A FIRST‐ORDER BSPDE FOR SWING OPTION PRICING

We study an optimal control problem related to swing option pricing in a general non‐Markovian setting in continuous time. As a main result we uniquely characterize the value process in terms of a first‐order nonlinear backward stochastic partial differential equation and a differential inclusion. Based on this result we also determine the set of optimal controls and derive a dual minimization problem.     

CONVERGENCE OF A LEAST‐SQUARES MONTE CARLO ALGORITHM FOR AMERICAN OPTION PRICING WITH DEPENDENT SAMPLE DATA

We analyze the convergence of the Longstaff–Schwartz algorithm relying on only a single set of independent Monte Carlo sample paths that is repeatedly reused for all exercise time‐steps. We prove new estimates on the stochastic component of the error of this algorithm whenever the approximation architecture is any uniformly bounded set of L² functions of finite Vapnik–Chervonenkis dimension (VC‐dimension), but in particular need not necessarily be either convex or closed. We also establish new overall error estimates, incorporating bounds on the approximation error as well, for certain nonlinear, nonconvex sets of neural networks.     

ANALYTIC APPROXIMATIONS FOR MULTI‐ASSET OPTION PRICING

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of subordinate multi‐ or single‐asset options. The underlying asset prices are assumed to follow lognormal processes, although our results can be extended to certain other price processes for the underlying. For some multi‐asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single‐asset option under a modified measure, and in such cases an almost exact analytic price exists. More generally, approximate analytic prices for multi‐asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the subordinate basket options. The analytic formulae for multi‐asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to subordinate multi‐asset options, and the deltas of these subordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations. Finally, a calibration algorithm is proposed and illustrated.     

OPTIMAL MULTIPLE STOPPING AND VALUATION OF SWING OPTIONS

The connection between optimal stopping of random systems and the theory of the Snell envelop is well understood, and its application to the pricing of American contingent claims is well known. Motivated by the pricing of swing contracts (whose recall components can be viewed as contingent claims with multiple exercises of American type) we investigate the mathematical generalization of these results to the case of possible multiple stopping. We prove existence of the multiple exercise policies in a fairly general set-up. We then concentrate on the Black–Scholes model for which we give a constructive solution in the perpetual case, and an approximation procedure in the finite horizon case. The last two sections of the paper are devoted to numerical results. We illustrate the theoretical results of the perpetual case, and in the finite horizon case, we introduce numerical approximation algorithms based on ideas of the Malliavin calculus.     

STOCHASTIC HYPERBOLIC DYNAMICS FOR INFINITE-DIMENSIONAL FORWARD RATES AND OPTION PRICING

We model the term-structure modeling of interest rates by considering the forward rate as the solution of a stochastic hyperbolic partial differential equation. First, we study the arbitrage-free model of the term structure and explore the completeness of the market. We then derive results for the pricing of general contingent claims. Finally we obtain an explicit formula for a forward rate cap in the Gaussian framework from the general results.     

A METHOD FOR PRICING AMERICAN OPTIONS USING SEMI‐INFINITE LINEAR PROGRAMMING

We introduce a new approach for the numerical pricing of American options. The main idea is to choose a finite number of suitable excessive functions (randomly) and to find the smallest majorant of the gain function in the span of these functions. The resulting problem is a linear semi‐infinite programming problem, that can be solved using standard algorithms. This leads to good upper bounds for the original problem. For our algorithms no discretization of space and time and no simulation is necessary. Furthermore it is applicable even for high‐dimensional problems. The algorithm provides an approximation of the value not only for one starting point, but for the complete value function on the continuation set, so that the optimal exercise region and, for example, the Greeks can be calculated. We apply the algorithm to (one‐ and) multidimensional diffusions and show it to be fast and accurate.     

OPTION PRICING AND HEDGING WITH EXECUTION COSTS AND MARKET IMPACT

This paper considers the pricing and hedging of a call option when liquidity matters, that is, either for a large nominal or for an illiquid underlying asset. In practice, as opposed to the classical assumptions of a price‐taking agent in a frictionless market, traders cannot be perfectly hedged because of execution costs and market impact. They indeed face a trade‐off between hedging errors and costs that can be solved by using stochastic optimal control. Our modeling framework, which is inspired by the recent literature on optimal execution, makes it possible to account for both execution costs and the lasting market impact of trades. Prices are obtained through the indifference pricing approach. Numerical examples are provided, along with comparisons to standard methods.     

