## Preference-Free Option Pricing with Path-Dependent Volatility: A Closed-Form Approach

Steven L. Heston and Saikat Nandi
Federal Reserve Bank of Atlanta
Working Paper 98-20
December 1998

This paper shows how one can obtain a continuous-time preference-free option pricing model with a path-dependent volatility as the limit of a discrete-time GARCH model. In particular, the continuous-time model is the limit of a discrete-time GARCH model of Heston and Nandi (1997) that allows asymmetry between returns and volatility. For the continuous-time model, one can directly compute closed-form solutions for option prices using the formula of Heston (1993). Toward that purpose, we present the necessary mappings, based on Foster and Nelson (1994), such that one can approximate (arbitrarily closely) the parameters of the continuous-time model on the basis of the parameters of the discrete-time GARCH model. The discrete-time GARCH parameters can be estimated easily just by observing the history of asset prices.

Unlike most option pricing models that are based on the absence of arbitrage alone, a parameter related to the expected return/risk premium of the asset does appear in the continuous-time option formula. However, given other parameters, option prices are not at all sensitive to the risk premium parameter, which is often imprecisely estimated.

JEL classification: G13

Key words: volatility, path-dependent, options, closed-form

The authors thank Peter Ritchken for helpful comments. The views expressed here are those of the authors and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors' responsibility.

Please address questions regarding content to Steven L. Heston, Goldman Sachs & Company, Fixed Income Arbitrage, 85 Broad Street, New York, New York, 10004, 212/902-3074, steven.heston@gs.com; or Saikat Nandi, Federal Reserve Bank of Atlanta, 104 Marietta Street, NW, Atlanta, Georgia 30303, 404/498-7094, saikat.u.nandi@atl.frb.org.