TAILIEUCHUNG - Optimal Investment with Derivative Securities

The Institute strongly urges the Commission to adjust the dollar thresholds to correct for this erosion. Section 413 of the Dodd-Frank Wall Street Reform and Consumer Protection Act required the Commission to exclude the value of private residences from the net worth calculations in the definition of “accredited investor,” which it did in 2011. That is certainly a positive first step towards modernizing the accredited investor definition, but much more needs to be done to respond to the concern that inflation and growth in wealth has inappropriately made a substantial number of investors eligible to invest in private. | Noname manuscript No. will be inserted by the editor Optimal Investment with Derivative Securities Aytac Ilhan1 Mattias Jonsson2 Ronnie Sircar3 1 Mathematical Institute University of Oxford Oxford OX1 3LB e-mail ilhan@ 2 Department of Mathematics University of Michigan Ann Arbor MI 48109-1109 e-mail mattiasj@ 3 Department of Operations Research Financial Engineering Princeton University E-Quad Prince- ton NJ 08544 e-mail sircar@ First draft January 2004 final version February 2005. Abstract We consider an investor who maximizes expected exponential utility of terminal wealth combining a static position in derivative securities with a traditional dynamic trading strategy in stocks. Our main result obtained by studying the strict concavity of the utility-indi erence price as a function of the static positions is that in a quite general incomplete arbitrage-free market there exists a unique optimal strategy for the investor. Key words utility maximization convex duality incomplete markets indi erence price JEL Classification G11 G13 Mathematics Subject Classification 2000 91B16 91B28 49N15 1 Introduction Consider an investor who tries to maximize expected utility from wealth at a certain time in the future from a self-financing trading strategy. If the instruments available to the investor are a finite number of risky assets stocks and one risk-free asset the money market account then the problem of finding the optimal strategy is classical and has been extensively studied we refer to 19 for a recent survey. In many cases however the investor may also be able to invest in derivative securities. As an example it is common for traders to buy options positions like butterfly spreads a particular combination of three call or three put options to indirectly trade volatility. Such investment opportunities are of particular interest in incomplete markets in which case the payoff cannot in general be replicated by a trading strategy in the

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