Recall that:

• c_T = Max {0, S_T - X}
• p_T = Max {0, X - S_TM}

Anything that affects the underlying price at expiration will affect the price of the option. For example, dividends that might be paid during the life of a stock option can obviously affect the stock price.

• We may regard a dividend as a repayment of a portion of the share's value to the shareholder.
• As such, we would expect the stock price to fall by the amount of the dividend payment.
• As a metaphor, we might think of the dividend on a stock as a leakage of value from the stock.

A drop in the underlying price due to a cash flow will have:

• An adverse effect on the price of a call.
• A beneficial effect on the price of a put.

To adjust for cash flows of the underlying, we will simply use S₀ - PV(CF, 0, T) in the Black-Scholes-Merton model instead of S₀, where PV(CF, 0, T) is the present value of the cash flows on the underlying over the life of the option.

As an example, consider call and put options with a common exercise price of $100 and 150 days until expiration. Assume that the underlying stock trades for $102, and that you expect the stock to pay a $3 dividend in 90 days. The continuously compounded interest rate is 9%, and the standard deviation for the stock is 0.30.

The present value of the $3 dividend is: $3 e ^{-r (90/365)} = $2.93.

According to this technique, we reduce the stock price now by the present value of the dividend, giving an adjusted stock price of $99.07. We then apply the Black-Scholes-Merton model in the usual way, except we use the adjusted stock price of $99.07 instead of the current price of $102. The following table shows the results of applying the adjusted and unadjusted models to the call and the put.

The difference in prices is substantial, amounting to almost 20%. Also, as we hypothesized, subtracting the present value of the dividends from the stock price reduces the value of the call and increases the value of the put.

Practice Question 1

Use the Black-Scholes-Merton model adjusted for cash flows on the underlying to calculate the price of a call option.

The underlying price: 125.
The exercise price: 100.
The continuously compounded risk-free rate: 5.25%.
The time to expiration: 2 years.
Volatility: 0.3.
The continuously compounded dividend yield is 2%.

Correct Answer: 36.38

Adjust the price of the underlying to S₀ = 125 ^{e -0.02 (2.0)} = 120.1.
d₁ = {ln(120.1/100) + [0.0525 + (0.3)²/2] 2.0} / [0.3 (2.0)^1/2] = 0.89132.
d₂ = 0.89132 - 0.3 (2.0)^1/2 = 0.4671.
N(d₁) = N(0.89132) = 0.8133.
N(d₂) = N(0.4671) = 0.6808.
c = 120.1 x 0.8133 - 100 ^{e-0.0525 (2)} x 0.6808 = 36.38.

Practice Question 2

A put expires in 150 days and has an exercise price of $100. The underlying stock is worth $105 and has a standard deviation of 0.2. The continuously compounded risk-free rate is 5.5 %. Assume that the stock will pay a dividend of $3 on day 50. The price of the option should be:

A. 3.19.
B. 7.44.
C. 2.10.

Correct Answer: A

The present value of the $3 dividend is: $3 e ^{-0.055 (50/365)} = $2.98.

Adjust the price of the underlying to S₀ = 105 - 2.98 = $102.02.
d₁ = {ln(102.02 /100) + [0.055 + (0.2)²/2] (150/365)} / [0.2 (150/365)^1/2] = 0.16.
d₂ = 0.16 - 0.2 (150/365)^1/2 = 0.03.
N(d₁) = N(0.16) = 0.5636.
N(d₂) = N(0.03) = 0.5120.
c = 102.02 x 0.5636 - 100 ^{e-0.055 (150/365)} x 0.5120 = 7.44.
p = 100 x e ^{-0.055 (150/365)} x [1 - 0.5120] - 102.02 x [1 - 0.5636] = 3.19.