Les Clewlow

The Dynamics of the S&P 500 Implied Volatility Surface

This empirical study is motivated by the literature on “smile-consistent” arbitrage pricing with stochastic volatility. We investigate the number and shape of shocks that move implied volatility smiles and surfaces by applying Principal Components Analysis. Two components are identified under a variety of criteria. Subsequently, we develop a “Procrustes” type rotation in order to interpret the retained components. The results have implications for both option pricing and hedging and for the economics of option pricing.

Optimal delta-hedging under transactions costs

This paper examines the problem of delta-hedging portfolios of options under transactions costs by maximising expected utility (or minimising a loss function on the replication error). We extend the work of Hodges and Neuberger (1989) to study the optimal strategy under a general cost function with fixed and proportional costs. A computational procedure for solving this problem is described and we develop an efficient computational method for the case of proportional transaction costs. We examine the nature of the solution close to the expiry date and using simulation we compare the performance of the optimal strategies with other common strategies.

The Dynamics of Implied Volatility Surfaces

Motivated by the papers of Dupire (1992) and Derman and Kani (1997), we want to investigate the number of shocks that move the whole implied volatility surface, their interpretation and their correlation with percentage changes in the underlying asset. This work differs from Skiadopoulos, Hodges and Clewlow (1998) in which they looked at the dynamics of smiles for a given maturity bucket. We look at daily changes in implied volatilities under two different metrics: the strike metric and the moneyness metric. Since we are dealing with a three dimensional problem, we fix ranges of days to maturity, we pool them together and we apply the Principal Components Analysis (PCA) to the changes in implied volatilities over time across both the strike (moneyness) metric and the pooled ranges of days to maturity. We find similar results for both metrics. Two shocks explain the movements of the volatility surface, the first shock being interpreted as a shift, while the second one has a Z-shape. The sign of the correlation of the first shock with percentage changes in the underlying asset depends on the metric that we look at, while the sign is positive under both metrics regarding the second shock. The results suggest that the number of shocks, their interpretation and the sign of their correlation with changes in the underlying asset is the same for the whole implied volatility surface as it is for the smile corresponding to a fixed maturity bucket.

Monte Carlo Valuation of Interest Rate Derivatives Under Stochastic Volatility

CHRIS STRICKLAND is with the Financial Options Research Centre at the University of Warwick. 0th Longstaff and Schwartz [1992] (LS) and Fong and Vasicek [1992] (FV) have developed two-factor stochastic volatility models of the term strucB ture in whch the two factors are the short rate and the variance of the short rate. The motivation behind these models is recogrution that the assumption of perfect correlation between rates implicit in one-factor models is too restrictive and that the volatihty of interest rates changes randomly over time and is correlated with the level of interest rates. The Longstaff-Schwartz model is developed in a general equilibrium framework, and the processes for the interest rate and the volathty of the interest rate are endogenously determined. LS are able to derive closedform solutions to the prices of dxount bonds and also to price ducount bond options “analytically” within their framework.’ The model of Fong and Vasicek is perhaps more intuitive. They start by assuming plausible stochastic processes for the short rate and short rate volatility. In their article, however, FV describe pricing only discount bonds, and the solution they present requires complex (as opposed to real) algebra, posing potential problems for practical implementation. Since then, Selby and Strickland [1995] have detailed a series solution for the bond price that can be implemented easily and computationally efficiently in a programming language or spreadsheet, avoiding the need to deal with complex numbers. The contribution of this article to this literature is twofold. First, we extend the work of Fong and Vasicek and show how to price a wide variety of interest rate derivatives withm their framework. We begin by showing how to price hscount bond options, and

A volatility decomposition control variate technique for Monte Carlo simulations of Heath Jarrow Morton models

The aim of this work is to develop a simulation approach to the yield curve evolution in the Heath, Jarrow and Morton [Econometrica 60 (1) (1992) 77] framework. The stochastic quantities considered as affecting the forward rate volatility function are the spot rate and the forward rate. A decomposition of the volatility function into a Hull and White [Rev. Financial Stud. 3 (1990) 573] volatility and a remainder allows us to develop an efficient Control Variate Method that makes use of the closed form solution of the Hull and White call option. This technique considerably speeds up the simulation algorithm to approximate call option values with Monte Carlo simulation.

