On the difference between locally risk-minimizing and delta hedging strategies for exponential Lévy models

Arai, Takuji; Imai, Yuto

doi:10.1007/s13160-017-0268-6

Japan J. Indust. Appl. Math.

2017

DOI: 10.1007/s13160-017-0268-6

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On the difference between locally risk-minimizing and delta hedging strategies for exponential Lévy models

Takuji Arai

Yuto Imai

Abstract: We discuss the difference between locally risk-minimizing and delta hedging strategies for exponential Lévy models, where delta hedging strategies in this paper are defined under the minimal martingale measure. We give firstly model-independent upper estimations for the difference. In addition we show numerical examples for two typical exponential Lévy models: Merton models and variance gamma models.

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Cited by 3 publications

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“…In addition, we set model parameters as C = 6.7910, G = 30.1807, and M = 33.1507, which are calibrated by the data set of European call options on the S&P 500 Index at 20 April 2016. Note that the above parameter set was used in Arai and Imai [2], and satisfies Assumptions 2.1 and 2.7. Figure 1 shows the values of ϑ H t .…”

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

Arai

Imai

2018

Applied Mathematical Finance

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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 ...

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

Arai

Imai

2018

Applied Mathematical Finance

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FFT-network for bivariate Lévy option pricing

Chiu

Wang

Wong

2020

Japan J. Indust. Appl. Math.

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Asymptotic expansion method for pricing and hedging American options with two-factor stochastic volatilities and stochastic interest rate

Zhang

2019

International Journal of Computer Mathematics

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On the difference between locally risk-minimizing and delta hedging strategies for exponential Lévy models

Cited by 3 publications

References 8 publications

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

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

FFT-network for bivariate Lévy option pricing

Asymptotic expansion method for pricing and hedging American options with two-factor stochastic volatilities and stochastic interest rate

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