  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 $0.5$.
