European and Asian Greeks for Exponential Lévy Processes

Abstract

In this paper we give easy-to-implement closed-form expressions for European and Asian Greeks for general L2-payoff functions and underlying assets in an exponential Lévy process model with nonvanishing Brownian motion part. The results are based on Hilbert space valued Malliavin Calculus and extend previous results from the literature. Numerical experiments suggest, that in the case of a continuous payoff function, a combination of Malliavin Monte Carlo Greeks and the finite difference method has a better convergence behavior, whereas in the case of discontinuous payoff functions, the Malliavin Monte Carlo method clearly is the superior method compared to the finite difference approach, for first- and second order Greeks. Reduction arguments from the literature based on measure change imply that the expressions for the Greeks in this paper also hold true for generalized Asian options in particular for fixed and floating strike Asian options.

greeks: Sensitivities of Prices of Financial Options and Implied Volatilities

“…greeks has been applied to investigate the performance of Monte Carlo Greeks for jump diffusion models from Hudde & Rüschendorf (2023). Furthermore, greeks can be used in (under-)graduate courses in financial mathematics to provide a better understanding of option prices and Greeks by interactive visualizations.…”

“…Classical formulas are implemented for European options in the Black-Scholes model, as is presented in Hull (2022). In the case of Asian options, Malliavin Monte Carlo Greeks are implemented, see Hudde & Rüschendorf (2023), or Lyuu et al (2019. For American options, the Binomial Tree method is implemented, as is presented in Hull (2022).…”

A Malliavin Calculus Computation of the Greeks Theta and Vega of Asian Option and Best of Asset Option

“…Asian option and Best of Asset option are option types whose payoff are defined with respect to multiple underlying assets. Asian option considered the average of the assets underlying the contract over a certain period of time to determine if there is profit when compared with the strike price [1,7]. Best of Asset option is the type that considered the maximum of the underlying assets prices in comparison with the strike price to determine the profitability of the contract.…”