PRICING AND HEDGING DERIVATIVE SECURITIES IN MARKETS WITH UNCERTAIN VOLATILITIES

We present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely but is assumed instead to lie between two extreme values min and max These bounds could be inferred from extreme values of the implied volatilities of liquid options or from high low peaks in historical stock or option implied volatilities They can be viewed as de ning a con dence interval for future volatility values We show that the extremal non arbitrageable prices for the derivative asset which arise as the volatility paths vary in such a band can be described by a non linear PDE which we call the Black Scholes Barenblatt equation In this equation the pricing volatility is selected dynamically from the two extreme values min max according to the convexity of the value function A simple algorithm for solving the equation by nite di erencing or a trinomial tree is presented We show that this model captures the importance of diversi cation in managing derivatives positions It can be used systematically to construct e cient hedges using other derivatives in conjunction with the underlying asset y Courant Institute of Mathematical Sciences Mercer st New York NY Institute for Advanced Study Princeton NJ J P Morgan Securities New York NY The uncertain volatility model According to Arbitrage Pricing Theory if the market presents no arbitrage opportunities there exists a probability measure on future scenarios such that the price of any security is the expectation of its discounted cash ows Du e Such a probability is known as a mar tingale measure Harrison and Kreps or a pricing measure Determining the appropriate martingale measure associated with a sector of the security space e g the stock of a company and a riskless short term bond permits the valuation of any contingent claim based on these securities However pricing measures are often di cult to calculate precisely and there may exist more than one measure consistent with a given market It is useful to view the non uniqueness of pricing measures as re ecting the many choices for derivative asset prices that can exist in an uncertain economy For example option prices re ect the market s expectation about the future value of the underlying asset as well as its projection of future volatility Since this projection changes as the market reacts to new information implied volatility uctuates unpredictably In these circumstances fair option values and perfectly replicating hedges cannot be determined with certainty The existence of so called volatility risk in option trading is a concrete manifestation of market incompleteness This paper addresses the issue of derivative asset pricing and hedging in an uncertain future volatility environment For this purpose instead of choosing a pricing model that incorporates a complete view of the forward volatility as a single number or a predetermined function of time and price term structure of volatilities or even a stochastic process with given statistics we propose to operate under the less stringent assumption that that the volatility of future prices is restricted to lie in a bounded set but is otherwise undetermined For simplicity we restrict our discussion to derivative securities based on a single liquidly traded stock which pays no dividends over the contract s lifetime and assume a constant interest rate The basic assumption then reduces to postulating that under all admissible pricing mea sures future volatility paths will be restricted to lie within a band Accordingly we assume that the paths followed by future stock prices are It o processes viz dSt St t dZt t dt where t and t are non anticipative functions such that