Optimal Hedging Under Robust-Cost Constraints
In this chapter, we consider an original problem: that of maximizing the best-case return of a trading strategy with a hard bound on the worst-case loss. We provide an explicit numerical algorithm for solving that problem.
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Optimal Hedging Under Robust-Cost Constraints
5.1 Introduction In this chapter we analyze hedging of a short position in a European call option by an optimal strategy in the underlying asset under a robust cost constraint (RCC), that is, under the restriction that the worst-case costs do not exceed a certain a priori given upper bound. This relates to a Value-at-Risk (VaR) condition, which is usually defined for stochastic models as the maximum costs for a specified confidence level. As compared to VaR, an RCC denotes a level of worst-case costs that cannot be exceeded within a given interval model. More specifically, the asset is modeled by an interval model Iu,d in N equal time steps from current time to expiry, cf. (3.9). Recall from Proposition 4.2 that a short position in the option, kept under a hedge strategy g, yields an outcome of costs in an interval I g . For discontinuous strategies this interval I g is not necessarily closed, and therefore best- and worst-case costs are defined as the infimum and supremum of costs: BCg := inf I g = inf Qg (S). S∈Iu,d
WC := sup I = sup Qg (S). g
g
S∈Iu,d
We refer to −BCg also as the maximum profit under g. The RCC condition simply limits the worst-case costs WCg . In this section we analyze the impact of such a restriction for the set of admissible hedging strategies and provide an algorithm to solve this constrained optimization problem. To that end we first introduce some notation. The set of all strategies with price paths S in some interval model is denoted by G. Thus G := {g = (g0 , . . . , gN−1 ) | g j : (S0 , . . . , S j ) → γ j ∈ R}. P. Bernhard et al., The Interval Market Model in Mathematical Finance, Static & Dynamic Game Theory: Foundations & Applications, DOI 10.1007/978-0-8176-8388-7 5, © Springer Science+Business Media New York 2013
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5 Optimal Hedging Under Robust-Cost Constraints
Let V denote the RCC limit; then the set of all admissible strategies under this RCC limit is defined by GV := {g ∈ G| WCg ≤ V }. Furthermore, by Δ j we will denote the delta-hedging strategy [see (3.4), (3.5)]
Δ j (S j ) = λ Δ j+1 (uS j ) + (1 − λ )Δ j+1(dS j ), with ΔN−1 (SN−1 ) =
[uSN−1 − X]+ − [dSN−1 − X]+ , (u − d)SN−1
where λ = uu(1−d) −d , and by f j (S j ) we will denote the corresponding Cox–Ross– Rubinstein option premium [see (3.2), (3.3)] fN (SN ) = [SN − X]+ , f j (S j ) = q f j+1 (uS j ) + (1 − q) f j+1(d j S j ), where q :=
(5.1)
1−d
u−d .
5.2 Effect of Cost Constraints on Admissible Strategies Since delta hedging yields the lowest upper bound of costs among all strategies in G (Theorem 4.5), we have the next result. Proposition 5.1. If V < f0 (S0 ), then GV is empty. If V ≥ f0 (S0 ), then the deltahedging strategy belongs to GV . So the arbitrage-free Cox–Ross–Rubinstein price of the option C in the binomial tree model Bu,d is the smallest RCC limit that is achievable for a hedged short position in the call option C with underlying asset S ∈ Iu,d . As may be expected, for RCC beyond this minimal level, the space of admissible strategies is
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