lecture 5: review of interest rate models - purdue university · 2014-02-06 · lecture 5: review...

Lecture 5: Review of interest rate models

Xiaoguang Wang

STAT 598W

January 30th, 2014

(STAT 598W) Lecture 5 1 / 46

Outline

1 Bonds and Interest Rates

2 Short Rate Models

3 Forward Rate Models

4 LIBOR and Swaps


Bonds and interest rates

Definition

A zero coupon bond with maturity date T , also called a T -bond, is acontract which guarantees the holder 1 dollar to be paid on the date T .The price at time t of a bond with maturity date T is denoted by p(t,T ).

We assume the following:

There exists a (frictionless) market for T -bonds for every T > 0.

The relation p(t, t) = 1 holds for all t.

For each fixed t, the bond price p(t,T ) is differentiable w.r.t time ofmaturity T .


Interest Rates

At time t, we can make a contract guaranteeing a riskless rate of interestover the future interval [S ,T ]. Such an interest rate is called a forwardrate.

Definition

The simple forward rate (or LIBOR rate) L, is the solution to the equation

1 + (T − S)L =p(t,S)

p(t,T )

whereas the continuously compounded forward rate R is the solution tothe equation

eR(T−S) =p(t,S)

p(t,T )


Interest Rates

Definition

The simple forward rate for [S ,T ] contracted at t, henceforth referredto as the LIBOR forward rate, is defined as

L(t; S ,T ) = −p(t,T )− p(t,S)

(T − S)p(t,T )

The simple spot rate for [S ,T ], or the LIBOR spot rate, is defined as

L(S ,T ) = − p(S ,T )− 1

(T − S)p(S ,T )


Interest Rates

The continuously compounded forward rate for [S ,T ] contracted at tis defined as

R(t; S ,T ) = − log p(t,T )− log p(t, S)

T − S

The continuously compounded spot rate, R(S ,T ) is defined as

R(S ,T ) = − log p(S ,T )

T − S

The instantaneous forward rate with maturity T , contracted at t, isdefined by

f (t,T ) = −∂ log p(t,T )

∂T

The instantaneous short rate at time t is defined by

r(t) = f (t, t)


Some useful facts

The money account is defined by

Bt = exp∫ t0 r(s)ds

For t ≤ s ≤ T we have

p(t,T ) = p(t, s)× exp

{−∫ T

sf (t, u)du

}and in particular

p(t,T ) = exp

{−∫ T

tf (t, s)ds

}


Relations between short rates, forward rates and zerocoupon bonds

Assume we havedr(t) = a(t)dt + b(t)dW (t)

dp(t,T ) = p(t,T )m(t,T )dt + p(t,T )v(t,T )dW (t)

df (t,T ) = α(t,T )dt + σ(t,T )dW (t)

Then we must have{α(t,T ) = vT (t,T )v(t,T )−mT (t,T )

σ(t,T ) = −vT (t,T )

and {a(t) = fT (t, t) + α(t, t)

b(t) = σ(t, t)


Continue

And we also should have

dp(t,T ) = p(t,T )

{r(t) + A(t,T ) +

1

2‖S(t,T )‖2

}dt

+p(t,T )S(t,T )dW (t)

where {A(t,T ) = −

∫ Tt α(t, s)ds

S(t,T ) = −∫ Tt σ(t, s)ds


Fixed Coupon Bonds

Fix a number of dates, i.e. points in time, T0, · · · ,Tn. Here T0 isinterpreted as the emission date of the bond, whereas T1, · · · ,Tn arecoupon dates.

At time Ti , i = 1, · · · , n, the owner of the bond receives thedeterministic coupon ci .

At time Tn the owner receives the face value K .

