2020/10/08 by Nishant Agrawal, Agrawal, Nishant, Yaozhong Hu +1 · 1 citation
Economics, Econometrics and Finance · Social Sciences · #34K50 #65C30 #91B25 #91B28 #91G20 #91G60 #FOS: Economics and business #Financial Risk and Volatility Modeling #Insurance, Mortality, Demography, Risk Management #Mathematical Finance (q-fin.MF) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2010.04287
openalex publication_date 2020/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 options is obtained together with the hedging portfolios. The option price is evaluated analytically at the last delayed period by using the Fourier transformation technique. But 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.