crude to be used in accurate calculationsas proposed by Rebonato. The reason why it has been presented is that it forms thebasis for more accurate predictor corrector method. We were using two more accuratemethods from Joshi [14]. In our numerical studies these two methods to approximatethe drift integral to price CMS type options had same accuracy. The first method usesthe fact that when computing the µi for forward rate Fi, we have already estimatedthe Fj(T )’s for j > i. In the base case the estimation is done for the terminal forwardFN , which is drift free under the terminal measure. Therefore we are not requiredto approximate the drift integral and thus sampling FN(T ) is straightforward from
(6.2). The procedure is repeated iteratively and the state dependent partFj(t)τj
1+Fjτjis
approximated by averaging it from the values at both endpoints:
Iterative predictor corrector method:
µi = −12
N∑
j=i+1
(Fj(S)τj
1 + Fj(S)τj+
Fj(T )τj
1 + Fj(T )τj
)∫ T
S‖γi(t)‖‖γj(t)‖ρi,j dt
Iterative method works only under terminal measure.
The second method is also a predictor corrector method, but it is not iterative. Inthe first step it uses Euler stepping to get an estimate sample from Fi(T ), denoted
FEuli (T ). This step is the predictor step. In the corrector step it corrects the estimate
by averaging the values at both endpoints using the predictor estimate at the final endpoint, that is:
Predictor Corrector method:
µi = −12
N∑
j=i+1
(Fj(S)τj
1 + Fj(S)τj+
FEulj (T )τj
1 + FEulj (T )τj
)∫ T
S‖γi(t)‖‖γj(t)‖ρi,j dt
Remark 6.2.1. In simulation, the marginal covariances Cov(S, T ) should be computedin advance because they do not change. The same holds for the corresponding pseudoroots.
47
Chapter 7
Numerical Results
In this chapter we present series of numerical tests that were performed to test theLibor market model in practise. The practical implementation was done in Matlabwith Financial, Statistical and Optimization toolboxes. Matlab is an ideal environmentfor the numerical computing as it contains many tweaked numerical algorithms. Inour purposes these were good calendar tools, random number generators and optimiza-tion routines. In the calibration we used fmincon to solve the nonlinear constrainedoptimization.
The outline of the chapter goes as follows: First we calibrate the model for the marketsituation on 22nd Jul 2008. We discuss in detail how the calibration was carriedout. After finding the optimal parameters for the model we will investigate the modelgoodness of fit with different low factor parametrizations. It is interesting to investigatehow the number of model factors affects the term structure of volatility and this is alsodone in the first section. As an application we will price the CMS spread call optionswith different swap tenors and make some empirical studies as to how the numberof model factors affect the option price. We should expect the low factor model togive relatively low prices as it ignores non perfect correlations between rates. Finallywe will propose a method to price more exotic CMS range accruals whose payoff isdependent on the daily market quotations for CMS spread.
After moving to the practical pricing in Libor Market Model, it is worth explaininghow the pricing is done in practise. The model should compute in advance as muchas possible to save the extra computational burden.
1. Calibrate the model for the market to find the unknown model parameters a,b,g∞ characterizing the instantaneous volatility and the parameters η1,η2 and ρ∞characterizing the model instantaneous correlation.
2. Compute the marginal covariance elements between the expiry-maturity datesT0, . . . , TN−1, namely Cov(Tk−1, Tk)ij :=
∫ Tk
Tk−1σInst
i (t)σInstj (t)ρi,j dt, for i, j =
1, . . . , N .
48
3. Express the option payoff in terms of model forward rates at price sensitiveevents.
4. Evolve the forward rates at price sensitive events. To do this in practise, assumethat the forward rates at time S are known and we are interested in knowingthe forward rates at time T > S. Assume that a model with n-factors is beingused. To evolve the forward rates for time T compute the marginal covari-ance matrix Cov(S, T )ij =
∫ T
SσInst
i (t)σInstj (t)ρi,j dt, between the rates. The
deterministic part (drift corrections and the quadratic variation) of the evolu-tion is fully characterized by the marginal covariance elements as explained inChapter 6. To sample from the stochastic part of the evolution find the pseudoroot A ∈ RN×n such that Cov(S, T ) = AAT . Make an independent sampleN (0n×1, In×n) from multi normal standard distribution. To sample from the
stochastic part∫ T
Sσi,1(t) dW1(t)+· · ·+σi,n(t) dWn(t) compute [AN (0n×1, In×n)]i.
Remark 7.0.2. One can use the pre computed matrices Cov(Tk−1, Tk) (if possible)to save some computational time. If using a n-factor model, the correlation matrixρi,j has a rank n. The marginal covariance matrix is obviously symmetric, thus ithas real and positive eigenvalues. In general it is possible, however, that the marginalcovariance matrix has a rank greater than n. To find the pseudo root of rank n, onemust use the eigenvalue zeroing taking into account only the n biggest eigenvalues formatrix A.
7.1 Model Calibration with Different Factors
The used input data is listed in Appendix B. Table B.1 lists the expiry maturity pairsand the corresponding zero coupon bonds. These zero coupons are used to compute thecorresponding forward rates today through the Eq. (3.1). Table B.2 lists the forwardrate caplet volatilities. As it was remarked earlier in Chapter 3, the market quotes onlythe flat volatilities for different cap tenors. The volatilities in Table B.2 are computedby bootstrapping the flat cap volatilities from the market, a method that is routinelydone by traders. Finally Table B.3 shows the ATM swaption volatilility matrix quotedby a market broker.
Calibration is first done for the full-factor case because the correlation matrix impliedby parameters ρ∞, η1, η2 is of full rank N and further the pseudo root of this is ofrank N .
In practise to find a globally good fit for the market swaptions, Schoenmakers in[22] suggests that the calibration should be done sequentially. It is first carriedout to the first row of the swaption matrix. That is, we will take the swaptionsexpiring in 1 year as the set of calibrated instruments. We find the optimal setof parameters a(1), b(1), g
(1)∞ , ρ
(1)∞ , η
(1)1 , η
(1)2 for pricing swaptions expiring in 1 year.
