# Relative Pricing of Options with Stochastic
Volatility

## Olivier Ledoit, Pedro Santa-Clara and Shu Yan

### Abstract

This paper offers a new approach for pricing options on assets with
stochastic volatility. We start by taking as given the prices of a few simple,
liquid European options. More specifically, we take as given the
"surface" of Black-Scholes implied volatilities for European options
with varying strike prices and maturities. We show that the Black-Scholes
implied volatilities of at-the-money options converge to the underlying asset's
instantaneous (stochastic) volatility as the time to maturity goes to zero.
Intuitively, one instant before the option expires, the effect of stochastic
volatility on the option price is negligible. Then, the Black-Scholes formula
accurately prices the option and, as a result, its implied volatility
corresponds to the instantaneous volatility of the underlying asset.

We model the stochastic processes followed by the implied volatilities of
options of all maturities and strike prices as a joint diffusion with the stock
price. In order for no arbitrage opportunities to exist in trading the stock and
these options, the drift of the processes followed by the implied volatilities
is constrained in such a way that it is fully characterized by the volatilities
of the implied volatilities. Finally, we use the 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. Our approach simply requires as inputs the stock price and the implied
volatilities at the time the exotic option is to be priced, as well as an
estimate of the volatilities of the implied volatilities.

UCLA
Finance Working Paper #9-98

Download full paper (Acrobat PDF - 236KB)

Back to research page

Back to home page