Semistatic and sparse variance‐optimal hedging

Abstract

We consider the problem of hedging a contingent claim with a “semistatic” strategy composed of a dynamic position in one asset and static (buy‐and‐hold) positions in other assets. We give general representations of the optimal strategy and the hedging error under the criterion of variance optimality and provide tractable formulas using Fourier integration in case of the Heston model. We also consider the problem of optimally selecting a sparse semistatic hedging strategy, i.e., a strategy that only uses a small subset of available hedging assets and discuss parallels to the variable‐selection problem in linear regression. The methods developed are illustrated in an extended numerical example where we compute a sparse semistatic hedge for a variance swap using European options as static hedging assets.

Semi-static variance-optimal hedging in stochastic volatility models with Fourier representation

“…In this paper we extend the results of [18,25] in two directions: First, we consider a very general setting of semimartingale factor models that is not limited to processes with independent increments or with affine structure. Second, in addition to classic variance optimal hedging, we also consider the variance-optimal semi-static hedging problem that we have introduced in [9]. The semi-static hedging problem combines dynamic trading in the underlying S with static (i. e. buy-and-hold) positions in a finite number of given contingent claims (η 1 , .…”

“…As shown in [9] and summarized in Section 2 below, the semi-static hedging problem can be solved under a variance-optimality criterion when also the covariances ε i j := E[L i T L j T ] of the residuals in the GKW-decompositions of all supplementary claims (η 1 , . .…”

“…As shown in [9] and summarized in Section 2 below, the semi-static hedging problem can be solved under a variance-optimality criterion when the covariances of the residuals in the GKW decompositions of all supplementary claims are also known. This leads to a natural extension of the questions investigated in [15,18]: If the model for S allows for explicit calculation of the moment-generating function of (such as Lévy models, the Heston model, or the 3/2 model), and each of the supplementary claims has a Fourier representation (such as European puts and calls), how can we compute the quantities ?…”

“…We now give a brief overview of the companion paper [9], where we develop numerical results connected with the variance-optimal semi-static hedging problem in the Heston model: We fix a d -dimensional basket of supplementary claims of European put and call options and set up a variance-optimal semi-static strategy for the variance swap. The quantities A , B , and C , whose analytic expressions are obtained in the present paper, are numerically computed in [9]. Furthermore, as we shall see below, the variance-optimal semi-static problem can be formulated by two nested minimization problems.…”

“…The outer minimization problem is interesting because it leads to questions related to optimal subset selection: Given , which subset of supplementary claims of cardinality minimizes the squared hedging error? We call this problem sparse variance-optimal semi-static hedging and we study it in [9] also comparing different optimal-subset-selection algorithms (e.g. greedy forward selection, Leaps and Bounds, and Lasso).…”