This parameter set is taken as an initial guess for the optimization which now takesinto account also the volatilities from the second row of the swaption matrix. Thesolution a(2), b(2), g
(2)∞ , ρ
(2)∞ , η
(2)1 , η
(2)2 for this optimization routine is again carried to
49
Swaptions # of full (40) factor parametersup to exp swaptions a b g∞ ρ∞ η1 η2 MRE MARE
1y 10 0 0.3455 1.1372 0.3693 0.9627 0.032 0.42% 1.30%2y 20 0 0.3399 1.0908 0.3325 0.9778 0.035 -0.45% 2.36%3y 30 0 0.3398 1.0909 0.3332 0.9778 0.035 -0.31% 2.03%4y 40 0 0.3397 1.0914 0.3344 0.9778 0.035 -0.16% 1.80%5y 50 0 0.3396 1.0918 0.3355 0.9779 0.035 -0.18% 1.65%7y 60 0 0.3395 1.0920 0.3364 0.9779 0.035 -0.22% 1.51%10y 70 0 0.3395 1.0920 0.3364 0.9779 0.035 -0.39% 1.57%
Table 7.1: Model parameters and mean vol errors through the sequential calibration
the next phase. When the whole swaption matrix was chosen for the set of cali-brated instruments and the parameter set a(7), b(7), g
(7)∞ , ρ
(7)∞ , η
(7)1 , η
(7)2 was found, we
computed the implied full rank correlation matrix ρi,j(ρ(7)∞ , η
(7)1 , η
(7)2 ). Now when we
reduced the number of model factors, the eigenvalue zeroing for the correlation matrixρi,j(ρ
(7)∞ , η
(7)1 , η
(7)2 ) was carried out. When we were calibrating the n-factor model
(n < N), N−n smallest eigenvalues of the correlation matrix were zeroed and the rank
n correlation matrix ρi,j(ρ(7)∞ , η
(7)1 , η
(7)2 ) was formed by the methods explained in section
5.3. This correlation matrix was fixed and the sequential optimization for parametersa, b, g∞ in n-factor model was carried out with initial guess a(7), b(7), g
(7)∞ . We
observed in the experiments that when trying to fit the humped shape volatility struc-ture, controlled by the parameter a, jointly with b and g∞ the a parameter convergedto zero independent of the initial guess feed for the algorithm. Thus at the first phaseof the calibration, we imposed a boundary condition for the parameter a to be zerowith the volatility structure being exponentially decaying. We feed the initial guess
[a(0), b(0), g(0)∞ , ρ(0)
∞ , η(0)1 , η
(0)2 ] = [0, 0.4, 0.7, 0.15, 1.1, 0.10]
It was found out that different initial guesses converged for different optimal sets ofparameters. Fortunately, it was the case that the mean relative errors with differentinitial guesses were the same, so the different optimal set of parameters were still lyingon the same contour curve in the parameter hyperspace.
Table 7.1 lists the optimal solutions in different phases of full-factor calibration andalso the mean of the relative volatility errors (MRE)
σBlackα,β − σModel
α,β
σBlackα,β
,
and the mean of absolute relative errors (MARE)∣∣∣∣∣σBlack
α,β − σModelα,β
σBlackα,β
∣∣∣∣∣
between the set of calibrated instruments. One can see from the Table 7.1, that theoptimal set of parameters found after 2 first phases do not change at all after themodel has been calibrated for the set of 2 two first rows of swaptions.
50
Exp/Mat 1y 2y 3y 4y 5y 6y 7y 8y 9y 10y1y 2.86% 1.22% 2.47% 1.93% 0.56% 0.60% 0.82% 1.27% 2.78% 3.62%2y -7.75% -6.08% -5.24% -4.08% -3.29% -3.00% -1.12% 0.52% 1.33% 3.30%3y -1.93% -3.19% -2.09% -1.23% -0.63% -0.33% 1.21% 1.83% 2.26% 2.86%4y -2.22% -0.61% -1.50% -1.11% -0.86% 0.04% 1.33% 1.74% 2.28% 2.79%5y -1.65% -0.37% -1.87% -1.62% -1.36% -0.92% 0.02% 0.64% 1.89% 2.95%7y -3.01% -1.06% -0.65% -0.88% -0.83% -0.74% -0.43% 0.61% 0.67% 1.66%10y -5.77% -2.98% -3.29% -1.89% -0.06% -0.91% -0.91% 0.13% 1.26% 2.41%
Table 7.2: Relative vol error for ATM swaptions in 5-factor model
Table 7.2 shows the relative errors along all the swaption volatilities for the 5-factormodel after the calibration. It can be observed that the relative error is highest in thefirst two elements in the leftmost upper corner of the table. The volatilites in the firstcolums of swaptions should behave much like 1 year cap volatilites, because an optionfor 1 year forward swap should be exactly a 1 year tenor interest rate caplet. 6 monthcaplet volatilities at the short end however are first low and increase rapidly before2 years and after that start decreasing. The Swap volatilities (in the first column)are, however, decreasing. This ”inconsistency” explains the worse fit in short maturityand short tenor swaptions which are dependent on short caplet volatilities throughEq. (5.8). More or less the relative error stays low in different regions of the swaptionmatrix and fit can be considered good. The error renders in the practical bid offerspreads that can be observed in the market. If compared to the swaption pricing errorsreported in Brigo et al. in [3] with different covariance parametrizations, relative errorsare smaller. Therein Brigo et al. have investigated the calibration through the anglesform earlier proposed in Chapter 5.
In the second phase of the calibration we feed the humped shape volatility by givingthe a parameter an initial guess a = 0.5. We found out that the fewer factors themodel was using, the more humped shape volatility structure it implied. This can beseen observing the optimal a parameter from Table 7.3(c). With the full-factor model,the volatility converged again to exponentially decaying.
