Option Pricing with Stochastic Volatility: Information-Time vs. Calendar-Time

Abstract

Empirical evidence has shown that subordinated processes represent well the price changes of stocks and futures. Using either transaction counts or trading volume as a proxy for information arrival, it supports the contention that volatility is stochastic in calendar-time because of random information arrival, and thus becomes stationary in information-time. This contention has also been supported later in theoretical models. In this paper we investigate the implication of this contention to option pricing. First we price the option in calendar-time where the return of the underlying asset follows a jump subordinated process. We extend Rubinstein's (Rubinstein, M. 1976. The valuation of uncertain income streams and the valuation of options. Bell J. Econom. Management Sci. 7 407–425.) and Ross's (Ross, S. 1989a. Information and volatility: The no-arbitrage martingale approach to timing and resolution irrelevancy. Finance 44 1–17.) martingale valuation methodology to incorporate the pricing of volatility risk. The resulting equilibrium formula requires estimating seven parameters upon implementation. We then make a stochastic time change, from calendar-time to information-time, in order to obtain a stationary underlying asset return process to price the option. We find that the isomorphic option has random maturity because the number of information arrivals prior to the option's calendar-time expiration date is random. We value the option using Dynkin's (Dynkin, E. B. 1965. Markov Processes, Vols. I and II. Springer-Verlag, Berlin and New York.) version of the Feynman-Kac formula that allows for a random terminal date. The resulting information-time formula requires estimating only one additional parameter compared to the Black-Scholes's in practical application. In this regard, the time change has reduced the computational complexity of the option pricing problem. Simulations show that the formula may outperform the Black-Scholes (Black, F., M. Scholes. 1973. The pricing of options and corporate liabilities. Political Econom. 81 637–659.) and Merton (Merton, Robert C. 1976a. Option pricing when underlying stock returns are discontinuous. Financial Econom. 3 125–143.) models in pricing currency options. As a first attempt to derive valuation relationships in the information-time economy, this investigation may suggest that the information-time approach is a functional alternative to the current calendar-time norm. It is especially suitable for deriving “volatility-free” portfolio insurance strategies.