Discrete Dividends in Risk-Neutral GBM and Forward-Based Simulation

Summary

The document examines how to model a stock under risk-neutral geometric Brownian motion when it pays both a continuous dividend yield and scheduled cash dividends. It gives a forward price that subtracts the dividends due before maturity, then proposes a terminal stock-price distribution and a normalized variable intended to absorb those cash dividends into the forward level.

The author is unsure whether this setup correctly allows simulation without explicitly modelling dividend jumps. The only response points to a paper on GBM with discrete dividends, without explaining the derivation or correcting the proposed equations. As a result, the post highlights a modelling question rather than providing a complete method or validated solution. Any use of the proposed distribution needs independent verification against the timing and treatment of discrete dividend payments.

Key ideas

• The setup combines a continuous dividend yield with cash dividends paid on specified dates.
• The proposed forward level subtracts dividends due before maturity.
• The author asks whether this adjustment permits simulation without modelling dividend jumps.
• The response recommends further reading but does not validate the proposed process.

Tags

# Discrete Dividend GBM process


# 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.

## Answer by StupidMan (score 2, accepted)

https://quant.stackexchange.com/a/59544

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

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