mathematical methods in financial modeling m ethodes math ...mathematical methods in financial...

Mathematical Methods in Financial ModelingMethodes Mathematiques en Finance

Organizers:

Marco AvellanedaCourant Institute of Mathematical Sciences

New York Universitye-mail [email protected]

Rama CONTCentre de Mathematiques Appliquees

CNRS - Ecole Polytechnique.e-mail [email protected]

Arbitrage pricing of equity basket options

Marco AvellanedaNew York University

e-mail [email protected]

With the current strong institutional and public interest in equity products, suchas Exchange Traded Funds (ETFs), equity indices such as the NASDAQ 100 Trust(QQQ) and HLDR Shares from the American Stock Exchange, there has been anincreased attention given to equity volatility trading and the pricing of options onbaskets. An important conceptual theme in this area is: can we price the impliedvolatility skew of an equity basket based on the implied volatility skews of the com-ponent stocks and ’correlation information’ available from market data? This talkpresents a new methodology for doing this, based upon sophisticated calibrationmethods for options on different underlying assets in a large-scale Monte Carlo sim-ulation. Calibration for baskets of up to 35 underlying assets with 25-30 optionsper asset and up to one year can be achieved within reasonable time on commonlyavailable platforms. The talk will present the methodology and some examples ofhow this method is able to capture the volatility structure of the basket and compareit with contemporaneous market quotes on index options.

Implied Volatility Models

P. BALLANDMerrill Lynch International, U.K.


In this talk, we analyze particular types of smile structures defined by assumingparticular dynamics for the implied volatility surface. First, we give a completecharacterization of the sticky-delta and of the sticky-strike implied volatility modelsintroduced recently in the litterature. Finally, we introduce the restricted and un-restricted stochastic implied volatility models based on the no-arbitrage diffusion ofthe smile-surface.

Non-Gaussian Ornstein-Uhlenbeck stochastic volatility modelling

Ole E. Barndorff-NielsenAarhus University, Denmark.


An overview will be given of stochastic volatility models, with particular focus oncases where the volatility process is of Ornstein-Uhlenbeck type or a superpositionof such processes. The talk is based on joint work with Neil Shephard and accountsof parts of this work are available at the web address:

http://www.nuff.ox.ac.uk/Users/SHEPHARD/levy.htm

A Geometric View of Interest Theory

Tomas BjorkDepartment of Finance, Stockholm School of Economics, SWEDEN


The purpose of this talk is to give an overview of some recent work concerningstructural properties of the evolution of the forward rate curve in an arbitrage freebond market. The main problems to be discussed are as follows.

• When is a given forward rate model consistent with a given family of forwardrate curves?

• When can the inherently infinite dimensional forward rate process be realizedby means of a finite dimensional state space model?

We consider interest rate models of Heath-Jarrow-Morton type, where the forwardrates are driven by a multidimensional Wiener process, and where he volatility isallowed to be an arbitrary smooth functional of the present forward rate curve.Within this framework we give necessary and sufficient conditions for consistency,as well as for the existence of a finite dimensional realization, in terms of the forwardrate volatilities. We also provide a general method for the actual construction offinite realizations, and we study the extension to stochastic volatility models.

Model Asymptotics, Calibration and PricingJerome BUSCA

Laboratoire de Mathematiques et de Physique TheoriqueUniversite Francois Rabelais

Parc de Grandmont37200 Tours, FRANCE

Starting from simple extensions of the Black-Scholes framework such as the localvolatility model, a number of models have been proposed in order to account forthe so-called smile effect due to the non-lognormality of prices, as well as othersmarket imperfections. A natural question then arises: how well do models perform,compared with each others, when it comes to calibrating them with market data?What is the right criterion to compare them?

It turns out that, broadly speaking, a key element in the understanding of apricing model are the qualitative properties of its asymptotics, as time to maturitygoes to zero or to infinity, or deep in or out of the money. I will establish suchasymptotics in several cases (local, stochastic volatility, jumps) and show how thesecan be used to try and answer the question above. The material of this talk originatesin a series of recent joint works with H. Berestycki (Paris VI) an d I. Florent (HSBC-CCF).

