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Register a new userNumerical analysis on local risk-minimization forexponential Levy models
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We illustrate how to compute local risk minimization (LRM) of call options for exponential Levy models. We have previously obtained a representation of LRM for call options; here we transform it into a form that allows use of the fast Fourier transform method suggested by Carr & Madan. In particular, we consider Merton jump-diffusion models and variance gamma models as concrete applications.
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We discuss the difference between locally risk-minimizing and delta hedging strategies for exponential Levy 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 Levy models: Merton models and variance gamma models.
We obtain explicit representations of locally risk-minimizing strategies of call and put options for the Barndorff-Nielsen and Shephard models, which are Ornstein--Uhlenbeck-type stochastic volatility models. Using Malliavin calculus for Levy processes, Arai and Suzuki (2015) obtained a formula for locally risk-minimizing strategies for Levy markets under many additional conditions. Supposing mild conditions, we make sure that the Barndorff-Nielsen and Shephard models satisfy all the conditions imposed in Arai and Suzuki (2015). Among others, we investigate the Malliavin differentiability of the density of the minimal martingale measure. Moreover, some numerical experiments for locally risk-minimizing strategies are introduced.
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. cite{AIS}, and Arai and Imai. Here normal inverse Gaussian process is a framework of Levy processes frequently appeared in financial literature. In addition, some numerical results are also introduced.
We derive representations of local risk-minimization of call and put options for Barndorff-Nielsen and Shephard models: jump type stochastic volatility models whose squared volatility process is given by a non-Gaussian rnstein-Uhlenbeck process. The general form of Barndorff-Nielsen and Shephard models includes two parameters: volatility risk premium $beta$ and leverage effect $rho$. Arai and Suzuki (2015, arxiv:1503.08589) dealt with the same problem under constraint $beta=-frac{1}{2}$. In this paper, we relax the restriction on $beta$; and restrict $rho$ to $0$ instead. We introduce a Malliavin calculus under the minimal martingale measure to solve the problem.
We consider option hedging in a model where the underlying follows an exponential Levy process. We derive approximations to the variance-optimal and to some suboptimal strategies as well as to their mean squared hedging errors. The results are obtained by considering the Levy model as a perturbation of the Black-Scholes model. The approximations depend on the first four moments of logarithmic stock returns in the Levy model and option price sensitivities (greeks) in the limiting Black-Scholes model. We illustrate numerically that our formulas work well for a variety of Levy models suggested in the literature. From a theoretical point of view, it turns out that jumps have a similar effect on hedging errors as discrete-time hedging in the Black-Scholes model.
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