Introduction

In the past two decades there has been an explosion in the use of derivative securities by investors, corporations, mutual funds, and financial institutions. Exchange traded derivatives have experienced unprecedented growth in volume while ‘exotic’ securities (i.e., securities with nonstandard payoff patterns) have become more common in the over-the-counter market. Using the most widely accepted financial models, there are many types of securities which cannot be priced in closed-form. This void has created a great need for efficient numerical procedures for security pricing.

Closed-form prices are available in a few special cases. One example is a European option (i.e., an option which can only be exercised at the maturity date of the contract) written on a single underlying asset. The European option valuation formula was derived in the seminal papers of Black & Scholes (1973) and Merton (1973). In the case of American options (i.e., options which can be exercised at any time at or before the maturity date) analytical expressions for the price have been derived, but there are no easily computable, explicit formulas currently available. Researchers and practitioners must then resort to numerical approximation techniques to compute the prices of these instruments. Further complications occur when the payoff of the derivative security depends on multiple assets or multiple sources of uncertainty. Analytical solutions are often not available for options with path-dependent payoffs and other exotic options.

Introduction

The purpose of most models is to explain the most commonly occurring phenomena. When designing a model and fitting parameter values, largedeviations situations are ignored, at least to begin with, and the success of the model is judged by the way it can explain common situations. Although models designed in this way will have implications for large deviations, those same large-deviations situations may reveal a richer structure not evident in ‘typical’ behaviour.

We are going to show how, in the framework of diffusion models (the most commonly-used of financial models), one may be led to take as constant quantities whose variable and random characteristics only appear in large motions of the market. We will give a general statement on the information likely to appear in large deviations of the market, and on other latent risks.

This applies naturally to models where volatilities or correlations are supposed constant, even though in reality this is not the case. Our results suggest an explanation of the ‘smile’ of the implied volatility curve (that is, overvaluation of options far from the money). Furthermore, the results display implied volatility smiles as a quite general phenomenon.

This is the reason why we will start by presenting this particular case, which will give an introduction and an intuition to the general results in Section 3.

Finally, in Section 4, we will show how the asymptotic estimations in Section 3 can be used in order to improve numerical calculation methods for out-of-the-money contingent claims.

Introduction

In this article we examine a general investment and consumption decision problem for a single agent. The investor consumes at a nonnegative rate and he distributes his current wealth between two assets. One asset is a bond, i.e. a riskless security with instantaneous rate of return r. The other asset is a stock, whose price is driven by a Wiener process.

When the investor makes a transaction, he pays transaction fees which are assumed to be proportional to the amount transacted. More specifically, let xt and yt be the investor's holdings in the riskless and the risky security prior to a transaction at time t. If the investor increases (or decreases) the amount invested in the risky asset to yt + ht (or yt - ht), the holding of the riskless asset decreases (increases) to xt - ht - λht (or yt + ht - µht). The numbers λ and µ are assumed to be nonnegative and one of them must always be positive. The control objective is to maximize, in an infinite horizon, the expected discounted utility which comes only from consumption. Due to the presence of the transaction fees, this is a singular control problem.

Our goals are to derive the Hamilton–Jacobi–Bellman (HJB) equation that the value function solves and to characterize the latter as its unique weak solution, to come up with numerical schemes which converge to the value function as well as the optimal investment and consumption rules and to perform actual numerical computations and compare the results to the ones obtained in closed form by Davis & Norman.

Introduction

The fundamental theorem of valuation by arbitrage reduces the pricing of any European contingent claim to the computation of the expectation of the discounted final reward under some probability measure, equivalent to the initial probability measure, under which the primitive assets processes are martingales. For some simple models and simple contingent claims one can hope to derive a closed form solution of such an expectation as in the seminal work by Black & Scholes (1973). However empirical work pointed out for a long time that such simple models do not fit financial asset price data. Therefore, it is important to develop some computational methods that can handle more complicated models.

Monte Carlo methods appear as a natural tool for such computations since they can deal with models involving many state variables and are well suited for the computation of path-dependent expectations which is the usual case in finance.

In this article, we present an application of Monte Carlo methods for the valuation of contingent claims in stochastic volatility models. In such models the primitive risky asset price process is driven by a bivariate diffusion. Therefore, even for expectations depending only on the terminal value of the process, deterministic methods based on the discretization of the partial differential equation satisfied by the expectation to be computed are timeconsuming. For some path-dependent expectations, deterministic methods can still be used by introducing a new state variable and therefore the dimensionality of the problem is increased (see Barles, Daher & Romano 1990).