Using principal component analysis, we can factor out the directions where the mostvariation in correlation matrix is located. The first 5 eigenvalues of the full rankcorrelation matrix are given by
[ 30.13 5.33 1.71 0.80 0.46 ]
The amount of variation explained in the directions spanned by the correspondingeigenvectors are
[ 75.3% 13.3% 4.3% 2.0% 1.2% ]
Thus the first 5 eigenvalues explain over 96% of the total correlation. We shouldexpect the pricing error for swaptions implied by the 3 to 5 factors be relatively samemagnitude as the one for the full-factor model reported in Table 7.2. This is indeedthe case as can be seen from tables 7.3(a), 7.3(b) 7.3(c). They show the optimalparameters for the model and also the average errors in volatility. We have plotted amodel implied swaption surface with comparison to the market. These can be seen inFigure 7.1.
51
Table 7.3: Optimal set of parameters with different factor models
(a) 1- and 2-factor model parameters and the mean pricing error in vols
1-factor 2-factora b g∞ MRE MARE a b g∞ MRE MARE
0.2941 0.9169 0.713 0.00% 1.57% 0.0793 1.0475 0.7369 -0.13% 1.508%
(b) 3- and 4-factor model parameters and the mean pricing error in vols
3-factor 4-factora b g∞ MRE MARE a b g∞ MRE MARE
0.1041 1.0137 0.8215 -0.31% 1.62% 0.0541 1.0057 0.8641 -0.38% 1.82%
(c) 5- and full-factor model parameters and the mean pricing error in vols
5-factor 40-factora b g∞ MRE MARE a b g∞ MRE MARE
0.0352 0.968 0.8999 -0.38% 1.88% 0 0.5916 1.0863 -0.31% 1.61%
24
68
10
1234567
0.1
0.12
0.14
0.16
0.18
0.2
0.22
expirymaturity
vola
tility
(a) Market ATM swaption vols
24
68
10
12
34
56
7
0.1
0.12
0.14
0.16
0.18
0.2
0.22
expirymaturity
vola
tility
(b) Model ATM swaption vols
2
4
6
8
10
12
34
56
7−0.01
−0.005
0
0.005
0.01
0.015
maturityexpiry
abso
lute
err
or in
vol
(c) absolute difference between model andmarket vols
Figure 7.1: Comparison between market and model implied volatilities
52
Figure 7.2 plots the model implied correlation surface after the eigenvalue zeroing hasbeen carried out for the semi-parametric full rank correlation matrix. Note that thecorrelations are always higher with low factor models if compared to the full-factorcorrelation surface in Subfigure 7.2(f). From the construction it is always the case thatthe classical 1-factor models imply a flat correlation surface as seen in Subfigure 7.2(a).Flat correlation thus shifts the yield curve in same direction.
1020
3040
10
20
30
400.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Cor
r
Forward rateForward rate
(a) 1-factor forward instanta-neous correlation
510
1520
2530
3540
10
20
30
400.3
0.4
0.5
0.6
0.7
0.8
0.9
Forward rateForward rate
Impl
ied
corr
elat
ion
(b) 2-factor forward instantaneouscorrelation
510
1520
2530
3540
10
20
30
400.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Forward rateForward rate
Impl
ied
corr
elat
ion
(c) 3-factor forward instantaneouscorrelation
510
1520
2530
3540
10
20
30
400.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Forward rateForward rate
Impl
ied
corr
elat
ion
(d) 4-factor forward instantaneouscorrelation
510
1520
2530
3540
10
20
30
40
0.4
0.5
0.6
0.7
0.8
0.9
1
Forward rateForward rate
Impl
ied
corr
elat
ion
(e) 5-factor forward instantaneouscorrelation
510
1520
2530
3540
10
20
30
400.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Forward rateForward rate
Impl
ied
corr
elat
ion
(f) full-factor forward instantaneouscorrelation
Figure 7.2: Model instantaneous correlations in different factor models
In Figure 7.3 and Figure 7.4 the term structure of volatilities are plotted. It canbe observed that increasing the number of model factors makes the term structureof volatility more time homogenous. For the 1-factor model the term structure isrelatively higher compared to the market situation, while it flattens down to the currentmarket situation when the number of model factors is increased. With the 5-factormodel the term structure of volatility is almost time homogenous.
53
4 6 8 10 12 14 16 18 20time
market term structureafter one yearafter 2 yearsafter 3 yearsafter 4 yearsafter 5 yearsafter 6 yearsafter 7 yearsafter 8 yearsafter 9 yearsafter 10 yearsafter 11 yearsafter 12 yearsafter 13 yearsafter 14 yearsafter 15 yearsafter 16 yearsafter 17 yearsafter 18 yearsafter 19 years
(a) 1-factor model term structure of volatility
4 6 8 10 12 14 16 18 20time
market term structureafter one yearafter 2 yearsafter 3 yearsafter 4 yearsafter 5 yearsafter 6 yearsafter 7 yearsafter 8 yearsafter 9 yearsafter 10 yearsafter 11 yearsafter 12 yearsafter 13 yearsafter 14 yearsafter 15 yearsafter 16 yearsafter 17 yearsafter 18 yearsafter 19 years
(b) 2-factor model term structure of volatility
4 6 8 10 12 14 16 18 20time
market term structureafter 1 yearafter 2 yearsafter 3 yearsafter 4 yearsafter 5 yearsafter 6 yearsafter 7 yearsafter 8 yearsafter 9 yearsafter 10 yearsafter 11 yearsafter 12 yearsafter 13 yearsafter 14 yearsafter 15 yearsafter 16 yearsafter 17 yearsafter 18 yearsafter 19 years
(c) 3-factor model term structure of volatility
Figure 7.3: Term structure of volatilities for 1-, 2-, and 3-factor models
54
4 6 8 10 12 14 16 18 20time
market term structureafter one year1after 2 yearsafter 3 yearsafter 4 yearsafter 5 yearsafter 6 yearsafter 7 yearsafter 8 yearsafter 9 yearsafter 10 yearsafter 11 yearsafter 12 yearsafter 13 yearsafter 14 yearsafter 15 yearsafter 16 yearsafter 17 yearsafter 18 yearsafter 19 years
(a) 4-factor model term structure of volatility
4 6 8 10 12 14 16 18 20time
market term structureafter 1 yearafter 2 yearsafter 3 yearsafter 4 yearsafter 5 yearsafter 6 yearsafter 7 yearsafter 8 yearsafter 9 yearsafter 10 yearsafter 11 yearsafter 12 yearsafter 13 yearsafter 14 yearsafter 15 yearsafter 16 yearsafter 17 yearsafter 18 yearsafter 19 years
(b) 5-factor model term structure of volatility
Figure 7.4: Term structure of volatilities for 4- and 5-factor models
55
7.2 Pricing CMS Spread Options
As an application we will use the model to price CMS spread options. CMS spreadoptions are largely used for speculative purposes in the over the counter (OTC)markets. Traders can take market views of the changes in different segments of theyield curve by buying or selling CMS spread options. For example if a trader believesthat the current 10 year rate is relatively low compared to the 2 year rate, he can spec-ulate this in different ways. One way is to buy this rate and go into a forward startingpayer swap and at the same time receive 2 year forward swap rate. Because markto market value of a swap is characterized by its basis point value through Eq. (3.3)and because long tenor swap has significantly higher basis point value than with ashort tenor swap, traders usually do this strategy risk neutrally by adjusting the swapnotionals appropriately. There is, however, theoretically unlimited downside risk inthis strategy if the trader’s views will be incorrect. This happens if the spread movesin unfavorable direction. This risk can be avoided with an extra cost of an optionality.In other words the future spread will be bought in case it is higher than predeterminedstrike level. This gives a natural need for market spread option.
