Option-Pricing in Incomplete Markets: The Hedging Portfolio plus a Risk Premium-Based Recursive Approach

Abstract

Consider a non-spanned security CT in an incomplete market. We study the risk/return tradeoffs generated if this security is sold for an arbitrage-free price bC0 and then hedged. We consider recursive “one-period optimal” self-financing hedging strategies, a simple but tractable criterion. For continuous trading, diffusion processes, the one-period minimum variance portfolio is optimal. Let C0(0) be its price. Self-financing implies that the residual risk is equal to the sum of the oneperiod orthogonal hedging errors, Pt=T Yt(0)er(T -t). To compensate the residual risk, a risk premium yt.t is associated with every Yt. Now let C0(y) be the price of the hedging portfolio, and Pt=T (Yt(y) + yt.t) er(T -t) is the total residual risk. Although not the same, the one-period hedging errors Yt(0) and Yt(y) are orthogonal to the trading assets, and are perfectly correlated. This implies that the spanned option payoff does not depend on y. Let bC0 = C0(y). A main result follows. Any arbitrage-free price, bC0, is just the price of a hedging portfolio (such as in a complete market), C0(0), plus a premium, bC0 - C0(0). That is, C0(0) is the price of the option’s payoff which can be spanned, and bC0 - C0(0) is the premium associated with the option’s payoff which cannot be spanned (and yields a contingent risk premium of Pyt.ter(T -t) at maturity). We study other applications of option-pricing theory as well.