General Black-Scholes models accounting for increased market volatility from hedging strategies
Increases in market volatility of asset prices have been observed and analysed in recent years and their cause has generally been attributed to the popularity of portfolio insurance strategies for derivative securities. The basis of derivative pricing is the Black-Scholes model and its use is so extensive that it is likely to influence the market itself. In particular it has been suggested that this is a factor in the rise in volatilities. A class of pricing models is presented that accounts for the feedback effect from the Black-Scholes dynamic hedging strategies on the price of the asset, and from there back onto the price of the derivative. These models do predict increased implied volatilities with minimal assumptions beyond those of the Black-Scholes theory. They are characterized by a nonlinear partial differential equation that reduces to the Black-Scholes equation when the feedback is removed. We begin with a model economy consisting of two distinct groups of traders: reference traders who are the majority investing in the asset expecting gain, and programme traders who trade the asset following a Black-Scholes type dynamic hedging strategy, which is not known a priori, in order to insure against the risk of a derivative security. The interaction of these groups leads to a stochastic process for the price of the asset which depends on the hedging strategy of the programme traders. Then following a Black-Scholes argument, we derive nonlinear partial differential equations for the derivative price and the hedging strategy. Consistency with the traditional Black-Scholes model characterizes the class of feedback models that we analyse in detail. We study the nonlinear partial differential equation for the price of the derivative by perturbation methods when the programme traders are a small fraction of the economy, by numerical methods, which are easy to use and can be implemented efficiently, and by analytical methods. The results clearly support the observed increasing volatility phenomenon and provide a quantitative explanation for it.
Volume (Year): 5 (1998)
Issue (Month): 1 ()
|Contact details of provider:|| Web page: http://www.tandfonline.com/RAMF20|
|Order Information:||Web: http://www.tandfonline.com/pricing/journal/RAMF20|
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.:
- Bick, Avi, 1987. "On the Consistency of the Black-Scholes Model with a General Equilibrium Framework," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(03), pages 259-275, September.
- Gerard Gennotte and Hayne Leland., 1989.
"Market Liquidity, Hedging and Crashes,"
Research Program in Finance Working Papers
RPF-192, University of California at Berkeley.
- Frey, Rüdiger & Alexander Stremme, 1995. "Market Volatility and Feedback Effects from Dynamic Hedging," Discussion Paper Serie B 310, University of Bonn, Germany.
- Brennan, Michael J & Schwartz, Eduardo S, 1989. "Portfolio Insurance and Financial Market Equilibrium," The Journal of Business, University of Chicago Press, vol. 62(4), pages 455-72, October.
- Frey, Rüdiger, 1996. "The Pricing and Hedging of Options in Finitely Elastic Markets," Discussion Paper Serie B 372, University of Bonn, Germany.
- Grossman, Sanford J, 1988.
"An Analysis of the Implications for Stock and Futures Price Volatility of Program Trading and Dynamic Hedging Strategies,"
The Journal of Business,
University of Chicago Press, vol. 61(3), pages 275-98, July.
- Sanford J. Grossman, 1987. "An Analysis of the Implications for Stock and Futures Price Volatility of Program Trading and Dynamic Hedging Strategies," NBER Working Papers 2357, National Bureau of Economic Research, Inc.
- Jarrow, Robert A., 1994. "Derivative Security Markets, Market Manipulation, and Option Pricing Theory," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 29(02), pages 241-261, June.
- 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.
- Hans Föllmer & Martin Schweizer, 1993. "A Microeconomic Approach to Diffusion Models For Stock Prices," Mathematical Finance, Wiley Blackwell, vol. 3(1), pages 1-23.
- Gregory Duffee & Paul Kupiec & Patricia White, 1990. "A primer on program trading and stock price volatility: a survey of the issues and the evidence," Finance and Economics Discussion Series 109, Board of Governors of the Federal Reserve System (U.S.).
- 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.
- 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.
When requesting a correction, please mention this item's handle: RePEc:taf:apmtfi:v:5:y:1998:i:1:p:45-82. 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.