Question

Question 4. Option Pricing with Black-Scholes-Merton Model

Today is January 12, 2017. The shares of XYZ Inc. are currently selling for $120 per share. The shares have an estimated volatility of 25%. XYZ Inc. is also expected to pay a dividend of $1.50 with an ex-dividend date of January 25, 2017. The risk-free rate is 6.17 percent per year with continuous compounding. Assume that one call option gives the holder the right to purchase one share.

a.       Use the Black-Scholes-Merton model to estimate the fair value of a European call option on XYZ shares, with exercise price of $125 and expiration date of March 21, 2017. (Note that 2017 is not a leap year.)                                                                    

b.       This European call option has a market price of $3.00. Is it correctly priced? If not, how can an investor use the put-call parity to take advantage of this arbitrage opportunity?                    

Answer 1

a)

The Black-Scholes call option price is given by

Here S(0) is the dividend adjusted stock price = 120-1.5*e^(-0.0617*13/365) =$118.5032

T is the days (in years) between today and maturity = between January 12, 2017 and January 25, 2017 ie. 68 days = 68/365

d1 = (ln(118.5032/125) + (0.0617+0.25*0.25/2)*(68/365))/(0.25*(68/365)^0.5)

d1 = -0.3342

d2 = -0.3342- (0.25*(68/365)^0.5) = -0.4421

N(d1) = 0.3691

N(d2) = 0.3292

c = 118.5032*0.3691- (125*e^(-0.0617*68/365))*0.3292

c = 3.06

Hence, he fair value of a European call option on XYZ shares = $3.06

b)

Since, the Black-Scholes call option price is $3.06, the call option is not correctly priced.

Using a put-call parity

c +X*e^(-r*t) = p + S(0)

c is the call option price

p is the put option price

X is the strike price =125

r is the risk-free rate = 0.0617

t is the time to maturity = 68/365

S(0) is the price of the stock today by adjusting dividend = 118.5032

Here, the LHS or the covered call equation is under-priced. This is because the calculated price of a call option if $3.06 and the actual price is $3.00. Hence, the $0.06 difference would make the covered call cheaper than a protective put. One can buy a covered call and short a protective put to arbitrage a gain of $0.06.