Stochastic volatility: the hedger’s perspective

Mark H.A. DavisImperial College, London, England


An adequate treatment of stochastic volatility is needed in the finance industryfor three distinct purposes: marking-to-market, computing hedge parameters andestimating Value-at-Risk. We argue that the model requirements in these threecases are very different, and we concentrate on the second. A ‘robustness’ propertyof the Black-Scholes model implies that it is quite possible to hedge successfully usingthe ‘wrong’ model. We must however, deal with volatility smiles in a systematic way.In this talk the general approach taken is analogous to the usual approach to interestrate modelling: we work directly in the risk-neutral measure in a complete marketmodel in which trade options are treated as assets in their own right. ‘Calibration’ isautomatic and the modelling problem comes down to a choice of volatility function,a purely empirical question.

Dynamic modeling of implied volatility surfaces

Rama CONTCentre de Mathematiques AppliqueesCNRS - Ecole Polytechnique, France.

e-mail [email protected] da FONSECA

ISFA, Universite de Lyon I, France

While theoretical extensions of the Black Scholes model have focused on in-finitesimal unobservable quantities such as the stochastic diffusion coefficient andjump intensities, market prices of options are usually described in terms of theirBlack-Scholes implied volatilities. The values of implied volatilities for call and putoptions on a given asset are represented by the implied volatility surface, indexedby strike price and maturity, whose evolution in time captures the evolution of theoptions market.

We describe a market-based modeling framework in which the implied volatilitysurface is directly used as the state variable to describe the stochastic evolution ofmarket prices of options. Based on an empirical study of time series of impliedvolatilities of SP500 and DAX options, we described the implied volatility surfaceas a randomly fluctuating surface driven by a small number of factors. We recoverand interpret the shape of these factors from a Karhunen-Loeve decomposition ofthe random volatility surface and study their correlation with the underlying asset.

The implied volatility surface is then modeled as a stationary random field witha covariance structure matching the empirical observations. This model extends andimproves the well known ”constant smile” model used by practitioners.

We show how our stochastic implied volatility model allows a simple descriptionof the time evolution of a set of options and provides a rationale for Vega hedging ofportfolios of options. This modeling approach also allows us to construct a MonteCarlo framework for simulating scenarios for the joint behavior of a portfolio ofcall or put options, leading to a considerable gain in computation time for scenariogeneration and VaR calculations for portfolios of options.

Presented by Rama Cont.

Superreplications in the presence of modeluncertainty: the continuous-time case

Laurent DENISUniversite du Maine, FRANCEe-mail [email protected]

Claude MARTINII.N.R.I.A. (Rocquencourt), FRANCE


The purpose of this work is to set a framework suitable for dealing with model un-certainty in mathematical finance, more precisely to handle the pricing of contingentclaims in this context. One approach is to try to set and solve the superreplicationproblem in the presence of model uncertainty. This has been undertaken by M.Avellaneda, A. Levy, A. Paras and T.J. Lyons by stochastic control techniques,in the case of standard European options with smooth payoffs, by specifying anuncertainty on the volatility (Uncertain Volatility (UVM) model). Nevertheless, ageneral framework was missing. A major difficulty is that one is faced with a familyof measures which are in general mutually singular-even worse, this family is typi-cally non-dominated in the statistical sense.The purpose of this paper is to provide a coherent framework in which one can setand solve the superreplication problem, at least for European contingent claims (in-cluding path-dependent ones), which encompasses the case of the UVM model.This framework is that of the space associated to a regular capacity defined by aset of martingale measures on the canonical space-the canonical process stands forthe underlying of the contingent claim at hand. We need an uniform upper boundfor the brackets of the canonical process under the probabilities of the family. Thiscondition seems to be pertaining from a practical point of view. In particular it isfull-filled in the UVM case.

Presented by L. DENIS

A ”Rating Diffusion” Based Model for Credit Derivatives

Raphael DouadyEcole Normale Superieure de Cachan, France


Monique JeanblancUniversite d’Evry, France


We present a model in which a bond B subject to possible default is assesseda ”continuous” rating RB(t) ∈ [0, 1) that follows a diffusion process, possibly withjumps. Default occurs when the rating reaches 0. Non-defaultable bonds have rating1 (unreachable). The value of the bond is the sum of that of its payments. For a givenevaluation time t and a given maturity T , the yield to maturity y of a zero-couponbond depends on the rating:

