How Continuous Dividends Affect Black–Scholes Call Values

# Answer by Winodd Dhamnekar (score 1)


# Prove from Black-Scholes that value of a European call option on an asset that pays continuous dividends less than a call without dividends












Black-Scholes gives us the following formulae for the prices of European calls on an underlying that does or doesn't pay continuous constant dividends (of proportion $D$):

$$C^E_D(S_t,t,K,T)=e^{-D(T-t)}S_t\Phi(d_D)-e^{-r(T-t)}K\Phi(d_D-\sigma \sqrt{(T-t)})$$ $$C^E(S_t,t,K,T)=S_t\Phi(d)-e^{-r(T-t)}K\Phi(d-\sigma \sqrt{(T-t)})$$

Where $d_D=\frac{\text{ln}\left(\frac{S_t}{K}\right)+(r-D+\frac{1}{2}\sigma^2)(T-t)}{\sigma \sqrt{(T-t)}}$ and $d=\frac{\text{ln}\left(\frac{S_t}{K}\right)+(r+\frac{1}{2}\sigma^2)(T-t)}{\sigma \sqrt{(T-t)}}$.

Now I know from common sense and no-arbitrage reasoning that we must have $C^E_D<C^E$, but am struggling to see how the formulae show this. Can it be proven mathematically?

i.e. something along the lines of: $$d_D<d\Rightarrow \Phi(d_D)<\Phi(d)$$

But this decrease works against us on the $K$ term...

## Answer by Winodd Dhamnekar (score 1)

https://quant.stackexchange.com/a/78544

Here's how we can prove that the Black-Scholes price of a European call option on an asset that pays a continuous dividend is less than the Black-Scholes price of a European call option on a similar asset that doesn't pay any dividend.

Understanding the Black-Scholes Model

The Black-Scholes model provides a way to calculate the theoretical price of a European-style call or put option using the following input variables:

- S: Current stock price

- K: Exercise/Strike price

- T: Time to expiration (in years)

- r: Risk-free interest rate

- σ: Volatility of the underlying asset

- q: Continuous dividend yield

The Impact of Dividends

In the presence of continuous dividend payments, the expected stock price at any point in the future is reduced. Here's why:

- When a stock pays a dividend, its price immediately decreases (on the ex-dividend date) by an amount roughly equal to the dividend payment.

- This dividend payment represents a cash outflow from the company to shareholders, reducing the assets held by the company and thereby reducing its stock price.

Effect on the Black-Scholes Formula

The Black-Scholes formula for a European call option on a stock that pays a continuous dividend yield is:

$C = Se^{(-qT)} N(d_1) - Ke^{(-rT)} N(d_2)$

Where:

$d_1 = [\ln{(S/K)} + (r - q + \sigma^²/2)T] / (\sigma\sqrt{T})$

$d_2 = d_1 - \sigma\sqrt{T}$

N(x) is the cumulative standard normal distribution function

Notice the term $e^{(-qT)}$ which reduces the value of `S` in the formula as opposed to the Black-Scholes formula for a non-dividend paying stock, where this term doesn't exist.

The Result

Since dividends effectively reduce the future value of the underlying stock, the call option becomes less valuable for the option holder for the same strike price. The potential upside from the option exercising is reduced because the stock price is likely to be lower due to the dividend payments. Consequently, the price of the call option on a dividend-paying stock will be lower than the price of a similar call option on a non-dividend-paying stock.

Conclusion

The presence of a continuous dividend yield reduces the price of a European call option, as calculated by the Black-Scholes model, compared to an identical option on a stock that doesn't pay any dividends.