General Black-Scholes models accounting for increased market volatility from hedging strategies

K. Ronnie Sircar and George Papanicolaou

Applied Mathematical Finance, 1998, vol. 5, issue 1, 45-82

Abstract: 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.

Keywords: Black-scholes Model; Dynamic Hedging; Feedback Effects; Option Pricing; Volatility, (search for similar items in EconPapers)
Date: 1998
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (30)

Downloads: (external link)
http://www.tandfonline.com/doi/abs/10.1080/135048698334727 (text/html)
Access to full text is restricted to subscribers.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:taf:apmtfi:v:5:y:1998:i:1:p:45-82

Ordering information: This journal article can be ordered from
http://www.tandfonline.com/pricing/journal/RAMF20

DOI: 10.1080/135048698334727

Access Statistics for this article

Applied Mathematical Finance is currently edited by Professor Ben Hambly and Christoph Reisinger

More articles in Applied Mathematical Finance from Taylor & Francis Journals
Bibliographic data for series maintained by Chris Longhurst ().