y(t, T, r) = y1(t, T ) +∫ 1

r

ϕ(t, T, s) ds

The non-default yield y1(t, T ) follows a traditional interest rate model (e.g. HJM,BGM, ”string”, etc.). The ”spread per unit of rating” ϕ(t, T, s) is a positive randomfield with respect to s and T , e.g. the square of a Gaussian field. It is constrainedby the fact that y(t, T, 0) is the recovery rate of the zero-coupon bond (possiblyitself stochastic). The random field ϕ being given, we compute a risk-neutral drift ofthe rating process RB(t). For several bonds, ratings are driven by correlated Wienerprocesses. In the case of pure diffusion processes, joint defaults have zero probability(though a default occurrence increases the intensity of others). Correlated jumpsinduce positive joint default probabilities. Credit derivatives are priced by Monte-Carlo simulation: non-standard credit default swaps (CDS), first n to default in abasket, etc. Hedge ratios are computed with respect to underlying bonds and CDS’s.

Major credit models (Merton, Jarrow-Turnbull, Duffie-Singleton, Hull-White)are particular cases of this model, which has been designed to ease calibration. Long-term statistics on yield spreads in each rating category provide the volatility andcorrelation structure of the random field ϕ . The rating process is, in a first step,statistically estimated, thanks to rating migration statistics from rating agencies(each agency rating is associated with a range for the continuous rating). Then itsdrift is replaced by the risk-neutral value, while the historical volatility is kept. Therating process being an abstract version of Merton’s firm value, we suggest to useissuers’ stock correlation for that of their rating processes.

Presented by Raphael Douady

Optimal Design of derivatives in illiquid markets.

Pauline BARRIEULaboratoire de Probabilites et Modeles Aleatoires.

Universite de Paris VI.Nicole EL KAROUI

Centre de Mathematiques Appliquees.Ecole Polytechnique, France.


The aim of this paper is to determine the optimal structure of derivatives writtenon an illiquid instrument, such as a weather event. The transaction involves twoagents : a bank who has a large exposure to the illiquid instrument, and an investorwhose objective is to diversify her exposures. Based on a utility maximization pointof view, we define an optimal profile and the value of derivative written on this illiquidinstrument which maximizes the bank’s utility given that the investor decides to takeposition in the product only if it increases her utility.

In the case of exponential utility, we show that the optimal profile is a linearfunction of the drift and give a non-linear pricing rule. We then proceed to quantifythe effect of diversification for the investor.

Stochastic Volatility Asymptotics

Jean-Pierre FouqueNorth Carolina State University, USA


In this talk we will present the approach to pricing and hedging under stochas-tic volatility developed in the recent book Derivatives in Financial Markets withStochastic Volatility by Fouque-Papanicolaou-Sircar (Cambridge University Press,2000). The emphasis will be put on the order of accuracy of the approximation andon an application to variance reduction in Monte Carlo computations.

A stochastic integral for self financing portfolios

Philippe HenrotteGroupe HEC, 78351 Jouy-en-Josas Cedex, France.


We define a continuous time stochastic integral which represents the gain fromtrade of self financed portfolios in a very general setting where security prices maynot follow semimartingales. Our results can be applied to study long memory priceprocesses such as fractional Brownian motions. We revisit the claim which statesthat absence of arbitrage implies that security prices must follow semimartingales.

Entropy and Information in the Interest Rate Term Structure

Dorje C. BrodyTheory Group, Blackett Laboratory, Imperial College, London SW7 2BZ, UK


Lane P. HughstonDepartment of Mathematics, King’s College London, Strand, London WC2R 2LS, UK


Associated with every positive interest term structure there is a probability den-sity function over the positive half line. This fact can be used to turn the problemof term structure analysis into a problem in the comparison of probability distribu-tions, an area well developed in statistics, known as information geometry. The keyidea is to take the square-root of the density function, which embeds the space ofdensities into a Hilbert space. As a consequence, Hilbert space operations can beemployed to study the structure of interest rate models. Some of the information-theoretic and geometric aspects of term structures thus arising will be illustrated.In particular, we introduce a new term structure calibration methodology based onmaximisation of entropy, and also present some new families of interest rate modelsarising naturally in this context.

Presented by Lane P. Hughston

Arbitrage With Fixed Costsand Interest Rate Models

Elyes JouiniCEREMADE, Paris, France & CRESTe-mail [email protected]

Clotilde NappCEREMADE, Paris, France & CRESTe-mail [email protected]

This paper studies foundational issues in discrete securities markets models withfixed costs of trading, i.e. transactions costs that are bounded regardless of thetransaction size, such as: fixed brokerage fees, investment taxes, operational andprocessing costs, or opportunity costs. We show that the absence of arbitrage in suchmodels is equivalent to the existence of a family of absolutely continuous probabilitymeasures for which the normalized securities price processes are martingales. Wealso show that the only arbitrage free pricing rules on the set of attainable contingentclaims are those that are equal to the sum of an expected value with respect to anyabsolutely continuous martingale measure and of a bounded fixed cost functional.

