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

Consider a non-spanned security C_{T} in an incomplete market. We study the risk/return trade-offs generated if this security is sold for an arbitrage-free price C₀ 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 one-period orthogonal hedging errors, ∑_{t≤T}Y_{t}(0)e^{r(T-t)}. To compensate the residual risk, a risk premium y_{t}Δt is associated with every Y_{t}. Now let C₀(y) be the price of the hedging portfolio, and ∑_{t≤T}(Y_{t}(y)+y_{t}Δt)e^{r(T-t)} is the total residual risk. Although not the same, the one-period hedging errors Y_{t}(0) and Y_{t}(y) are orthogonal to the trading assets, and are perfectly correlated. This implies that the spanned option payoff does not depend on y. Let C₀=C₀(y). A main result follows. Any arbitrage-free price, C₀, is just the price of a hedging portfolio (such as in a complete market), C₀(0), plus a premium, C₀-C₀(0). That is, C₀(0) is the price of the option's payoff which can be spanned, and C₀-C₀(0) is the premium associated with the option's payoff which cannot be spanned (and yields a contingent risk premium of ∑y_{t}Δte^{r(T-t)} at maturity). We study other applications of option-pricing theory as well