Static and semistatic hedging as contrarian or conformist bets

Abstract

In this paper, we argue that, once the costs of maintaining the hedging portfolio are properly taken into account, semistatic portfolios should more properly be thought of as separate classes of derivatives, with nontrivial, model‐dependent payoff structures. We derive new integral representations for payoffs of exotic European options in terms of payoffs of vanillas, different from the Carr–Madan representation, and suggest approximations of the idealized static hedging/replicating portfolio using vanillas available in the market. We study the dependence of the hedging error on a model used for pricing and show that the variance of the hedging errors of static hedging portfolios can be sizably larger than the errors of variance‐minimizing portfolios. We explain why the exact semistatic hedging of barrier options is impossible for processes with jumps, and derive general formulas for variance‐minimizing semistatic portfolios. We show that hedging using vanillas only leads to larger errors than hedging using vanillas and first touch digitals. In all cases, efficient calculations of the weights of the hedging portfolios are in the dual space using new efficient numerical methods for calculation of the Wiener–Hopf factors and Laplace–Fourier inversion.

Efficient inverse Z-transform: sufficient conditions

“…In applications to finance, typically, Y = Z ∆, ∆ > 0, is an increment of a Lévy process Z; the random walk appears implicitly when either a continuous time Lévy model is approximated or options with discrete monitoring are priced. We impose the symmetry condition used in [16] to justify the semi-static hedging of barrier options (see [10] for the discussion). Under the same condition, representations of KoBoL (CGMY) and Meixner processes as a subordinated Brownian motion is derived in [24]; in [12], the representation result was extended to wide classes of Stieltjes-Lévy processes (SL processes) and signed SL processes (sSL processes).…”

“…The following proposition is a refined form of Proposition 6.1. The proof is a straightforward modification of the proof of Proposition 6.1, in the same vein as the analog of Proposition 6.1 for Lévy processes in [6] is refined in [10]. Proposition 6.2.…”

Hedging option books using neural-SDE market models

“…In general, if an option can be statically hedged 9 by European vanilla options, then we can derive its partial derivatives with regard to S and ξ and therefore its hedge ratios. There is vast literature on static hedging of exotic options, to name a few recent developments, [8], [28] and [29].…”

“…However, these assumptions will not generally hold for neural-SDE market models. It remains to be investigated empirically how the violation of these assumptions impacts hedging performance using neural-SDE market models 8. A down-and-out call option is a vanilla call option if its barrier has not been hit by its expiry date.…”

Geometric step options and Lévy models: Duality, PIDEs, and semi-analytical pricing

“…and note that this transform has several desirable properties. 7 In particular, applying the Laplace-Carson transform in the context of mathematical finance allows to randomize the maturity of (certain) financial contracts, i.e. to switch from objects with deterministic maturity to corresponding objects with stochastic maturity.…”