CRITICAL PRICE NEAR MATURITY FOR AN AMERICAN OPTION ON A DIVIDEND-PAYING STOCK IN A LOCAL VOLATILITY MODEL

We consider an American put option on a dividend-paying stock whose volatility is a function of the stock value. Near the maturity of this option, an expansion of the critical stock price is given. If the stock dividend rate is greater than the market interest rate, the payoff function is smooth near the limit of the critical price. We deduce an expansion of the critical price near maturity from an expansion of the value function of an optimal stopping problem. It turns out that the behavior of the critical price is parabolic. In the other case, we are in a less regular situation and an extra logarithmic factor appears. To prove this result, we show that the American and European critical prices have the same first-order behavior near maturity. Finally, in order to get an expansion of the European critical price, we use a parity formula for exchanging the strike price and the spot price in the value functions of European puts.     

PRICING OF HIGH‐DIMENSIONAL AMERICAN OPTIONS BY NEURAL NETWORKS

Pricing of American options in discrete time is considered, where the option is allowed to be based on several underlyings. It is assumed that the price processes of the underlyings are given Markov processes. We use the Monte Carlo approach to generate artificial sample paths of these price processes, and then we use the least squares neural networks regression estimates to estimate from this data the so‐called continuation values, which are defined as mean values of the American options for given values of the underlyings at time t subject to the constraint that the options are not exercised at time t. Results concerning consistency and rate of convergence of the estimates are presented, and the pricing of American options is illustrated by simulated data.     

TUG‐OF‐WAR,MARKET MANIPULATION,AND OPTION PRICING

We develop an option pricing model based on a tug‐of‐war game. This two‐player zero‐sum stochastic differential game is formulated in the context of a multidimensional financial market. The issuer and the holder try to manipulate asset price processes in order to minimize and maximize the expected discounted reward. We prove that the game has a value and that the value function is the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the nonlinear and completely degenerate infinity Laplace operator.     

ANAYTICAL SOLUTIONS FOR THE PRICING OF AMERICAN BOND AND YIELD OPTIONS1

In this paper we use the Cox, Ingersoll, and Ross (1985b) single-factor, term structure model and extend it to the pricing of American default-free bond puts. We provide a quasi-analytical formula for these option prices based on recently established mathematical results for Bessel bridges, coupled with the optimal stopping time method. We extend our results to another interest rate contingent claim and provide a quasi-analytical solution for American yield option prices which illustrates the flexibility of our framework.     

A QUANTIZATION TREE METHOD FOR PRICING AND HEDGING MULTIDIMENSIONAL AMERICAN OPTIONS

We present here the quantization method which is well-adapted for the pricing and hedging of American options on a basket of assets. Its purpose is to compute a large number of conditional expectations by projection of the diffusion on optimal grids designed to minimize the (square mean) projection error ( Graf and Luschgy 2000 ). An algorithm to compute such grids is described. We provide results concerning the orders of the approximation with respect to the regularity of the payoff function and the global size of the grids. Numerical tests are performed in dimensions 2, 4, 5, 6, 10 with American style exchange options. They show that theoretical orders are probably pessimistic.     

OPTIMAL LIQUIDATION IN A LIMIT ORDER BOOK FOR A RISK‐AVERSE INVESTOR

In a limit order book model with exponential resilience, general shape function, and an unaffected stock price following the Bachelier model, we consider the problem of optimal liquidation for an investor with constant absolute risk aversion. We show that the problem can be reduced to a two‐dimensional deterministic problem which involves no buy orders. We derive an explicit expression for the value function and the optimal liquidation strategy. The analysis is complicated by the fact that the intervention boundary, which determines the optimal liquidation strategy, is discontinuous if there are levels in the limit order book with relatively little market depth. Despite this complication, the equation for the intervention boundary is fairly simple. We show that the optimal liquidation strategy possesses the natural properties one would expect, and provide an explicit example for the case where the limit order book has a constant shape function.     

THE DEPENDENCE STRUCTURE OF RUNNING MAXIMA AND MINIMA: RESULTS AND OPTION PRICING APPLICATIONS

We provide general results for the dependence structure of running maxima (minima) of sets of variables in a model based on (i) Markov dynamics; (ii) no Granger causality; (iii) cross-section dependence. At the time series level, we derive recursive formulas for running minima and maxima. These formulas enable to use a bootstrapping technique to recursively recover the pricing kernels of barrier options from those of the corresponding European options. We also show that the dependence formulas for running maxima (minima) are completely defined from the copula function representing dependence among levels at the terminal date. The result is applied to multivariate discrete barrier digital products. Barrier Altiplanos are simply priced by (i) bootstrapping the price of univariate barrier products; (ii) evaluating a European Altiplano with these values.     