Interest Rate Derivatives in a Duffie and Kan Model with Stochastic Volatility: An Arrow-Debreu Pricing Approach

Simple analytical pricing formulae have been derived, by different authors and for several derivatives, under the Gaussian Langetieg (1980) model. The purpose of this paper is to use such exact Gaussian solutions in order to obtain approximate analytical pricing formulas under the most general stochastic volatility specification of the Duffie and Kan (1996) model. Using Gaussian Arrow-Debreu state prices, first order stochastic volatility approximate pricing solutions will be derived only involving one integral with respect to the time-to-maturity of the contingent claim under valuation. Such approximations will be shown to be much faster than the existing exact numerical solutions, as well as accurate.

Mathematical Programming and Risk Management of Derivative Securities

In this paper we discuss the use of mathematical programming techniques linear, dynamic, and goal programming to the problem of the risk management of derivative securities (also known as contingent claims or options). We focus on the problem of the risk management of complex or exotic options in the presence of real market imperfections such as transaction costs. The advantages and disadvantages of the various approaches which have appeared in the literature are discussed including a new approach which we are developing.

The Evaluation of Gas Swing Contracts with Regime Switching

A typical gas swing contract is an agreement between a supplier and a purchaser for the delivery of variable daily quantities of gas, between specified minimum and maximum daily limits, over a certain period at a specified set of contract prices. The main constraint of such an agreement that makes them difficult to value are that there is a minimum volume of gas (termed take-or-pay or minimum bill) for which the buyer will be charged at the end of the period (or penalty date), regardless of the actual quantity of gas taken. We propose a framework for pricing such swing contracts for an underlying gas forward price curve that follows a regime-switching process in order to better capture the volatility behavior in such markets. With the help of a recombining pentanomial tree, we are able to efficiently evaluate the prices of the swing contracts and find optimal daily decisions in different regimes. We also show how the change of regime will affect the decisions.

A note on the efficiency of the binomial option pricing model

We discuss the efficiency of the binomial option pricing model for single and multivariate American style options. We demonstrate how the efficiency of lattice techniques such as the binomial model can be analysed in terms of their computational cost. For the case of a single underlying asset the most efficient implementation is the extrapolated jump-back method: that is, to value a series of options with nested discrete sets of early exercise opportunities by jumping across the lattice between the early exercise times and then extrapolating from these values to the limit of a continuous exercise opportunity set. For the multivariate case, the most efficient method depends on the computational cost of the early exercise test. However, for typical problems, the most efficient method is the standard step-back method: that is, performing the early exercise test at each time step.

THE EVALUATION OF MULTIPLE YEAR GAS SALES AGREEMENT WITH REGIME SWITCHING

A typical gas sales agreement (GSA), also called a gas swing contract, is an agreement between a supplier and a purchaser for the delivery of variable daily quantities of gas, between specified minimum and maximum daily limits, over a certain number of years at a specified set of contract prices. The main constraint of such an agreement that makes them difficult to value is that in each gas year there is a minimum volume of gas (termed take-or-pay or minimum bill) for which the buyer will be charged at the end of the year (or penalty date), regardless of the actual quantity of gas taken. We propose a framework for pricing such swing contracts for an underlying gas forward price curve that follows a regime switching process in order to better capture the volatility behavior in such markets. With the help of a recombining pentanomial tree, we are able to efficiently evaluate the prices of the swing contracts, find optimal daily decisions and optimal yearly use of both the make-up bank and the carry forward bank at different regimes. We also show how the change of regime will affect the decisions.