A numerically efficient closed-form representation of mean-variance hedging for exponential additive processes based on Malliavin calculus

Abstract

We focus on mean-variance hedging problem for models whose asset price follows an exponential additive process. Some representations of mean-variance hedging strategies for jump type models have already been suggested, but none is suited to develop numerical methods of the values of strategies for any given time up to the maturity. In this paper, we aim to derive a new explicit closed-form representation, which enables us to develop an efficient numerical method using the fast Fourier transforms. Note that our representation is described in terms of Malliavin derivatives. In addition, we illustrate numerical results for exponential Lévy models. strategies. Note that our representation is a closed-form one obtained by means of Malliavin calculus for Lévy processes. In addition, we develop a numerical method using the fast Fourier transforms (FFT); and show numerical results for exponential Lévy models.We consider throughout an incomplete financial market in which one risky asset and one riskless asset are tradable. Let T > 0 be the maturity of our market, and suppose that the interest rate of the riskless asset is 0 for sake of simplicity. The risky asset price process, denoted by S, is given as a solution to the following stochastic differential equation:, W is a one-dimensional standard Brownian motion, and N is the compensated version of a homogeneous Poisson random measure N.Here, α and β are deterministic measurable functions on [0, T], and γ is also deterministic and jointly measurable on [0, T] × R 0 . We assume γ > −1, which ensures the positivity of S. Then, S is given as an exponential of an additive process, that is, log(S) is continuous in probability and has independent increments. In addition, when α and β are given by a real number and a nonnegative real number, respectively, and γ t,z = e z − 1, we call S an exponential Lévy process. Let H be a square integrable random variable. We consider its value as the payoff of a contingent claim at the maturity T. In principle, since our market is incomplete, we cannot find a replicating strategy for H, that is, there is no pair (c, ϑ) ∈ R × Θ satisfyingwhere Θ is a set of predictable processes, which is considered as the set of all admissible strategies in some sense, and G(ϑ) denotes the gain process induced by ϑ, that is, G(ϑ) := · 0 ϑ u dS u . Note that each pair (c, ϑ) ∈ R × Θ represents a self-financing strategy. Instead of finding the replicating strategy, we consider the following minimization problem:and call its solution ( c H , ϑ H ) ∈ R × Θ the MVH strategy for claim H if it exists. In other words, the MVH strategy is defined as the self-financing strategy minimizing the corresponding L 2 -hedging error over R × Θ. Remark that c H gives the initial cost, which is regarded as the corresponding price of H; and ϑ H t represents the number of shares of the risky asset in the strategy at time t.In addition to MVH strategy, locally risk-minimizing (LRM) strategy has been studied well as alternative hedging method in quadratic way. Being different from ...

Numerical Analysis on Quadratic Hedging Strategies for Normal Inverse Gaussian Models

“…Remark that c F gives the initial cost, which is regarded as the corresponding price of F. In this paper, we propose numerical methods of LRM strategies ξ F and MVH strategies ϑ F for call options when the asset price process is given by an exponential NIG process, by extending results of [2] and [1]. Our main contributions are as follows:…”

“…However it is impossible to observe the trajectory of S continuously. Thus, [1] developed a numerical scheme to compute ϑ F t approximately using discrete observational data S t 0 , S t 1 , . .…”

“…Note that [3] obtained their representation for LRM strategies by means of Malliavin calculus for Lévy processes. As for MVH strategies, Arai and Imai [1] obtained a new closed-form representation for exponential additive models, and suggested a numerical scheme for VG models.…”

“…Our aim in this paper is to extend the results of [2] and [1] to normal inverse Gaussian (NIG) models. Note that an NIG process is a pure jump Lévy process described as a time-changed Brownian motion as well as a VG process is.…”

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The authors aim to develop numerical schemes of the two representative quadratic hedging strategies: locally risk minimizing and mean-variance hedging strategies, for models whose asset price process is given by the exponential of a normal inverse Gaussian process, using the results of Arai et al [2], and Arai and Imai [1]. Here normal inverse Gaussian process is a framework of Lévy processes frequently appeared in financial literature.

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