Then the price of the fixed coupon bond at time t < T1 is given by

p(t) = K × p(t,Tn) +n∑

i=1

ci × p(t,Ti )

And the return of the ith coupon is defined as ri :

ci = ri (Ti − Ti−1)K


Floating rate bonds

If we replace the coupon rate ri with the spot LIBOR rate L(Ti−1,Ti ):

ci = (Ti − Ti−1)L(Ti−1,Ti )K

If we set Ti − Ti−1 = δ and K = 1, then the value of the ith coupon attime Ti should be

ci = δ1− p(Ti−1,Ti )

δp(Ti−1,Ti )=

1

p(Ti−1,Ti )− 1

which further discounted to time t < T0 should be

p(t,Ti−1)− p(t,Ti )

Summing up all the terms we finally obtain the price of the floatingcoupon bond at time t

p(t) = p(t,Tn) +n∑

i=1

[p(t,Ti−1)− p(t,Ti )] = p(t,T0)

This also means that the entire floating rate bond can be replicatedthrough a self-financing portfolio. (Exercise for you)


Interest Rate Swap

An interest rate swap is a basically a scheme where you exchange apayment stream at a fixed rate of interest, known as the swap rate, for apayment stream at a floating rate (typically a LIBOR rate).Denote the principal by K , and the swap rate by R. By assumption wehave a number of equally spaced dates T0, · · · ,Tn, and payment occurs atthe dates T1, · · · ,Tn (not at T0). If you swap a fixed rate for a floatingrate (in this case the LIBOR spot rate), then at time Ti , you will receivethe amount

KδL(Ti−1,Ti )

and pay the amountKδR

where δ = Ti − Ti−1.


Pricing of IRS

Theorem

The price, for t < T0, of the swap above is given by

Π(t) = Kp(t,T0)− Kn∑

i=1

dip(t,Ti )

, wheredi = Rδ, i = 1, · · · , n − 1,

dn = 1 + Rδ

If, by convection, we assume that the contract is written at t = 0, and thecontract value is zero at the time made, then

R =p(0,T0)− p(0,Tn)

δ∑n

i=1 p(0,Ti )


Yield

In most cases, the yield of an interest rate product is the ”internal rate ofinterest” for this product. For example, the continuously compounded zerocoupon yield y(t,T ) should solve

p(t,T ) = e−y(T−t) × 1

which is given by

y(t,T ) = − log p(t,T )

T − t

For a fixed t, the function T → y(t,T ) is called the (zero coupon) yieldcurve.


Yield to maturity for coupon bonds

The yield to maturity, y(t,T ), of a fixed coupon bond at time t, withmarket price p, and payments ci , at time Ti for i = 1, · · · , n is defined asthe value of y which solves the equation

p(t) =n∑

i=1

cie−y(Ti−t)

For the fixed coupon bond above, with price p at t = 0, and yield tomaturity y , the duration, D is defined as

D =

∑ni=1 Ticie

−yTi

p

which can be interpreted as the ”weighted average of the coupon dates”.With the notations above we have

dp

dy= −D × p


Yield Curve

Definition

The zero-coupon curve (sometimes also referred to as ”yield curve”) attime t is the graph of the function

T 7→

{L(t,T ), t ≤ T ≤ t + 1 (years)

Y (t,T ), T > t + 1 (years)

where the L(t,T ) is the spot LIBOR rate and Y (t,T ) is the annuallycompounded spot interest rate.

Such a zero-coupon curve is also called the term structure of interestrates at time t. Under different economic environments, the shape ofzero-coupon curve can be very different, such as the ”normal curve”, ”flatcurve”, ”inverted curve”, ”steep curve” and so on.


Zero-bond curve

Definition

The zero-bond curve at time t is the graph of the function

T 7→ P(t,T ),T > t

which, because of the positivity interest rates, is a T -decreasing functionstarting from P(t, t) = 1. Such a curve is also referred to as termstructure of discount factors.


