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

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.