Nick Webber

Valuing path-dependent options in the variance-gamma model by Monte Carlo with a gamma bridge

The Variance-Gamma model has analytical formulae for the values of European calls and puts. These formulae have to be computed using numerical methods. In general, option valuation may require the use of numerical methods including PDE methods, lattice methods, and Monte Carlo methods. We investigate the use of Monte Carlo methods in the Variance-Gamma model. We demonstrate how a gamma bridge process can be constructed. Using the bridge together with stratified sampling we obtain considerable speed improvements over a plain Monte Carlo method when pricing path-dependent options. The method is illustrated by pricing lookback, average rate and barrier options in the Variance-Gamma model. We find the method is up to around 400 times faster than plain Monte Carlo. ∗Corresponding author. Claudia Ribeiro gratefully acknowledges the support of Fundacao para a Ciencia e a Tecnologia and Faculdade de Economia, Universidade do Porto. The paper has benefited from comments from Lynda McCarthy.

Valuing Bermudan options when asset returns are Lévy processes

Abstract Evidence from the financial markets suggests that empirical returns distributions, both historical and implied, do not arise from diffusion processes. A growing literature models the returns process as a Lévy process, finding a number of explicit formulae for the values of some derivatives in special cases. Practical use of these models has been hindered by a relative paucity of numerical methods which can be used when explicit solutions are not present. In particular, the valuation of Bermudan options is problematical. This paper investigates a lattice method that can be used when the returns process is Lévy, based upon an approximation to the transition density function of the Lévy process. We find alternative derivations of the lattice, stemming from alternative representations of the Lévy process, which may be useful if the transition density function is unknown or intractable. We apply the lattice to models based on the variance-gamma and normal inverse Gaussian processes. We find that the lattice is able to price Bermudan-style options to an acceptable level of accuracy.

Pricing Barrier Options with One-Factor Interest Rate Models

Pricing barrier options for interest rates and interest-dependent securities is harder than for equities, because interest rate processes typically require more complicated models than stock returns. Even single-factor models need to incorporate the initial yield curve in order to prevent apparent arbitrage opportunities among observed bond prices in the market. When closed-form solutions do not exist, numerical valuation techniques are required, and both lattice-based approaches and Monte Carlo simulation are in common use. But these methods are very computationally demanding, and lattice techniques also require careful handling of price nodes near the barrier. Kuan and Webber introduce a new approach, in which they estimate the first passage time to the barrier, and then value the option conditional on having hit the barrier. This leads to considerable improvement in performance, either less computational effort to achieve a given accuracy, or greater accuracy for a given amount of work.

Correcting for Simulation Bias in Monte Carlo Methods to Value Exotic Options in Models Driven by Lévy Processes

Lévy processes can be used to model asset returns distributions. Monte Carlo methods must frequently be used to value path dependent options in these models, but Monte Carlo methods can be prone to considerable simulation bias when valuing options with continuous reset conditions. This paper shows how to correct for this bias for a range of options by generating a sample from the extremes distribution of the Lévy process on subintervals. The method uses variance‐gamma and normal inverse Gaussian processes. The method gives considerable reductions in bias, so that it becomes feasible to apply variance reduction methods. The method seems to be a very fruitful approach in a framework in which many options do not have analytical solutions.

Interest Rate Models on Lie Groups

This paper examines an alternative approach to interest rate modeling, in which the nonlinear and random behavior of interest rates is captured by a stochastic differential equation evolving on a curved state space. We consider as candidate state spaces the matrix Lie groups; these offer not only a rich geometric structure, but—unlike general Riemannian manifolds—also allow for diffusion processes to be constructed easily without invoking the machinery of stochastic calculus on manifolds. After formulating bilinear stochastic differential equations on general matrix Lie groups, we then consider interest rate models in which the short rate is defined as linear or quadratic functions of the state. Stochastic volatility is also augmented to these models in a way that respects the Riemannian manifold structure of symmetric positive-definite matrices. Methods for numerical integration, parameter identification, pricing, and other practical issues are addressed through examples.

An EZI Method to Reduce the Rank of a Correlation Matrix in Financial Modelling

Reducing the number of factors in a model by reducing the rank of a correlation matrix is a problem that often arises in finance, for instance in pricing interest rate derivatives with Libor market models. A simple iterative algorithm for correlation rank reduction is introduced, the eigenvalue zeroing by iteration, EZI, algorithm. Its convergence is investigated and extension presented with particular optimality properties. The performance of EZI is compared with those of other common methods. Different data sets are considered including empirical data from the interest rate market, different possible market cases and criteria, and a calibration case. The EZI algorithm is extremely fast even in computationally complex situations, and achieves a very high level of precision. From these results, the EZI algorithm for financial application has superior performance to the main methods in current use.

Defusing volatility explosions with complex analysis

Ever since the first Black-Scholes implied volatilities were calculated almost thirty years ago practitioners have known that volatility changes randomly through time. At every level, vega-hedging has been as important if not more so than delta-hedging. A variety of models have appeared over the years that have tried to model the random nature of volatility. Some of these have been described in standard textbooks like Hull or Wilmott, and others, but there have been few books that focus just on stochastic volatility.

Valuation of financial models with non-linear state spaces

A common assumption in valuation models for derivative securities is that the underlying state variables take values in a linear state space. We discuss numerical implementation issues in an interest rate model with a simple non-linear state space, formulating and comparing Monte Carlo, finite difference and lattice numerical solution methods. We conclude that, at least in low dimensional spaces, non-linear interest rate models may be viable.