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

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 b 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 oneperiod orthogonal hedging errors, P t≤T Yt(0)e r(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 P t≤T (Yt(y) + yt∆t) e r(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 b C0 = C0(y). A main result follows. Any arbitrage-free price, b C0, is just the price of a hedging portfolio (such as in a complete market), C0(0), plus a premium, b C0 − C0(0). That is, C0(0) is the price of the option’s payoff which can be spanned, and b C0 − C0(0) is the premium associated with the option’s payoff which cannot be spanned (and yields a contingent risk premium of P yt∆te r(T−t) at maturity). We study other applications of option-pricing theory as well. ∗I am grateful to seminar participants at Universidad Carlos III and CEMFI (Madrid), Nova (Lisboa), USC and UCLA (Los Angeles), the 14th Derivative Securities Conference (New York), and to Alejandro Balbás, Tony Bernardo, Manuel Domínguez, Ming Huang, Francis Longstaff, Fernando Zapatero, and, especially, to Eduardo Schwartz for comments. Any remaining errors are of course my own. Part of this research was done when I was a 2003/04 Visiting Scholar in the Finance Department, the Anderson School at UCLA. †Dpto. de Administración. Instituto Tecnológico Autónomo de México, ITAM. Col Tizapán-San Angel. 01000 México D.F., México. Phone: + 52-55-56284000 (X 3415). Fax: + 52-55-56284049. e-mail: [email protected]. 1 Working Paper 05-81 Business Economics Series 21 January 2005 Departamento de Economía de la Empresa Universidad Carlos III de Madrid Calle Madrid, 126 28903 Getafe (Spain) Fax (34) 91 624 9608