Dynamic Hedging with Stochastic Differential Utility

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References [1] ADLER, Michael & DETEMPLE, Jerome B. On the Optimal Hedge of a Nontraded Cash Position. The Journal of Finance, vol.43,n. o 1, p.p. 143-153, 1988. [2] AIT-SAHALIA, Yacine, HANSEN, Lars P. & SCHEINKMAN, José A. Operator Methods for Continuous-Time Markov Models. Manuscript, University of Chicago, 2002. [3] ANDERSON, Ronald W. & DANTHINE, Jean-P. Cross-Hedging. Journal of Political Economy, vol.89,p.p. 1182-1196, 1981. [4] ANDERSON, Ronald W. & DANTHINE, Jean-P. The Time Pattern of Hedging and the Volatility of Futures Prices. Review of Economic Studies, vol.50,p.p. 249-266, 1983. [5] BREEDEN, D. Futures Markets and Commodity Options: Hedging and optimality in incomplete markets. Journal of Economic Theory, vol. 32, p.p. 275-300, 1984. [6] DUFFIE, Darrell. Futures Markets. Englewood Cliffs: Prentice Hall, 1989. [7] DUFFIE, Darrell & EPSTEIN, Larry G. Stochastic Differential Utility. Econometrica, vol.60,n. o 2, p.p. 353-394, 1992. [8] DUFFIE, Darrell & JACKSON, Matthew O. Optimal Hedging and Equilibrium in a Dynamic Futures Market. Journal of Economics Dynamics and Control, vol. 14, p.p. 21-33, 1990. 30
[9] EPSTEIN, Larry G. & ZIN, Stanley E. Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A theoretical framework. Econometrica, vol.57,n. o 4, p.p. 937-969, 1989. [10] ETHIER, Stwart N. & KURTZ, Thomas G. Markov Processes: Characterization and convergence. New York: Wiley, 1986. [11] HANSEN, Lars P. & SCHEINKMAN, José A. Semigroup Pricing. Manuscript, University of Chicago, 2002. [12] HO, Thomas S. Y. Intertemporal Commodity Futures Hedging and Production Decision. The Journal of Finance, vol. 39, n. o 2, p.p. 351- 376, 1984. [13] KRYLOV, Nicolai V. Controlled Diffusion Processes. New York: Springer-Verlag, 1980. [14] OKSENDAL, Berndt. Stochastic Differential Equations, 4. th ed. New York: Springer-Verlag, 1995. AppendixA-Infinitesimal Generators Here we present some technical concepts that we have used along the paper. First we present the infinitesimal generator approach, following closely Ait-Sahalia, Hansen and Scheinkman (2002). The HJB equation is easier to derive using this approach. Then we present the stochastic differential utility, accordingtotheauthors already cited. 31
Page 1 and 2: Dynamic Hedging with Stochastic Dif
Page 3 and 4: 1 INTRODUCTION In this paper we aim
Page 5 and 6: and losses are marked to market in
Page 7 and 8: time, whose prices are given by a K
Page 9 and 10: the von Newmann-Morgenstern index,
Page 11 and 12: 1 k(J) ¡ ¢ J 0 2 z t ΣJ zt =0,wh
Page 13 and 14: Under our assumptions about the uti
Page 15 and 16: Looking at the second part inside t
Page 17 and 18: equation of this case is a little b
Page 19 and 20: The next statement is a direct cons
Page 21 and 22: optimal hedging. 19 In order to sha
Page 23 and 24: Proof. Observe that · J w = −γE
Page 25 and 26: And substituting this expression in
Page 27 and 28: Lemma 1 Given the assumptions about
Page 29 and 30: Second, suppose u(w) = w1−γ , γ
Page 31: prices, mean-variance utility, and
Page 35 and 36: operator Z T t φ (x) =E [φ (y t )
Page 37 and 38: (2002). The domain of this second o
Page 39 and 40: In this paper, our expected-utility
Page 41 and 42: Now, let us analyze the RHS: RHS =
Page 43 and 44: Degenerated Additive Utility We app
Page 45 and 46: Now we follow the same procedure as
Page 47 and 48: 0 < J x = ¡ e r(T −t) − 1 ¢ E
Page 49: And the same holds for E (gq) . Sin

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