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

Abstract

Consider a non-spanned security $C_{T}$ in an incomplete market. We study the risk/return tradeoffs generated if this security is sold for an arbitrage-free price $\hat{C_{0}}$ 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 $C_{0}(0)$ be its price. Self-financing implies that the residual risk is equal to the sum of the one-period orthogonal hedging errors, $\sum_{t\leq T} Y_{t}(0) e^{r(T -t)}$. To compensate the residual risk, a risk premium $y_{t}\Delta t$ is associated with every $Y_{t}$. Now let $C_{0}(y)$ be the price of the hedging portfolio, and $\sum_{t\leq T}(Y_{t}(y)+y_{t}\Delta t)e^{r(T-t)}$ 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 $\hat{C_{0}}-C_{0}(y)$. A main result follows. Any arbitrage-free price, $\hat{C_{0}}$, is just the price of a hedging portfolio (such as in a complete market), $C_{0}(0)$, plus a premium, $\hat{C_{0}}-C_{0}(0)$. That is, $C_{0}(0)$ is the price of the option's payoff which can be spanned, and $\hat{C_{0}}-C_{0}(0)$ is the premium associated with the option's payoff which cannot be spanned (and yields a contingent risk premium of sum $y_{t}\Delta$t$ e^{r(T-t)}$ at maturity). We study other applications of option-pricing theory as well.