Outline


2 Short Rate Models


4 LIBOR and Swaps


Model set up

In this section we turn to the problem of how to model an arbitrage freefamily of zero coupon bond price process {p(·,T ); T ≥ 0}. To model that,we first assume that short rate under the objective probability measure P,satisfies the SDE

dr(t) = µ(t, r(t))dt + σ(t, r(t))dW̄ (t)

And the only exogenously given asset is the money account, with priceprocess B defined by the dynamics

dB(t) = r(t)B(t)dt

We further assume that there exists a market for zero coupon T -bond forevery value of T .


Incomplete Market

Question: Are bond prices uniquely determined by the P-dynamics of theshort rate r?Answer: No! The market is incomplete. The short rate, as an”underlying”, is not tradable.Pricing ideas:

Prices of bonds with different maturities will have to satisfy certaininternal consistency relations in order to avoid arbitragepossibilities on the bond market.

If we take the price of one particular ”benchmark” bond as giventhen the prices of all other bonds (with maturity prior to thebenchmark) will be uniquely determined in terms of the price of thebenchmark bond (and the r -dynamics).


Term Structure Equation: Assumptions

Assumption: We assume that there is a market for T -bonds for everychoice of T and that the market is arbitrage free. We assume furthermorethat, for every T , the price of a T -bond has the form

p(t,T ) = F (t, r(t); T )

where F is a smooth function of three real variables. Sometimes we writeFT (t, r) instead of F (t, r(t); T ). Apply Ito formula, then

dFT = FTαTdt + FTσTdW̄

where

αT =FTt + µFT

r + 12σ

2FTrr

FT

σT =σFT

r

FT


Term Structure Equation: Market price of Risk

Theorem

Assume that the bond market is free of arbitrage. Then there exists aprocess λ such that the relation

αT (t)− r(t)

σT (t)= λ(t)

holds for all t and for every choice of maturity time T .


Term Structure Equation

Theorem

In an arbitrage free bond market, FT will satisfy the term structureequation {

FTt + (µ− λσ)FT

r + 12σ

2FTrr − rFT = 0,

FT (T , r) = 1

For a general contingent claim X = Φ(r(T )). The price F (t, r(t)) willthen satisfy {

Ft + (µ− λσ)Fr + 12σ

2Frr − rF = 0,

FT (T , r) = Φ(r)


Risk neutral valuation

Bond prices are given by the formula p(t,T ) = F (t, r(t); T ) where

F (t, r ; T ) = EQt,r

[e−

∫ Tt r(s)ds

]Here the martingale measure Q and the subscripts t, r denote that theexpectation shall be taken given the following dynamics for the short rate

dr(s) = (µ− λσ)ds + σdW (s),

r(t) = r

For a general contingent claim, the valuation becomes

F (t, r ; T ) = EQt,r

[e−

∫ Tt r(s)ds × Φ(r(T ))

]The term structure, as well as the prices of all other interest ratederivatives, are completely determined by specifying the r-dynamics underthe martingale measure Q.


Popular r-dynamics

Vasicekdr = (b − ar)dt + σdW , (a > 0)

Cox-Intersoll-Ross (CIR)

dr = a(b − r)dt + σ√

rdW

Black-Derman-Toy

dr = Θ(t)rdt + σ(t)rdW

Ho-Leedr = Θ(t)dt + σdW

Hull-White

dr = (Θ(t)− a(t)r)dt + σ(t)dW , a(t) > 0

Hull-White (extended CIR)

dr = (Θ(t)− a(t)r)dt + σ(t)√

rdW (a(t) > 0)


Invert the yield curve

It is of key importance to pin down the r-dynamics under the martingalemeasure Q since all the derivatives pricing problems in a short rate modelframework depend on that. In order to accomplish that, we can use theso-called inverting yield curve approach:

Choose a particular model involving one or more parameters anddenote the entire parameter vector as α.

Solve, for every conceivable time of maturity T , the term structureequation for the T-bonds. Thus we have the theoretic term structureas

p(t,T ;α) = FT (t, r ;α)

Collect price date from the bond market. Denote the empirical termstructure by {p∗(0,T ); T ≥ 0}.Now choose the parameter vector α in such a way that the theoreticalcurve fits the empirical curve as well as possible. This gives us theestimated parameter α∗.


