MONTE CARLO METHODS FOR THE VALUATION OF MULTIPLE-EXERCISE OPTIONS

We discuss Monte Carlo methods for valuing options with multiple-exercise features in discrete time. By extending the recently developed duality ideas for American option pricing, we show how to obtain estimates on the prices of such options using Monte Carlo techniques. We prove convergence of our approach and estimate the error. The methods are applied to options in the energy and interest rate derivative markets.     

A NEW METHOD OF PRICING LOOKBACK OPTIONS

A new method for pricing lookback options (a.k.a. hindsight options) is presented, which simplifies the derivation of analytical formulas for this class of exotics in the Black-Scholes framework. Underlying the method is the observation that a lookback option can be considered as an integrated form of a related barrier option. The integrations with respect to the barrier price are evaluated at the expiry date to derive the payoff of an equivalent portfolio of European-type binary options. The arbitrage-free price of the lookback option can then be evaluated by static replication as the present value of this portfolio. We illustrate the method by deriving expressions for generic, standard floating-, fixed-, and reverse-strike lookbacks, and then show how the method can be used to price the more complex partial-price and partial-time lookback options. The method is in principle applicable to frameworks with alternative asset-price dynamics to the Black-Scholes world.     

A STATE-SPACE PARTITIONING METHOD FOR PRICING HIGH-DIMENSIONAL AMERICAN-STYLE OPTIONS

The pricing of American-style options by simulation-based methods is an important but difficult task primarily due to the feature of early exercise, particularly for high-dimensional derivatives. In this paper, a bundling method based on quasi-Monte Carlo sequences is proposed to price high-dimensional American-style options. The proposed method substantially extends Tilley's bundling algorithm to higher-dimensional situations. By using low-discrepancy points, this approach partitions the state space and forms bundles. A dynamic programming algorithm is then applied to the bundles to estimate the continuation value of an American-style option. A convergence proof of the algorithm is provided. A variety of examples with up to 15 dimensions are investigated numerically and the algorithm is able to produce computationally efficient results with good accuracy.     

TRUE UPPER BOUNDS FOR BERMUDAN PRODUCTS VIA NON-NESTED MONTE CARLO

We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic true stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale property. The so constructed martingale can be employed for computing tight dual upper bounds without nested simulation. In general, this martingale can also be used as a control variate for simulation of conditional expectations. In this context, we develop a variance reduced version of the nested primal-dual estimator. Numerical experiments indicate the efficiency of the proposed algorithms.     

PRICING A CLASS OF EXOTIC OPTIONS VIA MOMENTS AND SDP RELAXATIONS

We present a new methodology for the numerical pricing of a class of exotic derivatives such as Asian or barrier options when the underlying asset price dynamics are modeled by a geometric Brownian motion or a number of mean-reverting processes of interest. This methodology identifies derivative prices with infinite-dimensional linear programming problems involving the moments of appropriate measures, and then develops suitable finite-dimensional relaxations that take the form of semidefinite programs (SDP) indexed by the number of moments involved. By maximizing or minimizing appropriate criteria, monotone sequences of both upper and lower bounds are obtained. Numerical investigation shows that very good results are obtained with only a small number of moments. Theoretical convergence results are also established.     

STOCHASTIC VOLATILITY MODELS AND THE PRICING OF VIX OPTIONS

In this paper, we examine and compare the performance of a variety of continuous‐time volatility models in their ability to capture the behavior of the VIX. The “3/2‐ model” with a diffusion structure which allows the volatility of volatility changes to be highly sensitive to the actual level of volatility is found to outperform all other popular models tested. Analytic solutions for option prices on the VIX under the 3/2‐model are developed and then used to calibrate at‐the‐money market option prices.     

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.     

ACHIEVING HIGHER ORDER CONVERGENCE FOR THE PRICES OF EUROPEAN OPTIONS IN BINOMIAL TREES

A new family of binomial trees as approximations to the Black–Scholes model is introduced. For this class of trees, the existence of complete asymptotic expansions for the prices of vanilla European options is demonstrated and the first three terms are explicitly computed. As special cases, a tree with third-order convergence is constructed and the conjecture of Leisen and Reimer that their tree has second-order convergence is proven.     

PRICING SWAPTIONS AND COUPON BOND OPTIONS IN AFFINE TERM STRUCTURE MODELS

We propose an approach to find an approximate price of a swaption in affine term structure models. Our approach is based on the derivation of approximate swap rate dynamics in which the volatility of the forward swap rate is itself an affine function of the factors. Hence, we remain in the affine framework and well-known results on transforms and transform inversion can be used to obtain swaption prices in similar fashion to zero bond options (i.e., caplets). The method can easily be generalized to price options on coupon bonds. Computational times compare favorably with other approximation methods. Numerical results on the quality of the approximation are excellent. Our results show that in affine models, analogously to the LIBOR market model, LIBOR and swap rates are driven by approximately the same type of (in this case affine) dynamics.     

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.     

DEFAULTABLE OPTIONS IN A MARKOVIAN INTENSITY MODEL OF CREDIT RISK

This paper is a follow‐up to “Valuation and Hedging of Defaultable Game Options in a Hazard Process Model” by the same authors. In the present paper we give user friendly assumptions ensuring that the general conditions in the previous paper are satisfied. We also give a systematic procedure to construct suitable intensity models of credit risk, and, in the Markovian case, we provide a variational inequality approach to the pre‐default pricing problem. We finally illustrate our results on a study of defaultable convertible bonds.     