CMS spread option are options on the spread between two different swap rates. Thefirst leg in the swap is typically a long maturity swap (10 to even 30 years) and the otheris a short maturity one (typically in practise 2 years). More precisely let T0, . . . , TN bethe usual time structure and denote T−1 the spot date. A CMS spread option pays attime Tδ, δ ∈ 0, . . . , N the swap spread between two swap indices checked at a timeTα if the spread is above the strike level. CMS spread option pays the simple interest
max([G1Sα,γ(Tα)−G2Sα,β(Tα)]−X 0),
between the time interval [Tα, Tδ ] for a predetermined amount of cash called thecontract value. G1 and G2 are gearing ratios for both swap indices respectively andX is called a strike for the CMS spread. The spread option is said to be at the money(ATM) if the strike is chosen to be today’s forward starting spread, that is
X = G1S−1,γ−α−1(T−1)−G2S−1,β−α−1(T−1)
The above strike means that if nothing changes during the life of the option, it willjust stay at the money. We will derive the no arbitrage price for the option. Withoutloss of generality, we derive the price for unit notional. Denote the CMS spread optionprice at time t by CMSSO(t; Tα, Tδ, β, γ,X). It is obvious that time Tδ value of CMSspread option is given by
CMSSO(Tδ;Tα, Tδ, β, γ,X) = max([G1Sα,γ(Tα)−G2Sα,β(Tα)]−X , 0),
and as the payoff is already determined at Tα, it is obvious that
CMSSO(Tα;Tα, Tδ, β, γ,X) = max([G1Sα,γ(Tα)−G2Sα,β(Tα)]−X , 0)P (Tα;Tδ).
Now CMSSO(t; Tα, Tδ, β, γ,X) is a traded asset price and thus the normalized priceratio
CMSSO(t; Tα, Tδ, β, γ, X)P (t; TN )
56
Index 1 Index 2Index tenor 10y 2y, 5y, 8y
Payment frequency Annual vs Semi-annual Annual vs Semi-annualBasis 30/360 vs Act/360 30/360 vs Act/360
Gearing ratio 1.0 1.0Strike ATM
Notional 1 000 000Day convention Act/360
Table 7.4: CMS spread option structure
needs to be a martingale under the equivalent martingale measure QTN to avoid arbi-trage. It thus needs to satisfy the martingale condition
CMSSO(0;Tα, Tδ, β, γ, X)P (0; TN )
= EQTN
(CMSSO(Tα; Tα, Tδ, β, γ, X)
P (Tα; TN )
∣∣∣∣F0
)
= EQTN
(max ([G1Sα,γ(Tα)−G2Sα,β(Tα)]−X , 0)P (Tα;Tδ)
P (Tα;TN )
∣∣∣∣F0
).
It follows that the fair price today for the spread option is given by
CMSSO(0;Tα, Tδ, β, γ, X)
= P (0;TN )EQTN
(max ([G1Sα,γ(Tα)−G2Sα,β(Tα)]−X , 0)
N∏
k=δ+1
(1 + τkFk(Tα))
∣∣∣∣∣F0
).
(7.1)As can be seen from (7.1), the spread option price can be calculated as soon as theswap and forward rate distributions at terminal measure are known at time Tα. Wethus simulate the forward rates at time Tα and compute the corresponding par swaprates at future times through Eqs. (5.5) and (5.6). All the options considered have thestrike equal to the forward spread. This strike has been chosen to investigate whatkind of curve movements each factor model implies and thus how the option pricesdiffer from each other. If a model implies completely parallel curve movements it willimply ATM CMS option prices equal zero. Table 7.5(a) lists the calculated forwardswap rates and 7.5(b) spreads between chosen rates in basis points (1 BPS = 0.01%).We take all options having one year tenor, that is the payoff is paid 1 year after it hasbeen settled. Table 7.4 shows the precise option information.
The pricing has been carried out with 70 000 simulations for each contract. It wastested that even a number of 10 000 simulations converged enough to produce priceswhich were accurate in the second most significant digit. In trading this accuracy isusually enough in practise. We made an investigation how spread option prices aresensitive to the amount of factors in model. In Table 7.5(c) the ATM option prices aregiven. If one looks at the table, the spread option prices in each factor model have onespecific characteristic property. This is that the option prices decrease rapidly whenthe spread narrows in tenors. This truly makes intuitively sense because a 10 yearrate moves much more in parallel with an 8 year rate than with a 2 year rate.