We prove then that this result has important implications in interest rate the-ory. In particular, we start by proving that the quite surprising result obtained byDybvig-Ingersoll-Ross [1996], which asserts that, under the assumption of absence ofarbitrage, long zero-coupon and long forward rates can never fall, is no longer truein models with fixed costs. This permits then to consider arbitrage-free interest ratemodels where the long forward rate follows a diffusion like in Brennan and Schwartz[1979] or to generalize Cox, Ingersoll and Ross [1985] model in the spirit of Longstaff[1992].

Presented by Elyes Jouini

Hedging defaultable contingent claims

Jeanblanc, MoniqueUniversite d’Evry Val d’Essonne, France


Blanchet-Scaillet, ChristophetteUniversite d’Evry Val d’Essonne, Francee-mail [email protected]

One of the main problem in Finance is to determine prices for derivative products, called contingent claims. The price of the contingent claim is the initial valueof a self-financing portfolio (called hedging portfolio) which gives a terminal payoffequal to the contingent claim. Defaultable payoffs arise when a party is no more ableto respect the contract, hence do not deliver the prom ised payoff. In this paper wedeal with the case where a “default” arises at so me random time τ . If the defaultoccurs before the terminal time, the prom ised payoff is not paid and a compensationis paid at terminal date.The aim of papers on default risk is to compare prices of default-free conting entclaims and defaultable ones, hence the knowledge of the default free asset s’ dynam-ics and the default process is required. Following this methodology, we investigatethe links between the default time τ (a random time) and the default-free infor-mation F = (Ft, t ≥ 0), generally included i n the filtration of default-free assets’prices. It is generally assumed that the financial agents can observe the time ofdefault, therefore, the knowledge of the agent is the filtration G, which contains F

and the filtration generated by Dt = 11τ≤t. In the so-called structural approach, τ isa stopping time in the filtration FS generated by default-free assets’ prices and it isassumed that the agents have all the information contained on prices, i.e., F = FS ,whereas in the reduced-form approach, the default arrives “by surprise”, for exampleas in Cox process modelling.We investigate in particular the so-called Cox process approach, which enjoys thecharacteristic property that F-martingales are immerged in G-martingales. We es-tablish a representation theorem for martingales which arise in the pricing of de-faultable contingent claims. Obviously, we need a financial instrument to hedge thedefault. When the default-free market is complete, i.e. when an hedging portfolioexists for any contingent claim, we prove that a defaultable zero-coupon completesthe defaultable market and we make precise the hedging of defaultable contingentclaim using defaultable zero-coupons and default free assets.In a last part, we study the general case using progressive enlargement of filtrationframework.

Presented by Jeanblanc Monique

Dynamic risk management and liquidity effects

Jean-Michel LASRYCPR


Under some market and portfolio management conditions, the execution of dy-namic hedging and portfolio management can move market prices. Hence modelsshould be modified in order to include these effects.

We will show how to build models including these liquidity effects under rea-sonnable market elasticity hypotheses. This gives tools to compute analyticaly ornumericaly adequate modified hedges and portfolio management.

A large deviations approach to optimal long term investment

Huyen PhamLaboratoire de Probabilites et Modeles Aleatoires

Universite de Paris VIIe-mail [email protected]

We consider an investment model in where the objective is to overperform agiven benchmark or index. We study this portfolio management problem for a longterm horizon. This asymptotic criterion leads to a large deviation probability controlproblem. Its dual problem is an ergodic risk sensitive control problem on the optimallogarithmic moment generating function that is explicitly derived. A careful studyof its domain and its behavior at the boundary of the domain is required. We thenuse large deviations techniques for stating the optimal rate function of this measureof outperformance. This provides in turn an objective probabilistic interpretationof the usually subjective degree of risk aversion in CRRA utility function.

