# MonteCarlo simulation of stock prices using milstein scheme with dividend yield?

While performing a montecarlo simulation of stock prices using the milstein scheme is it possible to take into account the dividend yield into the simulation itself somehow, if we are given a continuous dividend yield?

Or is this something that has to be considered while valuing various derivatives using those simulated paths?

Am fairly new to quant finance, so pardon me if this turns out to be a stupid question.

• you build the dividend yield directly into your pricing model and evolve the paths. The derivatives will be priced off the final model. – Matthias Wolf Jan 7 '14 at 14:50

It is straightforward to include dividends into the model if it can be assumed that the dividend payment is a continuous dividend yield, $$q$$. Under $$Q$$ measure, in the Black-Scholes Model, Heston Model etc, $$r$$ is replaced by $$r − q$$., Here, We are going to simulate underlying asset in the Black-Scholes model by Milstein Method. Indeed, we assume underlying asset follows the Ito process as described by the following stochastic differential equation $$dS_t=(r-q)S_t dt+\sigma\, S_tdW_t$$ Apply Milstein discretization to this equation,we have $$S_{t+\Delta t} = S_t+(r-q)S_t \Delta t+\sigma\,S_t \sqrt{\Delta t}\,Z+\frac{1}{2}\sigma^2\Delta t(Z^2-1)$$ where $$Z$$ is distributed as standard normal. We then retain the last stock price from each stock price path and obtain the payoff of the European option at expiry, take the average over all stock price paths and discount back to time zero. Hence, for example, the call $$C(K)$$ and $$C(K,S_T,r,\sigma)$$ is $$C(K,S_T,r,\sigma)=e^{-rt}\frac{1}{N}\sum_{i=1}^{N}max\{S_T^{(i)}-K, 0\}$$