Introduction

Since the seminal article of Black and Scholes (1973) on option pricing, a vast amount of literature has been written concerning options. Most of these articles consider perfect and complete markets, where options can be replicated by self-financing strategies involving continual revisions in a portfolio containing the underlying security. The price is then shown to be the ‘riskneutral’ expected cash-flows discounted at the risk-free rate: see for instance the seminal articles of Harrison & Kreps (1979), Harrison & Pliska (1981, 1983), Karatzas (1988) and El Karoui & Rochet (1989).

However, very broadly speaking, in any financial model, the equilibrium price of any asset is found to aggregate the risk-aversions of the individuals and their demands for this asset taking into account its correlation with the state variables of the economy (see for instance the general equilibrium model of Cox, Ingersoll & Ross (1985)) or with a ‘well-defined’ variable (Breeden 1989). But expected utility is, in general, a function of all the moments of the distribution: rules and pricing involving only two moments or duplication are valid only for a limited class of utility functions, or for specific distributions. For several years, finance research has got rid of utility functions in option pricing through the inspired idea of duplication. If the option is simply, at each instant, a portfolio composed of two others assets, then of course its price is the simple sum of the prices of the two assets (the underlying and the risk-free assets), these two prices being determined elsewhere.

Introduction

The overwhelming majority of traded options are of American type. Yet their valuation, even in the standard case of a lognormal process for the underlying asset, remains a topic of active research. This situation stems from the nature of the solution which requires the determination of the optimal exercise strategy as well as the value of the option. In contrast the European option, which can only be exercised at its expiration date, has been valued by the celebrated Black–Scholes formula (Black & Scholes 1973) for the standard financial model.

Due to a lack of closed–form solutions to American option valuation problems, a vast array of approximation schemes has been advanced. The Broadie & Detemple article in this volume provides a summary of some experimental results. The present article is a detailed account of comparative experiments conducted with numerical schemes including the recent method of Carr & Faguet 1996. It is organized as follows: Section 2 reviews the basic Black–Scholes model, Section 3 presents the approximation approaches and Section 4 concludes with some benchmark comparisons.

The Standard Model

The prototypical definition of an American option is that of a contract giving its holder the right to buy (call option) or sell (put option) one unit of an underlying security (e.g. stock) at a pre–arranged price K. This right can be exercised at any time before an expiration date T. In contrast, a European option can be exercised at the expiration date only.

In April 1995, together with Alain Bensoussan and Agnès Sulem, we organized the session ‘Numerical Methods in Finance’ at the Isaac Newton Institute, within the framework of the 1995 Cambridge University programme on Financial Mathematics. We invited specialists in this area which is at the intersection of Probability Theory, Finance and Numerical Analysis.

Several participants worked in banks, which illustrates the needs of the practitioners for theoretical and/or numerical studies of the numerical methods they currently use (Monte Carlo procedures, approximation methods to solve PDEs appearing in option pricing, simulations, etc).

After the session, most of the lecturers agreed to write a paper on the subject of his or her talk. They also agreed not to write the paper as if it would be published in an ordinary volume of Proceedings. Each article presents the state of the art on a particular question of financial and numerical interest, with an extensive list of appropriate references, and then focuses on a new and original result (published elsewhere with complete proofs or complete numerical studies) with a pedagogical point of view: in particular, papers by mathematicians should be understandable by practitioners having a basic knowledge in the theory of stochastic processes.

To our knowledge, at the present time there does not exist any book presenting such a large variety of numerical methods in finance:

• computation of option prices, especially of American option prices, by finite difference methods;

• numerical solution of portfolio management strategies;

• […]

Introduction

Backward stochastic differential equations (BSDEs) were introduced by Pardoux & Peng (1990) to give a probabilistic representation for the solutions of certain nonlinear partial differential equations, thus generalizing the Feynman- Kac formula.

This sort of equation has also found many applications in finance, notably in contingent claim valuation when there are constraints on the hedging portfolios (see El Karoui & Quenez 1995, El Karoui, Peng & Quenez 1994, Cvitanic & Karatzas 1992) and in the definition of stochastic differential utility (see Duffie & Epstein 1992, El Karoui, Peng & Quenez 1994). A financial application of forward-backward SDEs can be found in Duffie, Ma & Yong (1993).

However little research has yet been performed on numerical methods for BSDEs. Here we give a review of three different contributions in that field.

In Section 2, we present a random time discretization scheme introduced by V. Bally to approximate BSDEs. The advantage of Bally's scheme is that one can get a convergence result with virtually no other regularity assumption than the ones needed for the existence of a solution to the equation. However that scheme is not fully numerical and its actual implementation would require further approximations.

In Section 3, we give an account of a four step algorithm developed by J. Ma and P. Protter to solve a class of more general equations called forwardbackward SDEs. It is based on solving the associated PDE by a deterministictype method and also makes use of the Euler scheme for stochastic differential equations.

