Pricing a European Put with a Discrete Dividend

Summary

This note asks how to price a three-month European put when a single cash dividend is expected before expiry. For the no-dividend case, the question supplies Black–Scholes inputs and a computed put value. The answer suggests approximating the dividend’s effect by converting its amount into a yield relative to the current share price, then using the risk-adjusted rate in the option formula.

As an alternative, it recommends a binomial tree for handling the discrete dividend. The response is brief and does not show the adjusted calculation or compare the resulting price with the original. Treating a cash dividend as a continuous yield is an approximation; a tree can represent the dividend’s timing more directly. The note therefore introduces two approaches without establishing their accuracy for this example.

Key ideas

• A discrete cash dividend affects the value of a European option on the underlying stock.
• One suggested approximation expresses the dividend as a yield relative to the current stock price.
• A binomial tree is offered as a more direct way to model the dividend’s timing.
• The answer provides no adjusted price or comparison, so it does not quantify the dividend’s effect.

Tags

# Calculate put price with Black-Scholes and one discrete dividend


# Calculate put price with Black-Scholes and one discrete dividend












I try to solve this exercise:

a) Calclculate the price of a 3-month European put option on a non-dividend-paying stock with a strike price of 45 when the current stock price is 40, the risk-free interest rate is 5% per annum, and the volatility is 40% per annum.

b) What dierence does it make to the option price if a dividend of 1.50 is expected in 2 months?

While I can solve a) im not able to solve b). My solution for a) is: $T=\frac{1}{4}, K=45, S_0=40,r=0.05,\sigma=0.4$ leads to $$d_{+}=\frac{\ln\left( \frac{S_0}{K}\right)-(r+\frac{\sigma^2}{2})T}{\sigma \sqrt T}=-0.7514$$ and $$d_{-}=-0.9514.$$ Hence the put option price is given by $$P_0=-S_0N(-d_{+})+Ke^{-rT}N(-d_{-})=5.9042.$$

Can anybody explain how b) works?

Greetings

## Answer by phdstudent (score 1, accepted)

https://quant.stackexchange.com/a/18782

One solution is to calculate the annual dividend yield implied by that. $Div_{yield}=\delta=1.5/40$ and then replace the $r$ on $d_+$ by $r-\delta$.

A cleaner way would be to compute it using a binomial tree.

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