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

Abstract

In a financial market model, we consider the variance-optimal semi-static hedging of a given contingent claim, a generalization of the classic variance-optimal hedging. To obtain a tractable formula for the expected squared hedging error and the optimal hedging strategy, we use a Fourier approach in a general multidimensional semimartingale factor model. As a special case, we recover existing results for variance-optimal hedging in affine stochastic volatility models. We apply the theory to set up a variance-optimal semi-static hedging strategy for a variance swap in both the Heston and the 3/2-model, the latter of which is a non-affine stochastic volatility model. * Corresponding author. Mail: Paolo.Di_Tella@tu-dresden.de. † MKR thanks Johannes Muhle-Karbe for early discussions on the idea of "Variance-Optimal Semi-Static Hedging". We acknowledge funding from the German Research Foundation (DFG) under grant ZUK 64 (all authors) and KE 1736/1-1 (MKR, MH) using the orthogonal component L in the GKW decomposition (2.3).Moreover, notice that, if L ∈ H 2 0 is orthogonal to L 2 (S) in the Hilbert space sense, then L is also orthogonal to S, i. e. LS is a martingale starting at zero and, in particular, L, S = 0. Therefore, from (2.3) we can compute the optimal strategy ϑ * by S, H = ϑ * · S, S = · 0

Semistatic and sparse variance‐optimal hedging

“…In the companion paper, Di Tella et al. (), we extend the results of Hubalek et al. (), Kallsen and Pauwels (), and Pauwels () to the semistatic hedging problem.…”

“…For our problem of interest, the semistatic hedging problem (5), the results of Hubalek et al (2006), Kallsen and Pauwels (2010), and Pauwels (2007) are not sufficient: To obtain the quantities , , and of Theorem 2.3, we also need to compute the covariances [ ] between the GKW residuals of different claims. In the companion paper, Di Tella et al (2019), we extend the results of Hubalek et al (2006), Kallsen and Pauwels (2010), and Pauwels (2007) to the semistatic hedging problem. Moreover, we show that the method can be used in any stochastic volatility models where the Fourier transform of the log-price is known (e.g., the Heston, the 3/2 or the Stein-Stein model, cf.…”

“…Proof Part (i) of the theorem is technically demanding and follows from theorems 4.5, 4.6, and 4.8 in the companion paper Di Tella et al. (). In order to show part (ii), let Y be a ‐valued continuous semimartingale and let be functions in .…”

“…Lewis, 2000). Here, we only need a special case of the more general results in Di Tella et al (2019), which is condensed into Theorem 4.1(i) below.…”

“…Here, we build on results from Kallsen and Pauwels (2010) which enable us to calculate the GKW decomposition "semi-analytically," i.e., in terms of Fourier integrals, in several models of interest, such as the Heston model. The results of Kallsen and Pauwels (2010), which focus on calculation of the strategy and the hedging error in a classic variance-optimal hedging framwork, are however not sufficient for the semistatic hedging problem and we draw from some extensions that are developed in the technical companion paper, Di Tella, Haubold, and Keller-Ressel (2019).…”