Mark S. Joshi

New and robust drift approximations for the LIBOR market model

We present four new methods for approximating the drift in the LIBOR market model when performing very long steps. These are compared with a variety of existing methods, including PPR, Glasserman–Zhao and predictor–corrector. We find that two of them, which use correlation adjustments to better approximate the drift, are more effective than existing methods.

Practical Policy Iteration: Generic Methods for Obtaining Rapid and Tight Bounds for Bermudan Exotic Derivatives Using Monte Carlo Simulation

We introduce a set of improvements which allow the calculation of very tight lower bounds for Bermudan derivatives using Monte Carlo simulation. These tight lower bounds can be computed quickly, and with minimal hand-crafting. Our focus is on accelerating policy iteration to the point where it can be used in similar computation times to the basic least-squares approach, but in doing so introduce a number of improvements which can be applied to both the least-squares approach and the calculation of upper bounds using the Andersen–Broadie method. The enhancements to the least-squares method improve both accuracy and efficiency.

A JOINT EMPIRICAL AND THEORETICAL INVESTIGATION OF THE MODES OF DEFORMATION OF SWAPTION MATRICES: IMPLICATIONS FOR MODEL CHOICE

We present a joint empirical/theoretical analysis of the changes in the implied volatility swaption matrix for two currencies (USD and DEM/EUR). We recognize the existence of a small number of recognizable shape patterns, and comment about the speed of transition between them. By Principal/Component/Analyzing the associated correlation and covariance matrices we highlight a non/trivial interpretation for the leading eigenvectors. We also compare the empirically obtained eigenvectors and eigenvalues with the corresponding quantities produced by the stochastic/volatility LIBOR market model of Joshi and Rebonato[10]. This allows us to perform a measure-independent comparison that is of intrinsic interest, and that can also provide a general blueprint for analyzing the realism of and choosing among similarly-fitting stochastic models. We find that mean reversion of the instantaneous volatility is a necessary condition in order to obatin the market-observed shape of the first eigenvector associated with the covariance matrix.

A Simple Derivation of and Improvements to Jamshidian's and Rogers' Upper Bound Methods for Bermudan Options

The additive method for upper bounds for Bermudan options is rephrased in terms of buyers and sellers prices. It is shown how to deduce Jamshidians upper bound result in a simple fashion from the additive method, including the case of possibly zero final pay‐off. Both methods are improved by ruling out exercise at sub‐optimal points. It is also shown that it is possible to use sub‐Monte Carlo simulations to estimate the value of the hedging portfolio at intermediate points in the Jamshidian method without jeopardizing its status as upper bound.

Fast and Accurate Greeks for the Libor Market Model

This paper derives the pathwise adjoint method for the predictor-corrector drift approximation in the displaced-diffusion LIBOR market model. We present a comparison of the Greeks between log-Euler and predictor-corrector, showing both methods have the same computational order but the latter to be much more accurate.

Using Monte Carlo Simulation and Importance Sampling to Rapidly Obtain Jump-Diffusion Prices of Continuous Barrier Options

The problem of pricing a continuous barrier option in a jump-diffusion model is studied. It is shown that via an effective combination of importance sampling and analytic formulas thatsubstantial speed ups can be achieved. These techniques are shown to be particularly effective for computing deltas.

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.

Fast Delta Computations in the Swap-Rate Market Model

We develop an efficient algorithm to implement the adjoint method that computes sensitivities of an interest rate derivative to different underlying rates in the co-terminal swap-rate market model. The order of computation per step of the new method is shown to be proportional to the number of rates times the number of factors, which is the same as the order in the LIBOR market model.

Monte Carlo Bounds for Game Options Including Convertible Bonds

We introduce two new methods to calculate bounds for zero-sum game options using Monte Carlo simulation. These extend and generalize upper-bound duality results to the case where both parties of a contract have Bermudan optionality. It is shown that the primal-dual simulation method can still be used as a generic way to obtain bounds in the extended framework, and we apply the new results to the pricing of convertible bonds by simulation. This paper was accepted by Wei Xiong, finance.

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 post-discretization 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.