OptionPricing in Incomplete Markets: The Hedging Portfolio Plus a Risk PremiumBased Recursive Approach
Posted: 31 Mar 2007
Date Written: February 2005
Abstract
Consider a nonspanned security C_T in an incomplete market. We study the risk/return tradeoffs generated if this security is sold for an arbitragefree price 'c0' and then hedged. We consider recursive oneperiod optimal selffinancing hedging strategies, a simple but tractable criterion. For continuous trading, diffusion processes, the oneperiod minimum variance portfolio is optimal. Let C_0(0) be its price. Selffinancing implies that the residual risk is equal to the sum of the oneperiod orthogonal hedging errors, \sum Y_t(0) . To compensate the residual risk, a risk premium y_t ?t is associated with every Y_t. Now let C_0(y) be the price of the hedging portfolio, and \sum (Y_t(y) + y_t ?t) is the total residual risk. Although not the same, the oneperiod 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=C_0(y). A main result follows. Any arbitragefree price, c0, is just the price of a hedging portfolio (such as in a complete market), C_0(0), plus a premium, c0C_0(0). That is, C_0(0) is the price of the option's payoff which can be spanned, and c0C_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 ?t at maturity). We study other applications of optionpricing theory as well.
Keywords: option pricing, incomplete markets, hedging, risk premium
JEL Classification: G12, G13, G22
Suggested Citation: Suggested Citation
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