Factorization of European and American Option Prices under Complete and Incomplete Markets

Abstract

In a standard option-pricing model, with continuous-trading and diffusion processes, this paper shows that the price of one European - style option can be factorized into two intuitive components: One robust, X_0, which is priced by arbitrage, and a second, Pi _0, which depends on a risk orthogonal to the traded securities. This result implies the following: 1) In an incomplete market, these parts represent the price of a hedging portfolio, which is unique, and a premium, which depends only on the risk premiums associated with the residual risk, respectively. 2) In a complete market, it allows factoring the contribution of the different sources of risk to the final option price. For example, in a stochastic volatility model, we can quantify the impact on the option price of volatility risk relative to market risk, Pi _0 and X_0, respectively. Hence, certain misspricings in option markets can be directly related to the premium, Pi _0. 3) Moreover, these results extend to American securities, which have a third component -an additional early-exercise premium.