High-dimensional problems frequently arise in the pricing of derivative securities – for example, in pricing options on multiple underlying assets and in pricing term structure derivatives. American versions of these options, ie, where the owner has the right to exercise early, are particularly challenging to price. We introduce a stochastic mesh method for pricing high-dimensional American options when there is a finite, but possibly large, number of exercise dates. The algorithm provides point estimates and confidence intervals; we provide conditions under which these estimates converge to the correct values as the computational effort increases. Numerical results illustrate the performance of the method. 1

This paper describes a practical algorithm based on Monte Carlo simulation for the pricing of multi-dimensional American (i.e., continuously exercisable) and Bermudan (i.e., discretelyexercisable) options. The method generates both lower and upper bounds for the Bermudan option price and hence gives valid confidence intervals for the true value. Lower bounds can be generated using any number of primal algorithms. Upper bounds are generated using a new Monte Carlo algorithm based on the duality representation of the Bermudan value function suggested independently in Haugh and Kogan (2001) and Rogers (2001). Our proposed algorithm can handle virtually any type of process dynamics, factor structure, and payout specification. Computational results for a variety of multi-factor equity and interest rate options demonstrate the simplicity and efficiency of the proposed algorithm. In particular, we use the proposed method to examine and verify the tightness of frequently used exercise rules in Bermudan swaption markets.

Abstract: We present the SPA framework, a novel approach to the modeling of the dynamics of portfolio default losses. In this framework, models are specified by a twolayer process. The first layer models the dynamics of portfolio loss distributions in the absence of information about default times. This background process can be explicitly calibrated to the full grid of marginal loss distributions as implied by initial CDO tranche values indexed on maturity, as well as to the prices of suitable options. We give sufficient conditions for consistent dynamics. The second layer models the loss process itself as a Markov process conditioned on the path taken by the background process. The choice of loss process is non-unique. We present a number of choices, and discuss their advantages and disadvantages. Several concrete model examples are given, and valuation in the new framework is described in detail. Among the specific securities for which algorithms are presented are CDO tranche options and leveraged super-senior tranches.

This paper is concerned with the implementation of the LIBOR market model and its extensions. It develops and tests a simple analytic approximation for calculating the volatilities used by the market to price European swap options from the volatilities used by the market to price interest rate caps. The approximation is found to be very accurate for the range of market parameters normally encountered. It enables swap option volatility skews to be implied from cap volatility skews. It also allows the LIBOR market model to be easily calibrated to broker quotes on caps and European swap options so that a wide range of non-standard interest rate derivatives can be valued. 1 FORWARD RATE VOLATILITIES, SWAP RATE VOLATILITIES, AND THE IMPLEMENTATION OF THE LIBOR MARKET MODEL The most popular over-the-counter interest rate options are interest rate caps/floors and European swap options. The standard market models for valuing these instruments are versions of Black's (1976) model. This mode...

This paper develops a simulation method for pricing path-dependent American options, and American options on a large number of underlying assets, such as basket options. Standard numerical procedures (lattice methods and nite difference methods) are generally inapplicable to such high-dimensional problems, and this has motivated research into simulation-based methods. The optimal stopping problem embedded in the pricing of American options makes this a nonstandard problem for simulation. This paper extends the stochastic mesh introduced in Broadie and Glasserman [5]. In its original form, the stochastic mesh method required knowledge of the transition density of the underlying process of asset prices and other state variables. This paper extends the method to settings in which the transition density is either unknown or fails to exist. We avoid the need for a transition density by choosing mesh weights through a constrained optimization problem. If the weights are constrained to correctly price su ciently many simple instruments, they can be expected to work well in pricing a more complex American option. We investigate two criteria for use in the optimization | maximum entropy and least squares. The methods are illustrated through numerical examples. 32 1

We present an extension of the LIBOR market model which allows for stochastic instantaneous volatilities of the forward rates in a displaced diffusion setting. We show that virtually all the powerful and important approximations that apply in the deterministic setting can be successfully and naturally extended to the stochastic volatility case. In particular we show that i) the caplet market can still be efficiently and accurately fit; ii) that the drift approximations that allow the evolution of the forward rates over time steps as long as several years are still valid; iii) that in the new setting the European swaption matrix implied by a given choice of volatility parameters can be efficiently approximated with a closed-form expression without having to carry out a Monte Carlo simulation for the forward-rate process; and iv) that it is still possible to calibrate the model virtually perfectly via simply matrix manipulations so that the prices of the coterminal swaptions underlying a given Bermudan swaption will be exactly recovered, while retaining a desirable behaviour for the evolution of the term structure of volatilities. We also show that, even after reducing the number of the possible fitting parameters, the market caplet surface across strikes and maturities can be well recovered. We notice the existence of what appears to be a systematic discrepancy for very low strikes, but we have refrained from attempting to recover this feature.

Abstract. We present an iterative procedure for computing the optimal Bermudan stopping time, hence the Bermudan Snell envelope. The method produces an increasing sequence of approximations of the Snell envelope from below, which coincide with the Snell envelope after finitely many steps. Then, by duality, the method induces a convergent sequence of upper bounds as well. In a Markovian setting the presented iterative procedure allows to calculate approximative solutions with only a few nestings of conditionals expectations and is therefore tailor-made for a plain Monte-Carlo implementation. The method presented may be considered generic for all discrete optimal stopping problems. The power of the procedure is demonstrated at Bermudan swaptions in a full factor LIBOR market model.

swaptions in a Libor market

This paper examines the static and dynamic accuracy of interest rate option pricing models in the U.S. dollar interest rate cap and floor markets. We evaluate alternative one-factor and two-factor term structure models of the spot and the forward interest rates on the basis of their out-of-sample predictive ability in terms of pricing and hedging performance. In addition, the models are evaluated based on the stability of their parameters, the presence of systematic biases, and their numerical complexity and computational efficiency. We conduct tests on daily data from March-December 1998, consisting of actual cap and floor prices across both strike rates and maturities. Results show that fitting the skew of the underlying interest rate distribution provides accurate pricing results within a onefactor framework. However, for hedging performance, introducing a second stochastic factor is more important than fitting the skew of the underlying distribution. Overall, the one-fa...

In the Libor market model framework of Brace-Gatarek-Musiela and Jamshidian (BGM/J) for the pricing of interest rate derivatives, the drifts of the underlying forward rates are state dependent. Wherever Monte Carlo methods can be used for the numerical calculation of discounted expectations, this poses no major difficulty since the computational effort necessary in order to achieve acceptable accuracy depends only weakly on the dimensionality of the sampling space. For the valuation of options that involve finding the optimal exercise strategy, however, this means that any finite-differencing method enabling us to make the pointwise comparison directly between intrinsic value and discounted expectation will have to cope with "the curse of dimensionality" whereby the number of evaluations explodes exponentially. This document is about the implementation of a non-recombining multi-factor tree algorithm with a minimal number of branches out of each node for the representation of the desired number of factors. This method can serve as a benchmark for simple test cases for the development of other approximations such as exercise-strategy parametrisations in a Monte Carlo setup. 1