Invert yield curve: Continued

Insert α∗ into µ and σ. Now we have pinned down exactly whichmartingale measure we are working with. Let us denote the result byµ∗ and σ∗ respectively.

We now can compute an interest rate derivative with final payoffX = Γ(r(T )). The price process Π(t; Γ) = G (t, r(t)) solves{

Gt + µ∗Gr + 12 [σ∗]2Grr − rG = 0

G (T , r) = Γ(r)

It is of great importance that the PDEs involved are easy to solve. And itturns out that some of the models above are much easier to deal withanalytically than the others, and this leads us to the subject of so calledaffine term structures.


Affine Term Structure

If the term structure {p(t,T ); 0 ≤ t ≤ T ,T > 0} has the form

p(t,T ) = F (t, r(t); T ),

where F has the form

f (t, r ; T ) = eA(t,T )−B(t,T )r

and where A and B are deterministic functions, then the model is said topossess an affine term structure (ATS).It turns out that the ATS is general enough to include lots of popularmodels. If the drift µ(t, r) and diffusion part σ(t, r) have the form{

µ(t, r) = α(t)r + β(t)

σ(t, r) =√γ(t)r + δ(t)

then the model admits an affine term structure.


Summary of short rate models

The main advantages with such models are as follows:

Specifying r as the solution of an SDE allows us to use Markovprocess theory, so we may work within a PDF framework.

In particular it is often possible to obtain analytical formulas for bondprices and derivatives

The main draw backs of short rate models:

From an economic point of view it seems unreasonable to assume thatthe entire money market is governed by only one explanatory variable.

It is hard to obtain a realistic volatility structure for the forward rateswithout introducing a very complicated short rate model.

As the short rate model becomes more realistic, the inversion of theyield curve described above becomes increasingly more difficult.


Outline


2 Short Rate Models


4 LIBOR and Swaps


Forward rate model

An obvious extending idea would, for example, be to present a prior modelfor the sort rate as well as for some long rate, and one could of coursemodel one or several intermediary interest rates. The method proposed byHeath-Jarrow-Morton is at the far end of this spectrum: they choose theentire forward rate curve as their (infinite dimensional) state variable.We assume that, for every fixed T > 0, the forward rate has a stochasticdifferential which under the objective measure P is given by

df (t,T ) = α(t,T )dt + σ(t,T )dW̄ (t)

f (0,T ) = f ∗(0,T )

where W̄ is a d-dimensional P-Wiener process whereas the α and σ areadapted process.


HJM drift condition

Theorem

Assume that the family of forward rates is given as above and that theinduced bond market is arbitrage free. Then there exists a d-dimensionalcolumn-vector process

λ(t) = [λ1(t), · · · , λd(t)]′

with the property that for all T ≥ 0 and for all t ≤ T , we have

α(t,T ) = σ(t,T )

∫ T

tσ(t, s)′ds − σ(t,T )λ(t)


HJM drift condition under martingale measure

Theorem

Under the martingale measure Q, the process α and σ must satisfy thefollowing relation, for every t and every T ≥ t.

α(t,T ) = σ(t,T )

∫ T

tσ(t, s)′ds

Thus we know that when we specify the forward rate dynamics (under Q)we may freely specify the volatility structure. The drift parameters arethen uniquely determined.


