# 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

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 $r-\delta$.
• And I use the exact same formula for $P_0$? I don't use $-S_0 e^{-\delta T}$ instead of $-S_0$? – user154085 Jul 10 '15 at 14:27
• Sorry for asking, but with "Yes, you are right." you mean that I have to use $-S_0e^{-\delta T}$? – user154085 Jul 10 '15 at 17:14
• Yes you have to use $-S_0e^{-\delta T}$ – phdstudent Jul 10 '15 at 17:29