A systematic and efficient simulation scheme for the Greeks of financial derivatives

Abstract

Greeks are the price sensitivities of financial derivatives and are essential for pricing, speculation, risk management, and model calibration. Although the pathwise method has been popular for calculating them, its applicability is problematic when the integrand is discontinuous. To tackle this problem, this paper defines and derives the parameter derivative of a discontinuous integrand of certain functional forms with respect to the parameter of interest. The parameter derivative is such that its integration equals the differentiation of the integration of the aforesaid discontinuous integrand with respect to that parameter. As a result, unbiased Greek formulas for a very broad class of payoff functions and models can be systematically derived. This new method is applied to the Greeks of (1) Asian options under two popular Lévy processes, i.e. Merton's jump-diffusion model and the variance-gamma process, and (2) collateralized debt obligations under the Gaussian copula model. Our Greeks outperform the finite-difference and likelihood ratio methods in terms of accuracy, variance, and computation time.

Keywords:

• Greeks
• Dirac delta function
• Variance-gamma processes
• Jump-diffusion processes
• Credit derivatives
• Monte Carlo simulation

JEL Classification:

• C15
• C53

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Huei-Wen Teng http://orcid.org/0000-0002-3651-9934

Additional information

Funding

The first author was supported in part by the Ministry of Science and Technology of Taiwan under grant 106-2221-E-002-049-MY3. The second author was supported in part by the Ministry of Science and Technology of Taiwan under grant 106-2118-M-008-004.