One can, however, note that 1- and 2-factor models imply much lower prices thatmodels with more factors. The relatively lowest prices for the 1-factor model can be
57
explained by an extrimely unrealistic flat correlations that the model implies as canbe seen from Figure 7.2(a). By construction this is always the case. This is actuallythe same problem that exists in earlier spot rate models, where the spot rate processis driven by only one Brownian driver. The first factor has an intuitive interpretationto produce parallel shifts in the yield curve.
A 2-factor model has non flat instantaneous correlation and this implies also increasedspread option prices as rates are more free to move in opposite directions. Two factorsstill lack to produce prices with the same magnitude for full amount of explainingfactors especially on interest spread between long and short maturity rates. Thepossible interpretation for the second factor is that it is changing the slope of the yieldcurve. With 2 explaining factors, forward rates give different responses to parallel andslope changes and the model produces curve scenarios that are widening the spread.It is still the case that the 2 factors cannot produce all the bending that are necessaryto invert the yield curve. It is crucial that a realistic interest rate model needs to toproduce them because they are possible in real market. This is especially the case inturbulent market conditions where central banks have made extraordinary changes inmonetary politics.
Observing the table further, 3 factors start to produce prices with the same magnitudefor full-factor model and thus the natural interpretation for the third factor is thatit is the change in curvature of the yield curve (”second derivative”). Thus differentforward rate responses for changes in curvature make it possible for the yield curve toinvert. The 5-factor model prices are practically the same compared to the full-factormodel and thus can be considered to produce same curve scenarios and same terminaldistributions for forward and swap rates. One can also make a second observationfrom Table 7.5(d). This is that the pricing error increases as a function of the optionexpiry. This is practically the case for all factor models up to 1 to 4. The 5-factormodel is the first that can produce same magnitude prices along all maturities up to10 years. If we are pricing options with relatively short maturities even a model with3 factors is accurate enough.
We have plotted the responsiveness of the forwards bi,j for the factors up to 5 inFigure 7.5. These plots are the the first 5 columns of the psuedo root of ρ(ρ∞, η1, η2).One can observe that the forward responses for the first factor have the greatest impactcompared to the others. However, responses are of the same sign and approximatelysame magnitude. Thus it is really the case that the first factor shifts the yield curve insame direction. Additionally the shifts are almost parallel. The forward responses forthe second factor are of the same magnitude, but they change sign at certain point.Second factor is thus effectively bending the yield curve through this point. Responsesfor the higher factors are smaller, but they give additional bending of the yield curvethrough different axis. One can make a second observation that through all factors,responses in long end are flat. This implies the long end of the yield curve to move inparallel which makes intuitively sense.
58
Tab
le7.
5:P
rici
ngre
sult
sdo
new
ith
the
Lib
orm
arke
tm
odel
(a)
Forw
ard
swap
rate
sfo
rsw
aps
wit
hdi
ffere
ntte
nors
Index
2y
5y
8y
10y
1y
5.1
499%
5.0
064%
4.9
856%
5.0
147%
2y
5.0
283%
4.9
300%
4.9
640%
5.0
021%
3y
4.9
311%
4.9
000%
4.9
718%
5.0
103%
4y
4.8
531%
4.9
049%
4.9
939%
5.0
346%
5y
4.8
483%
4.9
520%
5.0
353%
5.0
643%
7y
4.9
873%
5.0
942%
5.1
343%
5.1
204%
10y
5.1
982%
5.2
085%
5.1
534%
5.1
222%
(b)
Cor
resp
ondi
ngfo
rwar
dsw
apsp
read
sin
basi
spo
ints
Fw
dsp
read
/B
PS
10y-2
y10y-5
y10y-8
y1y
-13.5
20.8
32.9
12y
-2.6
27.2
23.8
13y
7.9
311.0
73.8
54y
18.1
512.9
44.0
85y
22.0
111.2
32.9
07y
13.3
12.6
3-1
.39
10y
-7.6
0-8
.63
-3.1
3
(c)
AT
MC
MS
spre
adop
tion
pric
esw
ith
diffe
rent
fact
orm
odel
sSta
rtEnd
1-facto
r2-facto
r3-facto
r4-facto
r5-facto
r40-facto
rdate
date
10y-2
y10y-5
y10y-8
y10y-2
y10y-5
y10y-8
y10y-2
y10y-5
y10y-8
y10y-2
y10y-5
y10y-8
y10y-2
y10y-5
y10y-8
y10y-2
y10y-5
y10y-8
yT1
(1y)
T3
(2y)
1143
458
138
1584
859
299
2026
979
300
2059
991
309
2060
981
312
2073
995
315
T3
(2y)
T5
(3y)
1350
546
158
2084
1096
367
2386
1121
366
2466
1232
386
2520
1216
390
2537
1226
385
T5
(3y)
T7
(4y)
1144
498
151
2090
1090
361
2121
1127
387
2419
1235
395
2490
1251
403
2496
1252
393
T7
(4y)
T9
(5y)
1028
471
149
1987
1038
332
2050
1153
397
2380
1216
399
2473
1257
405
2455
1234
403
T9(5
y)
T11
(6y)
990
450
165
1799
901
288
2057
1151
384
2220
1157
391
2370
1207
391
2364
1197
394
T13(7
y)
T15
(8y)
900
440
147
1293
639
206
1916
985
315
1977
1090
364
2062
1099
370
2070
1102
372
T19
(10y)
T21
(11y)
873
451
179
894
473
185
1146
572
204
1475
721
237
1653
836
267
1693
843
270
(d)
Abs
olut
epr
icin
ger
ror
inC
MS
spre
adop
tion
sco
mpa
red
for
the
full-
fact
orm
odel
Sta
rtEnd
1-facto
r2-facto
r3-facto
r4-facto
r5-facto
rdate
date
10y-2
y10y-5
y10y-8
y10y-2
y10y-5
y10y-8
y10y-2
y10y-5
y10y-8
y10y-2
y10y-5
y10y-8
y10y-2
y10y-5
y10y-8
yT1
(1y)
T3
(2y)
45
%54%
56%
23%
14%
5%
2%
2%
5%
0%
0%
2%
1%
1%
1%
T3
(2y)
T5
(3y)
47%
55%
58%
18%
11%
4%
6%
9%
4%
3%
0%
0%
1%
0%
1%
T5
(3y)
T7
(4y)
54%
60%
62%
16%
13%
8%
15%
10%
2%
3%
1%
0%
0%
0%
2%
T7
(4y)
T9
(5y)
58%
62%
63%
19%
16%
18%
17%
6%
1%
3%
1%
1%
1%
2%
1%
T9(5
y)
T11
(6y)
58%
62%
58%
24%
25%
27%
13%
3%
2%
6%
3%
1%
0%
1%
1%
T13(7
y)
T15
(8y)
56%
60%
60%
38%
42%
44%
7%
11%
15%
5%
2%
2%
1%
1%
1%
T19
(10y)
T21
(11y)
48%
47%
34%
47%
44%
31%
32%
32%
24%
13%
14%
12%
2%
1%
1%
59
0 5 10 15 20 25 30 35 40−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Forward rate
Res
pons
e
1 factor loadings2 factor loadings3 factor loadings4 factor loadings5 factor loadings
Figure 7.5: Forward rate responses for different factors
7.3 Pricing CMS Range Accrual Notes
CMS range accrual pays the daily CMS spread interest at the end of the optioncontract, when it is observed to stay in a certain predetermined range. The rangeconsidered here is the positive CMS spread, but it can be anything else too (boundedfrom above and/or below). At each day the spread between two spot par swap ratesin being checked. If this spread stays in the predetermined range, the interest equal tothe spread is paid between dates the spread is checked up again. If using the Act/360Basis the year fraction between observable date is 1/360 if there is no weekend, and3/360 if there is a weekend between.
