2
$\begingroup$

I have a stock process which I have decided to model as $$S_T=S_t\exp((r-q-\frac{1}{2}\sigma^2)(T-t)+\sigma(W_T-Wt))-D_T$$ where $D_T$ is a cash dividend at time $T$. This dividend is known. I then calculated its second moment as $$\mathbb{E}(S_T^2)=[S_t\exp((r-q)(T-t))]^2\exp(\sigma^2(T-t))-2D_TS_t\exp((r-q)(T-t))+D_T^2$$ My questions is, how would this expression change if there were say $n$ dividends (all different) between times 0 and T.

$\endgroup$
2
$\begingroup$

If $D_i$ is dividend paid at time $t_i \in [t, T]$ , then future value of all dividends payments (assuming payments are known in advance), $D$, is: $$D= \sum_{i=i}^{n} D_i e^{r(T-t_i)}$$

Since all payments are known in advance, $D$ is constant like $D_T$ ( in your example). So, just replace $D$ with $D_T$ in your expression of $\mathbb{E}(S_t^2)$, you will get desired result.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.