A simple Example

Now we illustrate a simple example. Set σ(t,T ) = σ. Then

α(t,T ) = σ

∫ T

tσds = σ2(T − t)

Then we have

f (t,T ) = f ∗(0,T ) +

∫ t

0σ2(T − s)ds +

∫ t

0σdW (s),

i.e.

f (t,T ) = f ∗(0,T ) + σ2t(

T − t

2

)+ σW (t)

In particular we see

r(t) = f (t, t) = f ∗(0, t) + σ2t2

2+ σW (t)

so the short rate dynamics is

dr(t) = (fT (0, t) + σ2t)dt + σdW (t)

which is exactly the Ho-Lee model, fitted to the initial term structure.(STAT 598W) Lecture 5 34 / 46

summary

The interest rate models based on infinitesimal interest rates like theinstantaneous short rate and the instantaneous forward rates are nice tohandle from a mathematical point of view, but they have two maindisadvantages:

The instantaneous short and forward rates can never be observed inreal life.

If you would like to calibrate your model to cap or swaption data,then this is typically very complicated from a numerical point of viewif you use one of the ”instantaneous” models.

For a very long time, the market practice has been to value caps,floors, and swaptions by using a formal extension of the Black(1976)model. Such an extension typically on one hand assumes that theshort rate at one point to be deterministic, while on the other handthe LIBOR rate is assumed to be stochastic, which is of courselogically inconsistent.


Outline


2 Short Rate Models


4 LIBOR and Swaps


Logically consistent models for LIBOR rates, caps, andswaptions

Instead of modeling the instantaneous interest rates, we modeldiscrete market rates like LIBOR rates in the LIBOR market models,or forward swap rates in the swap market models.

Under a suitable choice of numeraire, these market rates can in factbe modeled log normally.

The market models will thus produce pricing formulas for caps andfloors (LIBOR models), and swaptions (the swap market models)which are of the Black-76 type and this confirming with marketpractice.

By construction the market models are thus very easy to calibrate tomarket data for caps/floors and swaptions respectively.


Caplets and Caps

Consider a fixed set of increasing maturities T0,T1, · · · ,TN and we defineαi , by

αi = Ti − Ti−1, i = 1, · · · ,N

The number αi is known as the tenor, which often is a quarter of a year inpractice. Let pi (t) denote the zero coupon bond price p(t,Ti ) and letLi (t) denote the LIBOR forward rate contracted at t, for period [Ti−1,Ti ].A cap or cap rate R and resettlement dates T0, · · · ,TN is a contractwhich at time Ti givens the holder of the cap amount

Xi = αi ·max[Li (Ti−1)− R, 0]

for each i = 1, · · · ,N. The cap is thus a portfolio of the individual capletsX1, · · · ,XN .


Black-76 formula for Caplets

Theorem

The Black-76 formula for the caplet Xi = αi ·max[L(Ti−1,Ti )− R, 0] isgiven by the expression

CaplBi (t) = αi · pi (t){Li (t)N[d1]− RN[d2]}, i = 1, · · · ,N

where

d1 =1

σi√

Ti − t

[ln

(Li (t)

R

)+

1

2σ2i (T − t)

],

d2 = d1 − σi√

Ti − t

The constant σi is known as the Black Volatility for caplet No. i.Sometimes we also write CaplBi (t;σi ).


Implied Black Volatilities

In the market, cap prices are not quoted in monetary terms but instead interms of implied Black volatilities. And there are two types of impliedBlack volatilities, the flat volatilities and the spot volatilities (alsoknown as forward volatilities).First of all, it is easy to see that

Capli (t) = Capi (t)− Capi−1(t), i = 1, · · · ,NThen the implied flat volatilities σ̄1, · · · , σ̄N are defined as the solutions ofthe equations

Capmi (t) =

i∑k=1

CaplBk (t; σ̄i ), i = 1, · · · ,N

The implied spot volatilities σ̄1, · · · , σ̄N are defined as solutions of theequations

Caplmi (t) = CaplBi (t; σ̄i ), i = 1, · · · ,NA sequence of implied volatilities σ̄1, · · · , σ̄N (flat or spot) is called avolatility term structure.


