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

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  • Alfredo Ibañez

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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.

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Bibliographic Info

Paper provided by Universidad Carlos III, Departamento de Economía de la Empresa in its series Business Economics Working Papers with number wb058121.

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Date of creation: Jan 2005
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Handle: RePEc:cte:wbrepe:wb058121

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  1. Ross, Stephen A, 1978. "A Simple Approach to the Valuation of Risky Streams," The Journal of Business, University of Chicago Press, vol. 51(3), pages 453-75, July.
  2. Detemple, Jerome & Sundaresan, Suresh, 1999. "Nontraded Asset Valuation with Portfolio Constraints: A Binomial Approach," Review of Financial Studies, Society for Financial Studies, vol. 12(4), pages 835-72.
  3. Bergman, Yaacov Z & Grundy, Bruce D & Wiener, Zvi, 1996. " General Properties of Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1573-1610, December.
  4. Luenberger, David G., 2002. "A correlation pricing formula," Journal of Economic Dynamics and Control, Elsevier, vol. 26(7-8), pages 1113-1126, July.
  5. T. Clifton Green & Stephen Figlewski, 1999. "Market Risk and Model Risk for a Financial Institution Writing Options," Journal of Finance, American Finance Association, vol. 54(4), pages 1465-1499, 08.
  6. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
  7. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-47.
  8. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
  9. Mark Rubinstein, 1976. "The Valuation of Uncertain Income Streams and the Pricing of Options," Bell Journal of Economics, The RAND Corporation, vol. 7(2), pages 407-425, Autumn.
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Cited by:
  1. Ibáñez, Alfredo, 2008. "Factorization of European and American option prices under complete and incomplete markets," Journal of Banking & Finance, Elsevier, vol. 32(2), pages 311-325, February.

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