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Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application

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Author Info
Scott, Louis O.
Abstract

In this paper, we examine the pricing of European call options on stocks that have variance rates that change randomly. We study continuous time diffusion processes for the stock return and the standard deviation parameter, and we find that one must use the stock and two options to form a riskless hedge. The riskless hedge does not lead to a unique option pricing function because the random standard deviation is not a traded security. One must appeal to an equilibrium asset pricing model to derive a unique option pricing function. In general, the option price depends on the risk premium associated with the random standard deviation. We find that the problem can be simplified by assuming that volatility risk can be diversified away and that changes in volatility are uncorrelated with the stock return. The resulting solution is an integral of the Black-Scholes formula and the distribution function for the variance of the stock price. We show that accurate option prices can be computed via Monte Carlo simulations and we apply the model to a set of actual prices.

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Article provided by Cambridge University Press in its journal Journal of Financial and Quantitative Analysis.

Volume (Year): 22 (1987)
Issue (Month): 04 (December)
Pages: 419-438
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Handle: RePEc:cup:jfinqa:v:22:y:1987:i:04:p:419-438_01

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  1. Peter A. Abken & Saikat Nandi, 1996. "Options and volatility," Economic Review, Federal Reserve Bank of Atlanta, issue Dec, pages 21-35. [Downloadable!]
  2. Robert F. Engle & Joshua V. Rosenberg, 1995. "GARCH Gamma," NBER Working Papers 5128, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
    Other versions:
  3. Torben G. Andersen & Tim Bollerslev, 1997. "Answering the Critics: Yes, ARCH Models Do Provide Good Volatility Forecasts," NBER Working Papers 6023, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
  4. David S. Bates, 1993. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in thePHLX Deutschemark Options," NBER Working Papers 4596, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
  5. Frank Milne & Dilip Madan, 1991. "Option Pricing With V. G. Martingale Components," Working Papers 1159, Queen's University, Department of Economics. [Downloadable!]
  6. Jun Yu, 2006. "Temporal Aggregation and Risk-Return Relation," Working Papers 01-2007, Singapore Management University, School of Economics. [Downloadable!]
    Other versions:
  7. Jeroen V.K. Rombouts & Lars Stentoft, 2009. "Bayesian Option Pricing Using Mixed Normal Heteroskedasticity Models," CREATES Research Papers 2009-07, School of Economics and Management, University of Aarhus. [Downloadable!]
    Other versions:
  8. Pindyck, Robert S. & Solimano, Andres, 1993. "Economic instability and aggregate investment," Policy Research Working Paper Series 1148, The World Bank. [Downloadable!]
    Other versions:
  9. Bertholon, H. & Monfort, A. & Pegoraro, F., 2007. "Pricing and Inference with Mixtures of Conditionally Normal Processes," Documents de Travail 188, Banque de France. [Downloadable!]
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  10. David S. Bates, 1999. "Financial Markets' Assessment of EMU," NBER Working Papers 6874, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
  11. Jeroen V.K. Rombouts & Lars Stentoft, 2009. "Bayesian Option Pricing Using Mixed Normal Heteroskedasticity," Cahiers de recherche 0926, CIRPEE. [Downloadable!]
  12. Mercedes Arriojas & Yaozhong Hu & Salah-Eldin Mohammed & Gyula Pap, 2006. "A Delayed Black and Scholes Formula I," Quantitative Finance Papers math/0604640, arXiv.org. [Downloadable!]
  13. Darsinos, T. & Satchell, S.E., 2001. "Bayesian Forecasting of Options Prices: A Natural Framework for Pooling Historical and Implied Volatiltiy Information," Cambridge Working Papers in Economics 0116, Faculty of Economics, University of Cambridge. [Downloadable!]
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