# Discrete Dividend GBM process

I'm trying to derive the risk neutral process for a stock with both continuous and discrete dividends. In particular, suppose the forward level process at time, $$t$$ is given by $$F(S_t, t, T) = e^{(r-y)(T-t)}(S_t-d(t,T))$$, where $$r$$ is the risk free interest rate, $$y$$ is a continuous dividend yield and $$d(t,T)$$ is the sum of all dividends paid in $$(t, T]$$. Under the R.N. measure, the stock process is given by $$\dfrac{dS_t}{S_t}=(r-\mu-y)dt+\sigma dW_t$$. Then in distribution, we have that:

$$S_T=S_te^{(-0.5\sigma^2+r-\mu-y)(T-t) + \sigma\sqrt{T-t}Z}=(F(S_t,t,T)-d(t,T))e^{(-0.5\sigma^2)(T-t)+ \sigma\sqrt{T-t}Z}$$

If we put $$X_T:=\dfrac{S_T}{F(S_t,t,T)-d(t,T)}$$, then we have that:

$$X_T=e^{-0.5\sigma^2(T-t)+ \sigma\sqrt{T-t}Z}$$

Something doesn't seem correct. I'm trying to incorporate all the discrete divs into the forward process and skip the jumps in the simulation. Is this wrong? Please help or correct me. Thanks.

## 1 Answer

You may want to read this paper for GBM with discrete dividend.

• I was just about to post this :) Great paper. If dealing with proportional extrapolation and you have discrete fixed dividend estimates for 3Y is it better to calculate beta from the final year and replace the fixed with proportional in 3Y+ inclusive? Nov 24 '20 at 10:27