The simulation of option prices with application to LIFFE options on futures
Introduction
Whilst theoretical models of vanilla options can broadly be thought of as “Black and Scholes” models and their extensions, it is also well known that these models do not adequately describe the empirical data. This raises interesting issues in simulating option prices, a problem which we address in this paper.
The simulation of option prices can be a key input in important financial decision processes. It is one technique to assess the capital adequacy requirements for a Special Purpose Vehicle proposed by a finacial institution trading derivatives. Furthermore, an option price simulator could be employed by banks to test the risk management systems that the Basle Capital Accord allows banks to use. It could also be employed in the testing and preparation of trading models and systems, see Donney (1995). These issues are addressed by Jamshidian and Zhu (1997) who model FX portfolios involving credit risk. Although they consider derivatives in the form of swaps, they do not directly discuss options.
The purpose of this article is to discuss the simulation of option prices. It might be thought that this is a trivial matter as all that is required is to generate log-normal prices, apply the Black and Scholes (BS) formula and the problem is solved. This however does not take into account the role of implied volatility and the stochastic manner in which actual option prices differ from BS prices. Indeed it is this difference which is the phenomenon we are attempting to model. A second solution might be to use one of the many stochastic volatility option pricing models and generate prices through the solution of the appropriate partial differential equations. However, it is not clear that these models fit the data, particularly when they rest on unpalatable assumptions. For two excellent survey papers on option pricing with stochastic volatility see Hoggard (1995) and Frey (1997).
Our solution, which we present in Section 2, is to use statistical techniques to generate the option prices. We use a flexible model for a density, a two-sided switching Weibull, to capture the implied volatility. The reason we favour our model rather than some other model more closely linked to traditional option pricing is the ease with which we can simulate our data, the ability to impose no-arbitrage conditions on the simulated option prices and the relative ease of estimating parameters from data. The approach we outline concerns only European calls and puts but it does cover virtually all options traded on LIFFE (measured by volume of business) so it is of practical relevance. In Section 3we discuss the details of the estimation methodology. We apply our methodology in Section 4to LIFFE options on German government bond futures (Bund) and present hedging implications in Section 5. In Section 6we present our conclusions.
Section snippets
Generating option prices
The motivation for our procedure comes from the papers by Jarrow and Rudd, 1982, Jarrow and Wiggins, 1987 in which for a very wide class of distributions they show that it is possible to write the following relationship:
In Eq. (1), Oi is the actual option price with asset price Si, exercise price E and time to maturity τ, BSi(·) is the Black and Scholes price, σ(adj) is an adjusted volatility parameter and ϵt is an additive noise term.
Our proposal is for a
Estimation and specification testing
We use daily data from the London International Financial Options and Futures Exchange (LIFFE) for the period from September 1990 to February 1995. These concern call options on German government bond (Bund) futures. While there is a plethora of data, much of it is unreliable. For example, on data where there is little or no volume, the exchange uses artificially generated prices based on a system called Autoquote, which effectively uses the Black (1976) formula. We note that the Black and
Empirical results
In Section 3we have developed our estimation methodology. We estimate eight different versions of our model that correspond to the Weibull and Generalised Gamma distribution functions when truncated or not, both for overpriced (positive errors) and underpriced (negative errors) options. With regard to the practicalities of the estimation procedure, the nature of the data led us to estimate λ through an expression exp(λ) in the likelihood function but this does not affect the results. Empirical
Hedging and forecasting implications
It is interesting to examine6 how our modelling approach affects the “Greeks”; these are the first partial derivatives of the option price with respect to the associated factors as defined in the text and the second partial derivative with respect to S. Turning to Eq. (2)it is clear that if we do not assume any relationship between Vi and θ, these would have the formHowever, possibly of interest are the
Conclusion
In this paper we have outlined how to build a framework that models the deviation of observed option prices from Black and Scholes prices. Although empirically driven, the model allows us to (i) have asymmetries between under- and over-pricing, (ii) have asymmetries between puts and calls and (iii) generate prices that are subject to a variety of no-arbitrage constraints. Empirical results are encouraging, showing that the model is capable of capturing stylised facts in options markets. We find
Acknowledgements
The authors would like to thank Christian S. Pedersen, Steve Schaefer and two anonymous referees for helpful comments and suggestions.
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