THE EARLY EXERCISE PREMIUM REPRESENTATION OF FOREIGN MARKET AMERICAN OPTIONS1

The note deals with the pricing of American options related to foreign market equities. the form of the early exercise premium representation of the American option's price in a stochastic interest rate economy is established. Subsequently, the American fixed exchange rate foreign equity option and the American equity-linked foreign exchange option are studied in detail.     

FAST MONTE CARLO GREEKS FOR FINANCIAL PRODUCTS WITH DISCONTINUOUS PAY‐OFFS

We introduce a new class of numerical schemes for discretizing processes driven by Brownian motions. These allow the rapid computation of sensitivities of discontinuous integrals using pathwise methods even when the underlying densities postdiscretization are singular. The two new methods presented in this paper allow Greeks for financial products with trigger features to be computed in the LIBOR market model with similar speed to that obtained by using the adjoint method for continuous pay‐offs. The methods are generic with the main constraint being that the discontinuities at each step must be determined by a one‐dimensional function: the proxy constraint. They are also generic with the sole interaction between the integrand and the scheme being the specification of this constraint.     

MODEL UNCERTAINTY AND ITS IMPACT ON THE PRICING OF DERIVATIVE INSTRUMENTS

Uncertainty on the choice of an option pricing model can lead to "model risk" in the valuation of portfolios of options. After discussing some properties which a quantitative measure of model uncertainty should verify in order to be useful and relevant in the context of risk management of derivative instruments, we introduce a quantitative framework for measuring model uncertainty in the context of derivative pricing. Two methods are proposed: the first method is based on a coherent risk measure compatible with market prices of derivatives, while the second method is based on a convex risk measure. Our measures of model risk lead to a premium for model uncertainty which is comparable to other risk measures and compatible with observations of market prices of a set of benchmark derivatives. Finally, we discuss some implications for the management of "model risk."     

THE GARCH OPTION PRICING MODEL

This article develops an option pricing model and its corresponding delta formula in the context of the generalized autoregressive conditional heteroskedastic (GARCH) asset return process. the development utilizes the locally risk-neutral valuation relationship (LRNVR). the LRNVR is shown to hold under certain combinations of preference and distribution assumptions. the GARCH option pricing model is capable of reflecting the changes in the conditional volatility of the underlying asset in a parsimonious manner. Numerical analyses suggest that the GARCH model may be able to explain some well-documented systematic biases associated with the Black-Scholes model.     

ON UTILITY-BASED PRICING OF CONTINGENT CLAIMS IN INCOMPLETE MARKETS

We study the uniqueness of the marginal utility-based price of contingent claims in a semimartingale model of an incomplete financial market. In particular, we obtain that a necessary and sufficient condition for all bounded contingent claims to admit a unique marginal utility-based price is that the solution to the dual problem defines an equivalent local martingale measure.     

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.     

MCMC ESTIMATION OF LÉVY JUMP MODELS USING STOCK AND OPTION PRICES

We examine the performances of several popular Lévy jump models and some of the most sophisticated affine jump‐diffusion models in capturing the joint dynamics of stock and option prices. We develop efficient Markov chain Monte Carlo methods for estimating parameters and latent volatility/jump variables of the Lévy jump models using stock and option prices. We show that models with infinite‐activity Lévy jumps in returns significantly outperform affine jump‐diffusion models with compound Poisson jumps in returns and volatility in capturing both the physical and risk‐neutral dynamics of the S&P 500 index. We also find that the variance gamma model of Madan, Carr, and Chang with stochastic volatility has the best performance among all the models we consider.     

A TEST OF A GENERAL EQUILIBRIUM STOCK OPTION PRICING MODEL

An empirical version of the Cox, Ingersoll, and Ross (1985a) call option pricing model is derived, assuming execution price uncertainty in the options market. the pricing restrictions come in the form of moment conditions in the option pricing error. These can be estimated and tested using a version of the method of simulated moments (MSM). Simulation estimates, obtained by discretely approximating the risk-neutral processes of the underlying stock price and the interest rate, are substituted for analytically unknown call prices. the asymptotics and other aspects of the MSM estimator are discussed. the model is tested on transaction prices at 15-minute intervals. It substantially outperforms the Black-Scholes model. the empirical success of the Cox-Ingersoll-Ross model implies that the continuous-time interest rate implicit in synchronous transaction quotes of 90-day Treasury-bill futures contracts is an-albeit noisy-proxy for the instantaneous volatility on common stock. the process of the instantaneous volatility is found to be close to nonstationary. It is well approximated by a heteroskedastic unit-root process. With this approximation, the Cox-Ingersoll-Ross model only slightly overprices long-maturity options.     

THE COST OF ILLIQUIDITY AND ITS EFFECTS ON HEDGING

Though liquidity is commonly believed to be a major effect in financial markets, there appears to be no consensus definition of what it is or how it is to be measured. In this paper, we understand liquidity as a nonlinear transaction cost incurred as a function of rate of change of portfolio. Using this definition, we obtain the optimal hedging policy for the hedging of a call option in a Black‐Scholes model. This is a more challenging question than the more common studies of optimal strategy for liquidating an initial position, because our goal requires us to match a random final value. The solution we obtain reduces in the case of quadratic loss to the solution of three partial differential equations of Black‐Scholes type, one of them nonlinear.