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Option pricing under model and parameter uncertainty using predictive densities

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Abstract

The theoretical price of a financial option is given by the expectation of its discounted expiry time payoff. The computation of this expectation depends on the density of the value of the underlying instrument at expiry time. This density depends on both the parametric model assumed for the behaviour of the underlying, and the values of parameters within the model, such as volatility. However neither the model, nor the parameter values are known. Common practice when pricing options is to assume a specific model, such as geometric Brownian Motion, and to use point estimates of the model parameters, thereby precisely defining a density function.

We explicitly acknowledge the uncertainty of model and parameters by constructing the predictive density of the underlying as an average of model predictive densities, weighted by each model's posterior probability. A model's predictive density is constructed by integrating its transition density function by the posterior distribution of its parameters. This is an extension to Bayesian model averaging. Sampling importance-resampling and Monte Carlo algorithms implement the computation. The advantage of this method is that rather than falsely assuming the model and parameter values are known, inherent ignorance is acknowledged and dealt with in a mathematically logical manner, which utilises all information from past and current observations to generate and update option prices. Moreover point estimates for parameters are unnecessary. We use this method to price a European Call option on a share index.

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References

  • Avellaneda M., Levy A., and Paras A. 1995. Pricing and hedging deriva-tive securities in markets with uncertain volatilities. Applied Mathematical Finance 2: 73-88.

    Google Scholar 

  • Black F. and Scholes M. 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81: 637-654.

    Google Scholar 

  • Carlin B.P. and Louis T.A. 1996. Bayes and Empirical Bayes Methods for Data Analysis. Chapman and Hall, London.

    Google Scholar 

  • Cox D.R. and Hinkley D.V. 1974. Theoretical Statistics. Chapman and Hall.

  • Cox J. and Ross S. 1976. The value of options for alternative stochastic processes. Journal of Financial Economics January 3: 145-166.

    Google Scholar 

  • Draper D. 1995. Assessment and propagation of model uncertainty. Journal of the Royal Statistical Society Series B 57(1): 45-97.

    Google Scholar 

  • Emanuel D.C. and MacBeth J.D. 1982. Further results on the constant elasticity of variance call option pricing model. Journal of Financial and Quantitative Analysis 27(4): 533-554.

    Google Scholar 

  • Gilks W.R., Richardson S., and Spiegelhalter D.J. 1996. Introducing Markov Chain Monte Carlo. In: Gilks W.R., Richardson S., and Spiegelhalter D.J. (Eds.), Markov Chain Monte Carlo in Practice. Chapman and Hall, London.

    Google Scholar 

  • Harrison M. and Kreps D. 1979. Martingales and arbitrage in multi-period securities markets. Journal of Economic Theory June 20: 381-408.

    Google Scholar 

  • Harrison M. and Pliska S. 1981. Martingales and stochastic integrals in theory of continutous trading. Stochastic Processes and Their Applications 11: 313-316.

    Google Scholar 

  • Hoetig J., Madigan D., and Raftery A. 1999. Bayesian model averaging: A tutorial. Statistical Science 14(4): 382-417.

    Google Scholar 

  • Hull J. 1993. Options, Futures, and Other Derivative Securities. Prentice-Hall.

  • Jacquier E., Polson N.G., and Rossi P.E. 1994. Bayesian analysis of stochastic volatility models. Journal of Business and Economic Statistics 12(4): 371-389.

    Google Scholar 

  • Oksendal B. 1998. Stochastic Differential Equations, 5th edn. Springer-Verlag.

  • Pedersen A.R. 1995. A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations.Scandinavian Journal of Statistics 22: 55-71.

    Google Scholar 

  • Pitt M. and Shephard N. 1999. Filtering via simulation: Auxiliary particle filters. Journal of the American Statistical Association 94: 590-599.

    Google Scholar 

  • Schroder M. 1989. Computing the constant elasticity of variance option pricing formula. The Journal of Finance 44(1): 211-219.

    Google Scholar 

  • Silverman B.W. 1986. Density Estimation for Statistics and Data Analysis. Chapman and Hall.

  • Smith A.F.M. and Gelfand A.E. 1992. Bayesian statistics without tears: A sampling-resampling perspective. American Statistician 46: 84-88.

    Google Scholar 

  • The Economist Newspaper Ltd. 1998.

  • Wasserman L. 1997. Bayesian model selection and model averaging. Carnegie Mellon University Department of Statistics, Technical Report.

Download references

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Bunnin, F.O., Guo, Y. & Ren, Y. Option pricing under model and parameter uncertainty using predictive densities. Statistics and Computing 12, 37–44 (2002). https://doi.org/10.1023/A:1013116204872

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  • DOI: https://doi.org/10.1023/A:1013116204872

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