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Register a new userNumerical analysis on quadratic hedging strategies for normal inverse Gaussian models
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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. 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.
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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 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.
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.
We provide a new characterization of mean-variance hedging strategies in a general semimartingale market. The key point is the introduction of a new probability measure $P^{star}$ which turns the dynamic asset allocation problem into a myopic one. The minimal martingale measure relative to $P^{star}$ coincides with the variance-optimal martingale measure relative to the original probability measure $P$.
Deep hedging (Buehler et al. 2019) is a versatile framework to compute the optimal hedging strategy of derivatives in incomplete markets. However, this optimal strategy is hard to train due to action dependence, that is, the appropriate hedging action at the next step depends on the current action. To overcome this issue, we leverage the idea of a no-transaction band strategy, which is an existing technique that gives optimal hedging strategies for European options and the exponential utility. We theoretically prove that this strategy is also optimal for a wider class of utilities and derivatives including exotics. Based on this result, we propose a no-transaction band network, a neural network architecture that facilitates fast training and precise evaluation of the optimal hedging strategy. We experimentally demonstrate that for European and lookback options, our architecture quickly attains a better hedging strategy in comparison to a standard feed-forward network.
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