Subject 9. The effect of the underlying asset's cash flows

Recall that:

  • cT = Max {0, ST - X}
  • pT = Max {0, X - STM}

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.