Introduction

We have seen in El Karoui & Quenez (1997) that the pricing of European contingent claims, even in imperfect markets, can be formulated in terms of backward stochastic differential equations. However, the case of American options has not be considered. In this article, we will see that the price of an American option corresponds to the solution of a new type of backward equation called reflected BSDEs. The solution of such an equation is forced to stay above a given stochastic process, called the obstacle. An increasing process is introduced which pushes the solution upwards, so that it may remain above the obstacle. Recall that by definition the price of an American option is constrained to be greater than the payoff of the option (which corresponds to the obstacle). Furthermore, in a perfect market, it is well known (see Bensoussan (1984), Karatzas & Shreve (1995) and Karatzas (1988)) that such options cannot be perfectly hedged by a portfolio; in this case, the price process corresponds to the minimal ‘superhedging’ strategy for the option, that is a strategy with a so-called ‘tracking’ error which is an increasing process. We will see that this property can be generalized to imperfect markets.

In this article, we will refer often to the previous one by El Karoui & Quenez (1997) concerning the different results on BSDEs and also the notation. The problem is formulated in detail in Section 2.

Introduction

The aim of this article is twofold: on one hand, we describe a general convergence result which applies to a wide range of numerical schemes (‘monotone schemes’) for nonlinear possibly degenerate elliptic (or parabolic) equation; this type of equation arises naturally in Finance Theory as we will show first. This convergence result was obtained in an article written in collaboration with P.E. Souganidis (1991).

On the other hand, we present several simple numerical schemes for computing the price of different types of ‘simple’ options: American options, lookback options and Asian options. These schemes are all based on ‘splitting methods’ and we want to emphasize the fact that this allows also easy extensions for computing the price of more complex options with complicated contracts (cap, floor, … etc). These schemes also provide examples for which the convergence result of the first part applies. This second part reports on several works in collaboration with J. Burdeau, Ch. Daher & M. Romano (cf. references) which were done in connection with the Research and Development Department of the Caisse Autonome de Refmancement (CDC group).

The article is organized as follows: since the convergence result for numerical schemes relies strongly on the notion of ‘viscosity solutions’, which is a notion of weak solutions for nonlinear elliptic and parabolic equations, we are first going to present this notion of solutions. In order to introduce it, as a motivation, we examine in the first section several examples of equations arising in Finance Theory, and more particularly in options pricing, and we describe the theoretical difficulties in studying them.

Introduction

In the seminal paper of Black & Scholes (1973), the payoff on a European option is replicated perfectly by a dynamic strategy in the underlying security and a zero-coupon bond. Hence, the price of the replicating portfolio determines the no-arbitrage value for the option. Since that work of Black and Scholes, option pricing has been following this standard pattern of first specifying a perfect dynamic hedge, followed by an exploration of the pricing implications of the absence of arbitrage.

In Black–Scholes model, markets are complete in the sense that the payoff on the option is attainable through a dynamic trading strategy in the underlying stock and a money account. In practice, however, some of their assumptions can be proven wrong. Foremost, volatility is stochastic. In the absence of an asset that is instantaneously perfectly correlated with the volatility of the stock, market completeness is thereby invalidated. Likewise, actual rebalancing necessarily occurs over discrete intervals of time, in contrast with Black-Scholes continuous rebalancing.

If markets are incomplete, a blind application of complete-markets hedging strategies seems inappropriate. Most importantly, such policies do not self-correct even if a significant deviation from the target payoff is apparent. That immediately raises the question of whether there are hedging strategies that continuously correct in a way that facilitates error control. In other words, are there policies that adjust to past tracking errors such that the stochastic characteristics of the total tracking error become wellspecified?

Introduction

We consider a model of n risky and one risk-free asset in which the dynamics of the prices of the risky assets are governed by logarithmic Brownian motions (LBM). Risky assets are conventionally called Stocks while the risk-free asset is called a Bond or Bank. Modelling the stock prices by LBM goes back to Black & Scholes (1973). Since the paper by Merton (1971), in which a dynamic optimization consumption/investment model was introduced, there has been a lot of effort expanded to develop and perfect the LBM models.

One of the avenues of research deals with the transaction cost problem: Consider an investor who has an initial wealth invested in Stocks and Bonds and who has the ability to transfer funds between the assets. This transfer involves brokerage fees (transaction costs).

In one type of model the proceeds from these operations are used to finance the consumption and the objective is to maximize the cumulative expected utility of consumption over a finite or infinite time horizon (see Davis & Norman 1990, Akian, Menaldi & Sulem 1996, Fleming & Soner 1992, Shreve & Soner 1994, Zariphopoulou 1989). In Akian, Sulem & Séquier (1995), a similar problem is considered but on a finite horizon and without consumption. In Sethi & Taksar (1988) and Akian, Sulem & Taksar (1996), the objective is to maximize the long run average growth of wealth.

Mathematicaly, these problems can be formulated as singular stochastic control problems.