In this paper, we 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. 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 numerical 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.
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.
D
In this paper, we
R
option
options
A
,
A
,
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numerical
Euler-Maruyama