In LMM we can simulate only the par swap rates at the time points, which were chosenin the set of expiry maturity pairs T0, . . . , TN of modeled forward rates.
Let two time instants Ti−1 and Ti be fixed, and denote the business days between thesedates as sk, k = 1, + . . . n where s1 = Ti−1, and sn = Ti. Further denote here S1(t)the spot swap rate of the first leg in the spread, and S2(t) the spot swap rate of thesecond leg in the spread. In the Libor market model, we can simulate the values forspot swap rates at Ti−1 and Ti, namely Sl(Ti−1) and Sl(Ti) for l = 1, 2. Now betweenthe dates Ti−1 and Ti assume the simple GBM like dynamics with constant drift anddiffusion for the par swap rates
dSl(t) = µlSl(t) dt + σlSl(t)dWl(t− Ti−1), Ti−1 ≤ t ≤ Ti (7.2)
We want the Brownian motions dW1 and dW2 to be correlated by
d〈W1,W2〉 = ρ1,2dt, (7.3)
60
where ρ1,2 is the correlation between the two par swap rates. We also know that
lnSl(Ti)
Sl(Ti−1)= (µl − σ2
l /2)(Ti − Ti−1) + σlWl(Ti − Ti−1)
∼ N ((µl − σ2
l /2)(Ti − Ti−1) , σ2l (Ti − Ti−1)
)
By construction with the above assumptions ln Sl(Ti)Sl(Ti−1)
is normally distributed and we
can estimate the unknown drift and diffusion coefficients from the maximum likelihoodestimators for normal variables. We know that they are given by the sample meanand variance, that is
(µl − σ2l /2)(Ti − Ti−1) =
1M
M∑
j=1
lnSj
l (Ti)
Sjl (Ti−1)
(7.4)
σ2l (Ti − Ti−1) =
1
M
M∑
j=1
ln
(Sj
l (Ti)
Sjl (Ti−1)
)2
− (µl − σ2l /2)2(Ti − Ti−1)2
(7.5)
Replace µl and σl in (7.4) and (7.5) by µl and σl and solve them. One finds out themto be
σ2l =
1Ti − Ti−1
1
M
M∑
j=1
ln
(Sj
l (Ti)
Sjl (Ti−1)
)2
− 1
M
M∑
j=1
lnSj
l (Ti)
Sjl (Ti−1)
2 (7.6)
and
µl =1
Ti − Ti−1
1M
M∑
j=1
lnSj
l (Ti)
Sjl (Ti−1)
+ σ2l /2 (7.7)
The correlation ρ1,2 can be estimated using the sample correlation ρ1,2 from the real-izations of Sj
1(Ti) and Sj2(Ti). The maxium likelihood estimates for σl and µl in (7.2)
are given by (7.6) and (7.7) and ρ1,2 in (7.3) by ρ1,2. In the j:th realization it is possibleto solve the corresponding value of the Brownian motion that implies the correct valuefor the dynamics at both ends.
W jl (Ti − Ti−1) =
1σl
(ln
Sjl (Ti)
Sjl (Ti−1)
− (µl − σ2l /2)(Ti − Ti−1)
).
For each realization we have the correct value for the par swap rates at both endsand the value of the Brownian motion at both ends, namely Sj
l (Ti−1), Sjl (Ti) and
W jl (Ti − Ti−1). In each realization define two Brownian bridges by
BBjl (t− Ti−1) =
Bl(t− Ti−1)− t− Ti−1
Ti − Ti−1
(Bl(Ti − Ti−1)−W j
l (Ti − Ti−1))
, for Ti−1 ≤ t ≤ Ti,
where Bl’s are Brownian motions with instantaneous correlation d〈B1, B2〉 = ρi,2 dt.By construction in each realization BBj
l (0) = 0 and BBjl (Ti−Ti−1) = W j
l (Ti−Ti−1). Ifwe denote Tα = s1, s2, . . . , sn = Tβ the business days in [Tα, Tβ] and replace Wl(t−Ti−1)in (7.2) by BBj
l (t− Ti−1) it is possible to simulate in each realization the in betweenswap rates by
Sjl (sk) = Sj
l (Ti−1) exp((µl − σ2
l /2)(si − Ti− 1) + σlBB(sk)),
61
Table 7.6: Comparison of CMS range accrual prices with CMS spread option prices
(a) CMS Range accrual prices with 5-factor LiborMarket model
Maturity Expiry 10y-2y 10y-5y 10y-8yT1 T3 2120 1369 556T3 T5 2623 1724 596T5 T7 3135 1895 602T7 T9 3343 1860 577T9 T11 3409 1591 450T13 T15 2360 1057 280
(b) CMS spread options with strike 0 prices with5-factor Libor Market model
Maturity Expiry 10y-2y 10y-5y 10y-8yT1 T3 1440 1029 478T3 T5 2393 1610 601T5 T7 2883 1845 602T7 T9 3355 1930 613T9 T11 3446 1771 535T13 T15 1306 523 152
for k = 1, . . . n. For longer contracts where expiry and maturity are not adjacent timepoints in the set of expiry maturity pairs, the method is applied separately for eachadjacent time points.
