The simulation of option prices with application to LIFFE options on futures

https://doi.org/10.1016/S0377-2217(98)00254-9Get rights and content

Abstract

We build a framework for modelling the deviation of observed option prices from the Black & Scholes prices. We use a flexible model for a density, a two sided switching Weibull, to capture the implied volatility. The model can be used to generate prices, it can take into account no-arbitrage bounds for option prices and is capable of generating such stylised facts as the smile effect. We apply this methodology to LIFFE options on German government bond 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:Oi=OiSi,E,τ=BSiSi,e−r(T−t),σ(adj)i.

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 θ=S,τ,r,σ 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 formOiθ=BSiθexp(Vi).However, 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.

References (22)

  • F. Black

    The pricing of commodity contracts

    Journal of Financial Economics

    (1976)
  • R. Engle et al.

    Implied ARCH models for option prices

    Journal of Econometrics

    (1992)
  • R.A. Jarrow et al.

    Approximate option valuation for arbitrary stochastic processes

    Journal of Financial Economics

    (1982)
  • Bahra, B., 1997. Implied risk-neutral probability density functions from option prices: Theory and application, Working...
  • Bahra, B., 1998. Implied risk-neutral probability density functions from option prices: A central bank perspective. In:...
  • D.S. Bates

    The crash of 87: Was it expected? The evidence from option markets

    Journal of Finance

    (1991)
  • D.T. Breeden et al.

    Prices of state-contingent claims implicit in option prices

    Journal of Business

    (1978)
  • M. Brenner et al.

    On measuring the risk of common stocks implied by options prices: A note

    Journal of Financial and Quantitative Analysis

    (1984)
  • G.D. Donney

    The comparative role of simulated versus hypothetical trading

    Derivatives Use, Trading and Regulation

    (1995)
  • Frey, R., 1997. Derivative asset analysis in models with level-dependent and stochastic volatility, Manuscript,...
  • Hoggard, T., 1995. The development of a continuous time GARCH option pricing model, M.Sc. Finance Thesis, Birkbeck...
  • Cited by (2)

    • A generalized European option pricing model with risk management

      2020, Physica A: Statistical Mechanics and its Applications
      Citation Excerpt :

      One approach involves assessing the capital adequacy requirements of a Special Purpose Vehicle as proposed through financial institution trading derivatives. Furthermore, banks may use option price simulators to test risk management systems that the Basle Capital Accord allows banks to use [3]. Artificial control has clearly violated the market-oriented principle, and real market behaviours may vary.

    View full text