LIBOR Market Model: martingale measure

The standard risk neutral valuation for Capli (t) should be

Capli (t) = αiEQ[e−

∫ Ti0 r(s)ds max[Li (Ti−1)− R, 0]|Ft

]But it is much more natural to use the Ti forward martingale measure toobtain

Capli (t) = αipi (t)ETi [max[Li (Ti−1)− R, 0]|Ft ]

Furthermore, we have for every i = 1, · · · ,N, the LIBOR process Li is amartingale under the corresponding forward measure QTi , on the interval[0,Ti−1].


LIBOR Market Model

Set up:

A set of resettlement dates T0, · · · ,TN .

An arbitrage free market bond with maturities T0, · · · ,TN .

A k-dimensional QN -Wiener process W N .

For each i a deterministic function of time σi (t).

An initial nonnegative forward rate term structure L1(0), · · · , LN(0).

For each i, we define W i as the k-dimensional Q i -Wiener processgenerated by W N under the Girsanov transformation QN → Q i .

If the LIBOR forward rates have the dynamics

dLi (t) = Li (t)σi (t)dW i (t), i = 1, · · · ,N

where W i is Q i -Wiener as described above, then we say we have adiscrete tenor LIBOR market model with volatilities σ1, · · · , σN .


Terminal Measure Dynamics

Theorem

Consider a given volatility structure σ1, · · · , σN , where each σi is assumedto be bounded, a probability measure QN and a standard QN -Wienerprocess W N . Define L1, · · · , LN by

dLi (t) = −Li (t)

(N∑

k=i+1

αkLk(t)

1 + αkLk(t)σk(t)σ∗k(t)

)dt + Li (t)σi (t)dW N(t),

for each i where we use the convention∑N

N(· · · ) = 0. Then theQ i -dynamics of Li are given as above in the LIBOR market model. Thusthere exists a LIBOR model with the given volatility structure.


Pricing Caps in the LIBOR model

Theorem

In the LIBOR market model, the caplet prices are given by

Capli (t) = αi · pi (t){Li (t)N[d1]− RN[d2]}, i = 1, · · · ,N

where

d1 =1

Σi (t,Ti−1)

[ln

(Li (t)

R

)+

1

2Σ2i (t,Ti−1)

],

d2 = d1 − Σi (t,Ti−1)

with Σi defined below

Σ2i (t,T ) =

∫ T

t‖σi (s)‖2ds

We see that each caplet price is given by a Black type formula.


Calibration and Simulation

Suppose we want to price some exotic (not a cap or a floor) interest ratederivative, performing this with a LIBOR model means that we typicallycarry out the following to steps:

Use implied Black volatilities in order to calibrate the modelparameters to market data.

use Monte Carlo (or some other numerical methods) to price theexotic instrument.

Assume that we are given an empirical term structure of implied forwardBlack volatilities σ̄1, · · · , σ̄N for all caplets. In order to calibrate the modelwe have to choose the deterministic LIBOR volatilities σ1(·), · · · , σN(·)such that

σ̄i =1

Ti

∫ Ti−1

0‖σi (s)‖2ds, i = 1, · · · ,N

which is obviously a highly undetermined system. So in practice it iscommon to make some structural assumptions about the shape of thevolatility functions.


Shape of volatility functions

Popular specifications on the shape of volatility functions:

σi (t) = σi , 0 ≤ t ≤ Ti−1

σt(t) = σij , Tj−1 ≤ t ≤ Tj , for j = 0, · · · , i .σt(t) = σij = βi−j , Tj−1 ≤ t ≤ Tj , for j = 0, · · · , iσi (t) = qi (Ti−1 − t)eβi (Ti−1−t) where qi (·) is some polynomial and βiis a real number.

After the model has been calibrated, Monte Carlo simulation is thestandard tool for computing the prices of exotics. Since the SDEs in theLIBOR model are generally too complicated to allow analytical solutions,we have to resort to simulation of discretized versions of the equationsusing methods like Euler scheme.


lecture 5: review of interest rate models - purdue university · 2014-02-06 · lecture 5: review...

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