Relative Pricing of Options with Stochastic Volatility

University of California-Los Angeles Finance Working Paper 9-98

11 Pages Posted: 1 Sep 1998

See all articles by Olivier Ledoit

Olivier Ledoit

University of Zurich - Department of Economics

Pedro Santa-Clara

New University of Lisbon - Nova School of Business and Economics; National Bureau of Economic Research (NBER); Centre for Economic Policy Research (CEPR)

Date Written: March 1998

Abstract

This paper offers a new approach for pricing options on assets with stochastic volatility. We start by constructing the "surface" of Black-Scholes implied volatilities for (readily observable) liquid, European call options with varying strike prices and maturities. Then, we show that the implied volatility of an at-the-money call option with time-to-maturity going to zero is equal to the underlying asset's instantaneous (stochastic) volatility. We then model the stochastic processes followed by the implied volatilities of options of all maturities and strike prices jointly with the stock price, and find a no-arbitrage condition that their drift must satisfy. Finally, we use the resulting arbitrage-free joint process for the stock price and its volatility to price other derivatives, such as standard but illiquid options as well as exotic options using numerical methods.

The great advantage of our approach is that, when pricing these other derivatives, we are secure in the knowledge that the model values the hedging instruments - namely the stock and the simple, liquid options - consistently with the market.

Our approach can easily be extended to allow for stochastic interest rates and a stochastic dividend yield, which may be particularly relevant to the pricing of currency and commodity options. We can also extend our model to price bond options when the term structure of interest rates has stochastic volatility.

JEL Classification: G13

Suggested Citation

Ledoit, Olivier and Santa-Clara, Pedro, Relative Pricing of Options with Stochastic Volatility (March 1998). University of California-Los Angeles Finance Working Paper 9-98. Available at SSRN: https://ssrn.com/abstract=121257 or http://dx.doi.org/10.2139/ssrn.121257

Olivier Ledoit

University of Zurich - Department of Economics ( email )

Wilfriedstrasse 6
Z├╝rich, 8032
Switzerland

Pedro Santa-Clara (Contact Author)

New University of Lisbon - Nova School of Business and Economics ( email )

Lisbon
Portugal

HOME PAGE: http://docentes.fe.unl.pt/~psc/

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