Denote time t price of a CMS range accrual note in the same time span asCMSRA(t; Tα, Tβ, R). Assume for simplicity the contract notional for unity. Asusual χA is the indicator variable of a set A. The time Tβ value of a contract is givenby
CMSRA(Tβ; Tα, Tβ, R) =n−1∑i=1
χS1(si)−S2(si)−R≥0 (si+1 − si) .
In terminal measure, the contract time t value is given using the usual martingalearguments
CMSRA(0;Tα, Tβ, R)
= P (0; TN )EQTN
n−1∑
i=1
χS1(si)−S2(si)−R≥0 (si+1 − si)N∏
k=β+1
(1 + τkFk(Tβ))
∣∣∣∣∣∣F0
.
We priced 1 year tenor CMS range accruals with the 5-factor model. Table 7.6 liststhe prices and they are at the same time compared to a CMS spread option priceswhere the spread is checked only once at the start of the period. Looking at subtable7.6(a) and comparing it to subtable 7.6(b), it is evident that the model predicts thecurrent spread to widen, as the range accruals become more valuable. It makes highlysense that if the spread is checked only once at the start of the period and the trendis that the spread widens then the option must be cheaper compared to the one wherethe spread is checked continuously.
62
Chapter 8
Conclusion
We have tested the calibration method, where the instantaneous correlation wereparametrized by semi-parametric form, with relatively low amount of parameters. Thecalibration implied intuitively realistic correlations between forward rates that werediscussed in Chapter 5. At the same time using a semi-parametric approach fitting theinstantaneous correlations, it was found that error in swaption volatilities stayed rela-tively low. If compared to the classical angles form parametrization with test resultsin [3], our method seems to have a clear advantage. The amount of model factorswere found to imply slightly different term structure of volatilities with volatility termstructure being more time homogenous when more factors were used.
Pricing of CMS spread options was found out to be highly dependent on the numberof model factors used. This was because low rank models implied relatively more flatcorrelations and could not explain the full rank correlation matrix accurately enough.It was observed that reducing the number of model factors between 3 to 5, the modelgave relatively the same magnitude prices compared to the prices given by the full-factor model. For this behavior we gave an intuitive interpretation for the independentrandom factors. These were that model factors corresponded for parallel shifts, shiftsin slope and movements in curvature with higher order frequencies explaining thehigher derivatives. The explanative power with the first factor was biggest, whileadditional factors were correcting the curve movements by changing the slope andcurvature. It was found that when pricing very long expiring options with big timestep procedure, higher order factors were necessary being taken into account. With5 explaining factors even a long simulation step of 10 years was accurate comparedto the full-factor case. It was found that the proposed method to price CMS rangeaccruals gave consistent prices to the ordinary spread options.
63
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[22] John Schoenmakers. Calibration of LIBOR models to caps and swaptions: a wayaround intrinsic instabilities via parsimonious structures and a collateral marketcriterion. Weierstraβ-Institut fur Angewandte und Stochastik. WIAS Preprint740:Work presented at Risk Europe 2002, Paris 23 & 24 April 2002.URL: http://www.wias-berlin.de/people/schoenma/
[23] Lixin Wu, Fan Zhang. Libor Market Model with Stochastic Volatility. Journal ofIndustrial and Management Optimization Vol 2, No. 2, 2006. pp. 199-227
65
Appendix A
Derivation of Swaption VolatilityCoefficients
Rigorous proof is now given for the version of Hull and White formula that is used inswaption volatility approximation in Chapter 5.
Proof. The goal is to compute d ln Sα,β(t) in terms of the underlying forwards: Duringthe calculation we do not mention the drift terms explicitly (even though they appearin d ln Sα,β(t) but drop out out when forming the quadratic variation process). ByIto’s lemma
d ln Sα,β(t) =1
Sα,β
β∑
i=α+1
wi(t)dFi(t) + Fi(t)dwi(t) + dwi(t)dFi(t) + dt terms
=1
Sα,β
β∑
i=α+1
wi(t)dFi(t) +
β∑
