tailieunhanh - Optimal investment in derivative securities

While the demand for options can arise from many sources, our focus on jumps stems from fundamental considerations regarding the nature of price pro- cesses in an arbitrage-free economy. Recently, Madan [19] has argued that all arbitrage-free continuous time price processes must be both semi-martingales and time-changed Brownian motion. Furthermore, it is argued that if the time change is not locally deterministic, then the resulting price process is discontinuous. If the price is modelled as a pure jump L´ evy process with infinity activity, then the need for a continuous martingale component is obviated, since there will be an infinite number of small jumps in any time interval | Finance Stochast. 5 33-59 2001 Finance and Stochastics Springer-Verlag 2001 Optimal investment in derivative securities Peter Carr1 Xing Jin2 Dilip B. Madan2 1 Banc of America Securities 9 West 57th Street 40th floor New York . 10019 USA e-mail carrp@ 2 Robert H. Smith School of Business Van Munching Hall University of Maryland College Park MD. 20742 USA e-mail xjin@ dbm@ Abstract. We consider the problem of optimal investment in a risky asset and in derivatives written on the price process of this asset when the underlying asset price process is a pure jump Levy process. The duality approach of Karatzas and Shreve is used to derive the optimal consumption and investment plans. In our economy the optimal derivative payoff can be constructed from dynamic trading in the risky asset and in European options of all strikes. Specific closed forms illustrate the optimal derivative contracts when the utility function is in the HARA class and when the statistical and risk-neutral price processes are in the variance gamma VG class. In this case we observe that the optimal derivative contract pays a function of the price relatives continuously through time. Key words Levy process market completeness stochastic duality option pricing variance gamma model JEL Classification G11 C61 Mathematics Subject Classification 1991 60G44 60J75 49L20 1 Introduction In a classic paper Merton 23 derived the optimal consumption and investment rules for investors maximizing the expected utility of their consumption in an economy consisting of a riskless asset and risky assets whose prices follow geometric Brownian motion. He showed that when investors have HARA hyperbolic absolute risk aversion utility functions then one can solve for the optimal consumption and investment rules in closed form. Merton s analysis Manuscript received November 1999 final version received February 2000 34 P. Carr et al. relied on Markov state processes and sought

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