Second Moments of Stock Prices with Multiple Known Dividends

Summary

The document asks how to extend a stock-price second-moment calculation when there are multiple known cash dividends before the horizon. Its starting model represents the terminal stock value using a lognormal price component and a cash dividend adjustment. The answer aggregates the intervening dividend payments by carrying each known payment forward to the horizon at the risk-free rate.

Because the aggregated future value of the dividends is treated as a constant, the response says to substitute that total for the single dividend in the existing second-moment expression. This provides a compact extension of the one-dividend formula. The result relies on dividends being known in advance and on the stated deterministic carry convention; it does not address uncertain dividends, alternative reinvestment assumptions, or other corporate actions.

Key ideas

• Multiple known dividends can be aggregated into a single value carried forward to the horizon.
• The aggregation discounts or accrues each payment according to the time remaining until the horizon.
• Treating the aggregate as deterministic allows it to replace the single dividend term in the second-moment expression.
• The approach assumes dividend amounts are known in advance.

Tags

# Second Moment of Stock Process


# Second Moment of Stock Process












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.

## Answer by Neeraj (score 2)

https://quant.stackexchange.com/a/24605

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.