Question

Q-3 (25p)        Consider a European put option on a dividend paying stock CCC. Current stock price is $60, the exercise price is 58, the risk-free interest rate is 5% p.a., the volatility is 30% p.a., and the time to maturity is three months. Let’s assume that the CCC stock’s ex-dividend is in 2 months. Expected dividend is $0.90. (we discussed the meaning of ex-dividend day in class. Think that dividend will be paid in 2 month)

1. What is the European put option value/price now?

2. If the company had been paying $1.25 dividend instead of $0.90, do you think the put value you found would be higher or lower?                           Why? (NO CALCULATION, just explanation in one-two sentences!)

Answer 1

The formula for European call & Put option using Black Scholes model is given by

Call = S0 x e^(-d x t) x N(d1) - K x e^(-rxT) x N(d2)

Put = K x e^(-r xT) x N(-d2)- (S0 x e^(-d x t) x N(-d1))

Where,

S0 = Price of the underlying, eg stock price.

K = Exercise price.

r = Risk free interest rate

t = Time to expiry

σ = Standard deviation of the underlying asset, eg stock.

d = Dividend yield

N(d1) =  standard normal cumulative distribution function using value of d1

N(d2) =  standard normal cumulative distribution function using the value of d2

The value of d1 & d2 is calculated using the below formula

d1 = (Ln(So/X) + ((r - d + σ ^2)/2) x t)) / ((σ^2) x t)^1/2)

d2 = d1 – σ x (t)^1/2

Given here,

S0 = 60, K = 58, r = 5%, t = 3/12 = 0.25 years, σ = 30% , d = Dividend yield = $0.90 / 60 = 1.50%

d1 = (Ln(60/58) + ((0.05 - 0.015 + 30% ^2)/2) x 0.25)) / (30%^2) x 0.25)^1/2)

= (0.0339 + 0.02) / 0.15

= 0.359

d2 = 0.359 - 30% x 0.25^ 0.5 = 0.209

N(d1) = 0.6403 , N(d2) = 0.583, N(-d1) = 0.36 , N(-d2) = 0.4171

The price of Put = K x e^(-r xT) x N(-d2) - (S0 x e^(-d x t) x N(-d1))

= 58 x e^(-0.5 x 0.5) x 0.4171 - 60 x e^(-0.015 x 0.25) x 0.36

= 23.89 - 21.50

Price of Put = 2.39

ii) If the dividend yield goes up, the price of Put goes up

Theoritically dividend yield will impact Normal distributioon function as well as the second part of Black scholes formula

Practically, if the dividend yield goes up, market will discount it in the price of put as the fall in ex dividend price would be high, and there is demand for Puts craeting a higher price.