Everything built in Chapters 18–20 rests on one foundation: the discounted stock price e^(−rt)S(t) is a Martingale under the risk-neutral measure ℙ̃. When a stock pays a dividend, it literally transfers cash out of the company and into your account — making the stock price drop on the payment date. A falling stock cannot be a Martingale. The models break. This chapter explains the fix.
The price drop at the ex-dividend date breaks the Martingale property. The fix: track total wealth (price + reinvested dividends) instead of price alone.
The resolution is a subtle but important shift in perspective. Stop tracking the stock price. Start tracking the wealth process — the total value of the stock plus all dividends, immediately reinvested back into more shares. The stock price leaks, but the reinvested wealth does not. The wealth process is the true Martingale.
For indices and ETFs — like NIFTY 50 or a mutual fund — dividends flow continuously at a known annual rate δ. Every instant, a tiny fraction δ·dt of the stock's value is paid out. The stock price itself drifts downward relative to a non-dividend stock, but the reinvested wealth — where every dividend payment instantly buys more shares — still grows at the risk-free rate r under ℙ̃.
Under ℙ̃: dS = (r − δ)S dt + σS dW̃
The stock grows at r − δ, not r. It leaks δ per year. e^(−rt)S(t) is NOT a Martingale anymore.
Let ψ(t) = e^(δt)·S(t) (reinvested account).
Under ℙ̃: e^(−rt)ψ(t) IS a Martingale. The leakage is cancelled by the reinvested dividends.
The dividend stock (red) drifts below the no-dividend path. But reinvesting every dividend immediately (gold dots) keeps total wealth on the Martingale track.
Economically: S(0)e^(−δT) is the "forward price" of the stock adjusted for dividends — what you would expect to receive at time T if you started with S(0) today and leaked δ per year. Plugging this into BSM instead of S(0) is all that is needed to correctly price options on dividend-paying stocks.
Most real companies pay dividends quarterly, not continuously. On the ex-dividend date, the stock price falls by the dividend amount — a discrete jump. This looks alarming on a chart, but your total wealth does not change: the price drop in your shares is exactly offset by the dividend cash landing in your account.
The stock price jumps down (left). The wealth account stays smooth (right) — because the cash received exactly offsets the price drop. The risk-neutral formula remains valid on the wealth process.
The mathematical statement: at time tⱼ, the stock price jumps to S(tⱼ) = S(tⱼ⁻)·(1 − aⱼ) where aⱼ is the fractional dividend. But the portfolio value satisfies X(tⱼ) = X(tⱼ⁻) — no jump. The Δ(tⱼ) shares each lose aⱼ·S in price, but the Δ(tⱼ)·aⱼ·S dividend payment exactly compensates. Portfolio continuity is maintained throughout.
If you are pricing an NSE option and know that dividends a₁, a₂, … will be paid before expiry, you cannot simply use the current stock price S(0) in the BSM formula — the option will be mispriced. The correct approach: compute the dividend-adjusted forward price and use that as the effective stock price input.
Three quarterly dividends reduce the effective stock price to 94.63% of S(0) by expiry. Use this adjusted price in BSM — not the raw current price.
The backtesting warning: On the ex-dividend date, a stock that pays ₹2 on a ₹100 share will show as a 2% drop in the price series. If your algorithm treats this as a price crash and generates a "sell" signal — or if your mean-reversion Z-score treats this as a deviation — you are trading on an artefact, not a signal. Always use dividend-adjusted price series for backtesting. Most data vendors (NSE, Bloomberg, Refinitiv) provide adjusted close prices that account for all historical dividends.
| Feature | No Dividends | Continuous δ | Lump aⱼ |
|---|---|---|---|
| Risk-neutral drift of S | r | r − δ | r (between jumps) |
| What is the Martingale? | e^(−rt)S(t) | e^(−rt)·e^(δt)·S(t) | e^(−rt)·X(t) (wealth) |
| BSM input change | None — use S(0) | Use S(0)·e^(−δT) | Use S(0)·∏(1−aⱼ) |
| Call option impact | Baseline price | Lower (leakage reduces S) | Lower (jumps reduce S) |
| Ex-dividend date visible? | N/A | No — continuous drain | Yes — discrete price jump |
| Backtesting adjustment | None needed | Use total return index | Use adjusted close prices |
Price a European call option with and without dividends. See how continuous leakage or lump payments reduce the option value. Observe that the put price rises when dividends are present — because dividends make it more likely the stock will fall below the strike.
Green = call price across stock prices | Purple = put price | Dashed = no-dividend baseline | Dividends shift both curves downward for calls, upward for puts
Chapter 21 — dividends and the wealth process:
Chapter 5 complete. You have now covered the full arc of Risk-Neutral Pricing: Girsanov's Theorem, the Martingale Representation Theorem, the Two Fundamental Theorems, and dividends. The next chapter connects this probabilistic machinery back to Partial Differential Equations — showing that the two approaches (PDE and expectation) are two sides of the same coin.