# Utility Maximization When Shorting American Options

To appear in SIAM Journal on Financial Mathematics

27 Pages Posted: 15 Oct 2019 Last revised: 6 Oct 2020

Date Written: October 5, 2019

### Abstract

Abstract. An investor initially shorts a divisible American option f and dynamically trades stock S to maximize her expected utility. The investor faces the uncertainty of the exercise time of f, yet by observing the exercise time she would adjust her dynamic trading strategy accordingly. We thus investigate the robust utility maximization problem V (x) = sup(H,c) infη E[U(x+H·S−c(η(f)−p))], where H is the dynamic trading strategy for S, c represents the amount of f the investor initially shorts, η is the liquidation strategy for f, and p is the initial price of f. We mainly consider two cases: In the ﬁrst case the investor shorts a ﬁxed amount of f, i.e., w.l.o.g., c = 1 and p = 0; in the second case she statically trades f, i.e., c can be any nonnegative number.

We ﬁrst show that in both cases V (x) = sup(H,c) infτ E[U(x+H ·S−c(fτ −p))] = infρ sup(H,c) E[U(x + H · S − c(fρ − p))], where τ is a pure stopping time, ρ is a randomized stopping time, and H satisﬁes certain non-anticipation condition. Then in the ﬁrst case (i.e., c = 1), we show that when U is exponential, V (x) = infτ supH E[U(x+H·S−fτ)]; for general utility this equality may fail, yet can be recovered if we in addition let τ be adapted to H in certain sense. Finally, in the second case (c ∈ [0,∞)) we obtain a duality result for the robust utility maximization on an enlarged space.

**Keywords:** Shorting American options, robust utility maximization, semi-static trading strategy, non-anticipative strategy, liquidating strategy, rondomized stopping time

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