Jump Models with Delay—Option Pricing and Logarithmic Euler–Maruyama Scheme

Nishant Agrawal and Yaozhong Hu
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Nishant Agrawal: Department of Mathematical and Statistical Sciences, University of Alberta at Edmonton, Edmonton, AB T6G 2G1, Canada
Yaozhong Hu: Department of Mathematical and Statistical Sciences, University of Alberta at Edmonton, Edmonton, AB T6G 2G1, Canada

Mathematics, 2020, vol. 8, issue 11, 1-21

Abstract: In this paper, we obtain the existence, uniqueness, and positivity of the solution to delayed stochastic differential equations with jumps. This equation is then applied to model the price movement of the risky asset in a financial market and the Black–Scholes formula for the price of European option is obtained together with the hedging portfolios. The option price is evaluated analytically at the last delayed period by using the Fourier transformation technique. However, in general, there is no analytical expression for the option price. To evaluate the price numerically, we then use the Monte-Carlo method. To this end, we need to simulate the delayed stochastic differential equations with jumps. We propose a logarithmic Euler–Maruyama scheme to approximate the equation and prove that all the approximations remain positive and the rate of convergence of the scheme is proved to be 0.5 .

Keywords: Lévy process; hyper-exponential processes; Poisson random measure; stochastic delay differential equations; positivity; options pricing; Black–Scholes formula; logarithmic Euler–Maruyama scheme; convergence rate (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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