Abstract
Although many methods for computing the Greeks of discrete-time Asian options are proposed, few methods to calculate the Greeks of continuous-time Asian options are known. In this paper, we develop an integration by parts formula in the multi-dimensional Malliavin calculus, and apply it to obtain the Greeks formulae for continuous-time Asian options in the multi-asset situation. We combine the Malliavin method with the quasi-Monte Carlo method to calculate the Greeks in simulation. We discuss the asymptotic convergence of simulation estimates for the continuous-time Asian option Greeks obtained by Malliavin derivatives. We propose to use the conditional quasi-Monte Carlo method to smooth Malliavin Greeks, and show that the calculation of conditional expectations analytically is viable for many types of Asian options. We prove that the new estimates for Greeks have good smoothness. For binary Asian options, Asian call options and up-and-out Asian call options, for instance, our estimates are infinitely times differentiable. We take the gradient principal component analysis method as a dimension reduction technique in simulation. Numerical experiments demonstrate the large efficiency improvement of the proposed method, especially for Asian options with discontinuous payoff functions.
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Notes
In fact, this space is an irreducible Gaussian space \((\varOmega ,{{\mathscr {F}}},{\mathbb {P}};H)\). Here, \((\varOmega ,{{\mathscr {F}}},{\mathbb {P}})\) is a complete probability space, H is a real separable Hilbert space, \(\{U_h\}_{h\in H}\) is a family of Gaussian random variables satisfying \({\mathbb {E}}[U_h]=0\) and \({\mathbb {E}}[U_hU_g]=(h,g)_H\) (the inner product of h and g in H) for all \(h,g\in H\), and \({{\mathscr {F}}}\) is the completion of the \(\sigma \)-algebra generated by \(\{U_h\}_{h\in H}\) with respect to \({\mathbb {P}}\). We refer to Huang and Yan (2000) for more details.
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The authors are very grateful to the editors and the anonymous referee for their helpful suggestions and comments.
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The work is funded by the National Natural Science Foundation of China (No. 72071119).
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Yu, C., Wang, X. Quasi-Monte Carlo-Based Conditional Malliavin Method for Continuous-Time Asian Option Greeks. Comput Econ 62, 325–360 (2023). https://doi.org/10.1007/s10614-022-10257-3
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DOI: https://doi.org/10.1007/s10614-022-10257-3