A direct solution method for pricing options in regime‐switching models

Pricing financial or real options with arbitrary payoffs in regime‐switching models is an important problem in finance. Mathematically, it is to solve, under certain standard assumptions, a general form of optimal stopping problems in regime‐switching models. In this article, we reduce an optimal stopping problem with an arbitrary value function in a two‐regime environment to a pair of optimal stopping problems without regime switching. We then propose a method for finding optimal stopping rules using the techniques available for nonswitching problems. In contrast to other methods, our systematic solution procedure is more direct as we first obtain the explicit form of the value functions. In the end, we discuss an option pricing problem, which may not be dealt with by the conventional methods, demonstrating the simplicity of our approach.     

ON THE TIMING OPTION IN A FUTURES CONTRACT

The timing option embedded in a futures contract allows the short position to decide when to deliver the underlying asset during the last month of the contract period. In this paper we derive, within a very general incomplete market framework, an explicit model independent formula for the futures price process in the presence of a timing option. We also provide a characterization of the optimal delivery strategy, and we analyze some concrete examples.     

A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES

We propose a model which can be jointly calibrated to the corporate bond term structure and equity option volatility surface of the same company. Our purpose is to obtain explicit bond and equity option pricing formulas that can be calibrated to find a risk neutral model that matches a set of observed market prices. This risk neutral model can then be used to price more exotic, illiquid, or over‐the‐counter derivatives. We observe that our model matches the equity option implied volatility surface well since we properly account for the default risk in the implied volatility surface. We demonstrate the importance of accounting for the default risk and stochastic interest rate in equity option pricing by comparing our results to Fouque et al., which only accounts for stochastic volatility.     

NO‐ARMAGEDDON MEASURE FOR ARBITRAGE‐FREE PRICING OF INDEX OPTIONS IN A CREDIT CRISIS

In this work, we consider three problems of the standard market approach to credit index options pricing: the definition of the index spread is not valid in general, the considered payoff leads to a pricing which is not always defined, and the candidate numeraire for defining a pricing measure is not strictly positive, which leads to a nonequivalent pricing measure. We give a solution to the three problems, based on modeling the flow of information through a suitable subfiltration. With this we consistently take into account the possibility of default of all names in the portfolio, that is neglected in the standard market approach. We show on market inputs that, while the pricing difference can be negligible in normal market conditions, it can become highly relevant in stressed market conditions, like the situation caused by the credit crunch.     

BOUNDARY EVOLUTION EQUATIONS FOR AMERICAN OPTIONS

We consider the problem of finding optimal exercise policies for American options, both under constant and stochastic volatility settings. Rather than work with the usual equations that characterize the price exclusively, we derive and use boundary evolution equations that characterize the evolution of the optimal exercise boundary. Using these boundary evolution equations we show how one can construct very efficient computational methods for pricing American options that avoid common sources of error. First, we detail a methodology for standard static grids and then describe an improvement that defines a grid that evolves dynamically while solving the problem. When integral representations are available, as in the Black–Scholes setting, we also describe a modified integral method that leverages on the representation to solve the boundary evolution equations. Finally we compare runtime and accuracy to other popular numerical methods. The ideas and methodology presented herein can easily be extended to other optimal stopping problems.     

A MULTINOMIAL APPROXIMATION FOR AMERICAN OPTION PRICES IN LÉVY PROCESS MODELS

This paper gives a tree-based method for pricing American options in models where the stock price follows a general exponential Lévy process. A multinomial model for approximating the stock price process, which can be viewed as generalizing the binomial model of Cox, Ross, and Rubinstein (1979) for geometric Brownian motion, is developed. Under mild conditions, it is proved that the stock price process and the prices of American-type options on the stock, calculated from the multinomial model, converge to the corresponding prices under the continuous time Lévy process model. Explicit illustrations are given for the variance gamma model and the normal inverse Gaussian process when the option is an American put, but the procedure is applicable to a much wider class of derivatives including some path-dependent options. Our approach overcomes some practical difficulties that have previously been encountered when the Lévy process has infinite activity.