Monte Carlo valuation of American options

L. C, G. RogersUniversity of Bath, England


This paper introduces a ‘dual’ way to price American options, based on simulat-ing the path of the option payoff, and of a judiciously-chosen Lagrangian martingale.Taking the pathwise maximum of the payoff less the martingale provides an upperbound for the price of the option, and this bound is sharp for the optimal choiceof Lagrangian martingale. As a first exploration of this method, three examplesare investigated numerically; the accuracy achieved with even very simple-mindedchoices of Lagrangian martingale is surprising. The method also leads naturally tocandidate hedging policies for the option, and estimates of the risk involved in usingthem.

Performance Analysis of Funds with Benchmarks:A Large Deviations Approach

Michael StutzerUniversity of Iowa, USA


Recent attention has focused on fund managers who measure their performancerelative to a benchmark portfolio. This paper provides performance analysts with asimple method of ranking portfolios’ probabilities of outperforming a time averagedbenchmark portfolio return. It does not require specifications for either the managedor benchmark portfolios’ return processes, although knowledge of these can be uti-lized. We show how a ranking by outperformance probability agrees with the rankingby a generalized expected utility dependent on the return of the portfolio in excessof the benchmark. This provides an objective probability-based generalization of aheretofore purely subjective, managerial objective assumed in recent models of fundmanager behavior, as well as a critique of those models from the perspective of fundinvestors.

We illustrate the implementation of this method using the S&P 500 index port-folio as a benchmark for 42 mutual funds that invested almost solely in domestic USequities. We show that only 7 of the 42 funds would almost surely outperform thetime averaged S&P500 return, and provide the relative ranking of those 7 based ontheir outperformance probabilities.

A Libor Market Model with Default Risk

Philipp J. SchonbucherDepartment of Statistics, Bonn University


In this paper a new credit risk model for credit derivatives is presented. Themodel is based upon the ‘Libor market’ modelling framework for default-free inter-est rates. We model effective default-free forward rates and effective forward creditspreads as lognormal diffusion processes, and recovery is modelled as a fraction ofthe par value of the defaulted claim. The newly introduced survival-based pricingmeasures are a valuable tool in the pricing of defaultable payoffs and allow a straight-forward derivation of the no-arbitrage dynamics of forward rates and forward creditspreads. The model can be calibrated to the prices of defaultable coupon bonds, as-set swap rates and default swap rates for which closed-form solutions are given. Foroptions on default swaps and caps on credit spreads, approximate solutions of highaccuracy exist. This pricing formula for options on default swaps is made exact ina modified modelling framework using an analogy to the swap measure, the defaultswap measure.

Finite dimensional Realizationsin HJM-framework

Josef TeichmannTechnical University of Vienna


Damir FilipovicETH Zurich, Switzerland


This article discusses finite-dimensional (Markovian) realizations (FDRs) forHeath–Jarrow–Morton interest rate models. We consider a d-dimensional drivingBrownian motion and stochastic volatility structures that are non-degenerate smoothfunctionals of the current forward rate. In a recent paper, Bjork and Svensson givesufficient and necessary conditions for the existence of FDRs within a particularHilbert space setup. We extend their framework, provide new results on the ge-ometry of the implied FDRs and classify all of them. In particular, we prove theirconjecture that every short rate realization is 2-dimensional. More general, we showthat all generic FDRs are at least (d + 1)-dimensional and that all generic FDRsare affine. As an illustration we sketch an interest rate model, which goes wellwith the Svensson curve-fitting method. These results cannot be obtained in theBjork–Svensson setting.

A substantial part is devoted to the analysis on Frechet spaces, where we derivea Frobenius theorem. Though we only consider stochastic equations in the HJM-framework many of the results carry over to a more general setup.

Presented by Josef Teichmann

Hedging, target reachability,and geometric flows

H.M. SonerKoc University, Istanbul, Turkey


N. TouziCERMSEM, Universite Paris 1, France


Starting from the classical hedging problem in finance, we define the target reach-ability problem. This can be seen as an extension of forward-backward SDE’s. Thesolution of the hedging problem in the financial literature is obtained via duality.Instead, we prove a geometric dynamic programming principle (DP) directly statedon the target reachability problem. The Hamilton-Jacobi-Bellman equation in thediscontinuous viscosity sense is derived by use of this DP. An important consequenceis that the target reachability problem turns out to provide a stochastic representa-tion of a large class of geometric flows, including the mean curvature and the inversemean curvature flows. For smooth geometric flows, this connection can be provedby simple arguments without involving the viscosity theory.

Presented by N. Touzi

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