Stochastic volatility, smile & asymptotics
We consider the pricing and hedging problem for options on stocks whose volatility is a random process. Traditional approaches, such as that of Hull and White, have been successful in accounting for the much observed smile curve, and the success of a large class of such models in this respect is guaranteed by a theorem of Renault and Touzi, for which we present a simplified proof. Motivated by the robustness of the smile effect to specific modelling of the unobserved volatility process, we introduce a methodology that does not depend on a particular stochastic volatility model. We start with the Black-Scholes pricing PDE with a random volatility coefficient. We identify and exploit distinct time scales of fluctuation for the stock price and volatility processes yielding an asymptotic approximation that is a Black-Scholes type price or hedging ratio plus a Gaussian random variable quantifying the risk from the uncertainty in the volatility. These lead us to translate volatility risk into pricing and hedging bands for the derivative securities, without needing to estimate the market's value of risk or to specify a parametric model for the volatility process. For some special cases, we can give explicit formulas. We outline how this method can be used to save on the cost of hedging in a random volatility environment, and run simulations demonstrating its effectiveness. The theory needs estimation of a few statistics of the volatility process, and we run experiments to obtain approximations to these from simulated stock price and smile curve data.
If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.
Volume (Year): 6 (1999)
Issue (Month): 2 ()
|Contact details of provider:|| Web page: http://www.tandfonline.com/RAMF20|
|Order Information:||Web: http://www.tandfonline.com/pricing/journal/RAMF20|
References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Stephen J. Taylor, 1994. "Modeling Stochastic Volatility: A Review And Comparative Study," Mathematical Finance, Wiley Blackwell, vol. 4(2), pages 183-204.
- Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-52.
- N. Hofmann & E. Platen & M. Schweizer, 1992.
"Option Pricing under Incompleteness and Stochastic Volatility,"
Discussion Paper Serie B
209, University of Bonn, Germany.
- Norbert Hofmann & Eckhard Platen & Martin Schweizer, 1992. "Option Pricing Under Incompleteness and Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 2(3), pages 153-187.
- MacBeth, James D & Merville, Larry J, 1979. "An Empirical Examination of the Black-Scholes Call Option Pricing Model," Journal of Finance, American Finance Association, vol. 34(5), pages 1173-86, December.
- Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
- Ball, Clifford A. & Roma, Antonio, 1994. "Stochastic Volatility Option Pricing," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 29(04), pages 589-607, December.
- Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
- Mark. B. Garman., 1976. "A General Theory of Asset Valuation under Diffusion State Processes," Research Program in Finance Working Papers 50, University of California at Berkeley.
- Merton, Robert C., 1976.
"Option pricing when underlying stock returns are discontinuous,"
Journal of Financial Economics,
Elsevier, vol. 3(1-2), pages 125-144.
- Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
- K. Ronnie Sircar & George Papanicolaou, 1998. "General Black-Scholes models accounting for increased market volatility from hedging strategies," Applied Mathematical Finance, Taylor & Francis Journals, vol. 5(1), pages 45-82.
- Latane, Henry A & Rendleman, Richard J, Jr, 1976. "Standard Deviations of Stock Price Ratios Implied in Option Prices," Journal of Finance, American Finance Association, vol. 31(2), pages 369-81, May.
- Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
- Rubinstein, Mark, 1994. " Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
- Eric Renault & Nizar Touzi, 1996. "Option Hedging And Implied Volatilities In A Stochastic Volatility Model," Mathematical Finance, Wiley Blackwell, vol. 6(3), pages 279-302.
- Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
- Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
- Marc Romano & Nizar Touzi, 1997. "Contingent Claims and Market Completeness in a Stochastic Volatility Model," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 399-412.
- M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
- Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
- Canina, Linda & Figlewski, Stephen, 1993. "The Informational Content of Implied Volatility," Review of Financial Studies, Society for Financial Studies, vol. 6(3), pages 659-81.
When requesting a correction, please mention this item's handle: RePEc:taf:apmtfi:v:6:y:1999:i:2:p:107-145. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Michael McNulty)
If references are entirely missing, you can add them using this form.