Stewart D. Hodges

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.

How Large are the Benefits from Using Options

The paper explores the economic value of being able to span market outcomes through the use of options. We model an economy with a single risky asset. Consumption takes place at one date, corresponding to the horizon of all investors. Options on the consumption good are not redundant securities in the economy because volatility is uncertain. The model enables us to examine the benefits to investors of using options to optimize their investments. Within this model, the gains from the use of options appear to be relatively minor.

Simulating the Evolution of the Implied Distribution

Motivated by the implied stochastic volatility literature (Britten–Jones and Neuberger, forthcoming; Derman and Kani, 1997; Ledoit and Santa–Clara, 1998) this paper proposes a new and general method for constructing smile–consistent stochastic volatility models. The method is developed by recognising that option pricing and hedging can be accomplished via the simulation of the implied risk neutral distribution. We devise an algorithm for the simulation of the implied distribution, when the first two moments change over time. The algorithm can be implemented easily, and it is based on an economic interpretation of the concept of mixture of distributions. It can also be generalised to cases where more complicated forms for the mixture are assumed.

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.

Volatility Cones and Their Sampling Properties

Determining the volatility of the underlying asset is perhaps the single most important issue in practical option pricing. Many forecasting techniques exist, using historical returns data, implied volatility parameters from observed option prices, normal and non-normal probability distributions and more complex stochastic processes, non-market information, and more. But one of the most important aspects of the problem is seldom formally examined: estimation error. Even under ideal conditions, both measured volatility in a historical sample of returns and future realized volatility over an options lifetime are subject to sampling error. Shorter samples have larger estimation error. A “volatility cone” is a plot of the range of volatilities within a fixed probability band around the true parameter, as a function of sample length. In this article, Hodges and Tompkins examine volatility cones under different assumptions about the true returns process. One important contribution is a bias correction for estimation using an overlapping data sample that produces unbiased estimates and a substantial gain in efficiency. They then apply the analysis empirically to S&P 500 index futures returns and conclude that the observed volatility behavior is consistent with a stochastic volatility process that has fat-tailed innovations.

Term-Structure Slope Risk: Convexity Revisited

The article provides a new and practical approach to measuring the risk of changes in the slope, and curvature of the term structure, as well as its level. This is related to the principal components commonly found in term structure movements, but obviates the need to estimate them. The technique may be applied to construct hedges and to reconcile how value changes have stemmed from term structure movements. The article establishes how the new measures are related to the existing measures of duration and convexity, and to the (generalized) moments of the (present value weighted) distribution of cash flow dates.

Communicating the risk of injury in schoolboy rugby: using Poisson probability as an alternative presentation of the epidemiology

Background The communication of injury risk in rugby and other sports is underdeveloped and parents, children and coaches need to be better informed about risk. Method A Poisson distribution was used to transform population based incidence of injury into average probabilities of injury to individual players. Results The incidence of injury in schoolboy rugby matches range from 7 to 129.8 injuries per 1000 player-hours; these rates translate to average probabilities of injury to a player of between 12% and 90% over a season. Conclusion Incidence of injury and average probabilities of injury over a season should be published together in all future epidemiological studies on school rugby and other sports. More research is required on informing and communicating injury risks to parents, staff and children and how it affects monitoring, decision making and prevention strategies.

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.