Question

Question 3 (9 marks) Consider the following information about a non-dividend paying stock: The current stock price is $36, and its return standard deviation is 30% per year. The continuous compound risk-free rate is 3% per year. Assume that the Black-Scholes model correctly prices all the options written on the stock given the information above. a). A call option written on the stock expires in 1 year and has an exercise price of $36. Calculate the Black-Scholes value of the call option using the provided cumulative normal density table with arguments rounded to two decimal places. (3 marks) I b). A put option written on the stock expires in 1 year and has an exercise price of $36. Find the value of the put option using your answer in part a). (2 mark) c). You find that a risk-less zero-coupon bond that pays $36 in 1 year is currently priced at $35.50. Assume that you can buy and short sell i) the stock at $35.50 and ii) the call and put options at the prices you calculated in parts a) and b). Further assume that any fraction of the bond can be bought or short sold - that is, if you buy (short sell) N bonds, then you would pay (receive) $35.50XN today and receive (pay) $36XN in 1 year for any real number N. Design an arbitrage strategy using the bond, stock, call, and put option, and show that the strategy has a zero cost with a sure profit at the end of Year 1. (4 marks)

Answer 1

a) The call option value (C) is given by

C=S*N(d1)-K*e^(-r*t) * N(d2)

where S is the current stock price = 36

K is the strike price = 36

r is the continuously compounded risk free rate = 3% = 0.03

t is the time in years = 1

d1= ( ln(S/K) + (r + s^2/2) *t ) / (s*t^0.5) where s is the annual standard deviation

= 0.25  

N(d1) = (area under normal distribution curve till z=0.25) from the table

= 0.5987

d2=  ( ln(S/K) + (rd- s^2/2) *t ) / (s*t^0.5) = -0.05

N(d2) = (area under normal distribution curve till z=-0.05)  

= 0.4801

So, putting all the values

C = 4.7804

So, the Black Scholes value of the call option is $4.78

b) From put call parity , put option price on the same stock with the same strike price and the same expiry is given by

C+ K* exp(-r*t) = P+S

P =  C+ K* exp(-r*t) - S

= 4.781991+36*exp(-0.03*1) -36

= 3.718

So, the Black Scholes value of the Put option is $3.72

c) The return from the bond = 36/35.5 -1 = 0.0140845 or 1.41%

As the bond has less return than the 3% used to calculate the option premiums, one should short the bond and along with it the call option

Arbitrage is as follows:

1.Buy the stock and the put option for $ 3.72 and $36 and short the call option for $4.78. The remaining $34.94 can be obtained by shorting $34.94/35.5 = 0.984225 units of bond. So, the net cost of the strategy today is 0

2. After 1 year

if stock price > $36, the call option will be exercised and the put option will be worthless

Sell the stock at $36 and repay the bond amount of 0.984225*$36 = $35.43 and make an arbitrage profit of $0.57

  

if stock price < $36, the put option will be exercised and the call option will be worthless

Sell the stock at $36 and repay the bond amount of 0.984225*$36 = $35.43 and make an arbitrage profit of $0.57

So, the strategy results in a sure profit of $0.57 at the end of year 1