Risk-Neutral Stock Dynamics with Dividends and Payment Lags

Summary

The document poses a modeling question about the risk-neutral dynamics of a stock when dividends have distinct ex-dividend and payment dates, equity cash flows settle after a market payment lag, and both the risk-free and repo rates may be stochastic. The author wants to know what local martingality condition is required for a risk-neutral measure and asks how to justify discounting the stock using rates shifted by the payment lag between ex-dividend dates.

The discussion includes no answer, derivation, references, or numerical evidence. It therefore frames a pricing and measure-consistency problem rather than presenting a resolved method. Any application would need to account carefully for dividend entitlement, payment timing, funding through repo, and the chosen numeraire; the document itself does not specify the resulting condition or establish that the proposed discounted process is a martingale.

Key ideas

• Dividend entitlement dates and dividend payment dates are distinct in the proposed stock model.
• A market payment lag shifts the timing of equity cash flows relative to stock trading.
• Both risk-free and repo rates may be stochastic in the setup.
• The author asks whether a lag-adjusted discounted stock price is a martingale between ex-dividend dates.
• No solution or evidence is provided, so the risk-neutral condition remains unresolved.

Tags

# Local Martingality Condition for a Stock with Discrete Dividends, Repo Rate, Market Payment Lag


# Local Martingality Condition for a Stock with Discrete Dividends, Repo Rate, Market Payment Lag












I have a specific question I have not been able to solve nor to find in classic books.

Assume I have a stock price process $S$ with a given discrete dividends stochastic process such that the dividend $D_i$ has ex-dividend date $t^e_i$ but is paid at a given payment date $t^p_i$. Moreover, the cash flows associated to the equity are delayed by a market payment lag $m$ (notice that $t^p_i \ge t^e_i + m)$. Finally, I have potentially stochastic riskfree rate $(r_t)$ and repo rate $(q_t)$.

I want to know the local-martingality condition on this stock under $\mathbb{Q}$ such that $\mathbb{Q}$ is a risk-neutral probability.

It matters for me as I have a stock price model which requires that $\left(e^{-\int_0^{t+m} (r_s - q_s) ds} S_t\right)$ should be a $\mathbb{Q}$-martingale between two ex-dividend dates. I want to know where does it comes from, and if it is in accordance with the definition of a risk-neutral probability.

I know how to derive the condition under no dividends / continuous dividend yield and no market payment lag, but not in this case...

Thank you for your help !

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