j=α+1
Fi(t)∂wi
∂Fj(t)dFj(t)
+ dt terms
=1
Sα,β
β∑
j=α+1
(β∑
i=α+1
wj(t)δi,j + Fi(t)∂wi
∂Fj
)dFj(t) + dt terms,
where on last line the order of summation is swapped. Define now the coefficient ζi(t)by
ζi(t) = wj(t) +
β∑i=α+1
Fi(t)∂wi
∂Fj
(t) = wj(t) + yj(t),
where it has been separated for Rebonato’s part wj(t), and the part, which comes fromthe Hull White correction yj(t) defined as
yj(t) =
β∑i=α+1
Fi(t)∂wi
∂Fj
(t)
Thus the Ito differential of ln Sα,β(t) can be computed in compact form
d ln Sα,β(t) =1
Sα,β
β∑i=α+1
ζj(t)dFj(t) + dt terms
66
Using that weights wi can be calculated from forwards Fj, j = α + 1, . . . , β by
wi(t) =τi
∏βk=i+1(1 + τkFk)
∑β−α2
l=1 τ ′2l+α
∏βk=2l+α+1(1 + τkFk)
, (A.1)
it is possible to work out the tedious partial differentiation ∂wi
∂Fjcarefully but straight-
forwardly
∂wi
∂Fj=
τj
1 + τjFj
χj≥i+1τiτj
1+τjFj
∏βk=i+1(1 + τkFk)
(∑β−α2
l=1 τl+α∏β
k=2l+α+1(1 + τkFk))
(∑β−α2
l=1 τl+α∏β
k=2l+α+1(1 + τkFk))2
− τj
1 + τjFj
∑β−α2
l=1
(τ ′l+αχj≥2l+α+1
∏βk=2l+α+1(i + τkFk)
)(τi
∏βk=i+1(1 + τkFk)
)
(∑β−α2
l=1 τl+α∏β
k=2l+α+1(1 + τkFk))2
Now multiplying ∂wi
∂Fjby Fi, writing the products in terms of zero coupon fractions and
summing over α + 1 to β and at the same time taking into account that sum takeseffect only when the indicators variables χj≥i+1 and χj≥2l+α+11 are effective, thatis when i ≤ j − 1 and l ≤ ⌊
j−α−12
⌋. It thus implies that
yi(t) =τj
1 + τjFj
(∑j−1i=α+1 τiFiPi
)(∑β−α2
l=b j−α−12 c+1
τ ′l+αP2l+α
)
(∑β−α2
l=1 τ ′l+αP2l+α
)2
− τj
1 + τjFj
∑βi=j τiFiPi
∑b j−α−12 c
l=1 P2l+ατl+α(∑β−α2
l=1 τ ′l+αP2l+α
)2
(A.2)
Now work out the quadratic variation for ln Sα,β(t), thus giving
d 〈ln Sα,β(t), ln Sα,β(t)〉 =1
Sα,β(t)2
β∑i=α+1
β∑j=α+1
ζi(t)ζj(t)Fi(t)Fj(t)σi(t)σj(t)ρi,j(t)dt
To get an approximation for the swaption volatility, we need to freeze the stochasticquantities, namely Fi’s and the coefficients ζi’s
σ2α,β(t)dt = d 〈ln Sα,β(t), ln Sα,β(t)〉 ≈ 1
Sα,β(0)2
β∑
i=α+1
β∑
j=α+1
ζi(0)ζj(0)Fi(0)Fj(0)σi(t)σj(t)ρi,j(t)dt
(A.3)
The coefficients are computed from (A.1) and (A.2)
67
Appendix B
Used Model Input Data
(c) yield curve for first 10 years
Index Dates Ti P (0;Ti)0 23.1.2009 0.5041 0.97431 23.7.2009 1 0.95022 25.1.2010 1.5096 0.92463 23.7.2010 2 0.90304 24.1.2011 2.5068 0.88075 25.7.2011 3.0055 0.85926 23.1.2012 3.5041 0.83867 23.7.2012 4.0027 0.81868 23.1.2013 4.5068 0.79919 23.7.2013 5.0027 0.780610 23.1.2014 5.5068 0.762311 23.7.2014 6.0027 0.744612 23.1.2015 6.5068 0.727013 23.7.2015 7.0027 0.710114 25.1.2016 7.5123 0.693015 25.7.2016 8.0110 0.676516 23.1.2017 8.5096 0.660217 24.7.2017 9.0082 0.644118 23.1.2018 9.5096 0.628219 23.7.2018 10.0055 0.6128
(d) yield curve for 10.5 to 20.5 years
Index Dates Ti P (0;Ti)20 23.1.2019 10.5096 0.597321 23.7.2019 11.0055 0.582522 23.1.2020 11.5096 0.567823 23.7.2020 12.0082 0.553724 25.1.2021 12.5178 0.539625 23.7.2021 13.0082 0.526226 24.1.2022 13.5151 0.512727 25.7.2022 14.0137 0.499828 23.1.2023 14.5123 0.487329 24.7.2023 15.0110 0.475330 23.1.2024 15.5123 0.463831 23.7.2024 16.0110 0.445332 23.1.2025 16.5151 0.441433 23.7.2025 17.0110 0.430834 23.1.2026 17.5151 0.420335 23.7.2026 18.0110 0.410236 25.1.2027 18.5205 0.400137 23.7.2027 19.0110 0.390738 24.1.2028 19.5178 0.381339 24.7.2028 20.0164 0.372340 23.1.2029 20.0164 0.3636
Table B.1: Expiry-Maturity pairs and corresponding Zero Coupons
68
Index σBlacki+1
0 15.49%1 18.60%2 22.82%3 20.49%4 21.00%5 18.22%6 17.46%7 17.00%8 16.30%9 15.89%10 15.19%11 15.26%12 14.70%13 14.38%14 13.77%15 13.43%16 12.95%17 12.60%18 11.98%19 12.71%
Index σBlacki+1
20 12.36%21 12.00%22 11.65%23 12.09%24 11.75%25 11.44%26 11.15%27 10.91%28 10.79%29 11.80%30 11.60%31 11.40%32 11.20%33 11.01%34 10.82%35 10.62%36 10.43%37 10.25%38 10.06%39 13.63%
Table B.2: 6M-tenor Caplet volatilities
Exp/Mat 1y 2y 3y 4y 5y 6y 7y 8y 9y 10y1y 21.1% 20.4% 19.4% 18.3% 17.2% 16.5% 15.9% 15.4% 15.1% 14.8%2y 19.2% 17.9% 17.0% 16.3% 15.7% 15.1% 14.8% 14.5% 14.2% 14.1%3y 17.5% 16.5% 15.9% 15.4% 14.9% 14.4% 14.1% 13.8% 13.5% 13.3%4y 16.3% 15.8% 15.1% 14.6% 14.1% 13.7% 13.5% 13.2% 13.0% 12.8%5y 15.3% 14.9% 14.3% 13.8% 13.3% 13.0% 12.8% 12.6% 12.5% 12.4%7y 13.7% 13.4% 12.9% 12.6% 12.3% 12.1% 11.9% 11.8% 11.7% 11.7%10y 11.9% 11.8% 11.6% 11.5% 11.4% 11.3% 11.2% 11.2% 11.2% 11.2%
Table B.3: ATM swaption volatilities
69