How do you adjust the Black-Scholes-Merton model for stocks that pay a continuous dividend yield?

Question

AcadiFi · Accepted Answer

The Merton extension of BSM for continuous dividends is essential for CFA Level II because it applies to stock index options, currency options, and futures options.

## The Adjustment

In the standard BSM model, replace the stock price S with **S x e^(-qT)**, where q is the continuous dividend yield.

**Call:** c = S x e^(-qT) x N(d1) - K x e^(-rT) x N(d2)

**Put:** p = K x e^(-rT) x N(-d2) - S x e^(-qT) x N(-d1)

Where:
- d1 = [ln(S/K) + (r - q + σ²/2)T] / (σ√T)
- d2 = d1 - σ√T

## Intuition

The dividend yield reduces the expected growth rate of the stock. During the option's life, the stock continuously "leaks" value through dividends that the call holder does not receive. So the effective stock price for option pricing purposes is lower.

## Worked Example — Kensington Index Options

Price a 3-month European call on the Kensington Equity Index:
- Index level: S = 4,200
- Strike: K = 4,300
- Risk-free rate: r = 5.0%
- Dividend yield: q = 1.8%
- Volatility: σ = 18%
- T = 0.25

**Without dividends (q = 0):**
- d1 = [ln(4200/4300) + (0.05 + 0.0162)0.25] / (0.18 x 0.5) = [-0.02353 + 0.01655] / 0.09 = -0.0776
- d2 = -0.0776 - 0.09 = -0.1676
- N(d1) = 0.4691, N(d2) = 0.4335
- c = 4200(0.4691) - 4300 x e^(-0.0125)(0.4335) = 1970.22 - 1838.84 = **$131.38**

**With dividends (q = 1.8%):**
- d1 = [ln(4200/4300) + (0.05 - 0.018 + 0.0162)0.25] / 0.09 = [-0.02353 + 0.01205] / 0.09 = -0.1276
- d2 = -0.1276 - 0.09 = -0.2176
- N(d1) = 0.4492, N(d2) = 0.4139
- c = 4200 x e^(-0.0045)(0.4492) - 4300 x e^(-0.0125)(0.4139)
- c = 4181.1(0.4492) - 4246.5(0.4139) = 1878.58 - 1757.76 = **$120.82**

The dividend yield reduces the call value by about $10.56, or roughly 8%.

## Impact on Greeks

| Greek | Effect of Dividends |
|---|---|
| Delta (call) | Lower (e^(-qT) x N(d1) < N(d1)) |
| Delta (put) | More negative |
| Gamma | Slightly lower |
| Rho | Unchanged in direction |

**Exam tip:** CFA Level II often gives you the no-dividend BSM value and asks how dividends would change it. Calls decrease in value; puts increase.

Practice more BSM problems in our CFA Level II question bank.

How do you adjust the Black-Scholes-Merton model for stocks that pay a continuous dividend yield?

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