Question

Consider a two-period binomial model in which a stock trades currently at $44. The stock price can go up 6% or down 6% each period. The risk free rate is 2% per period.

A) Calculate the price of a call option expiring in two periods with an exercise price of $45.

B) Calculate the price of a put option expiring in two periods with an exercise price of $45.

C) Based on your answer in A), calculate the number of units of the underlying stock that would be needed at t=0 in the binomial tree to construct a risk-free hedged portfolio which includes 10,000 calls.

Answer 1

The stock price either goes up or goes down by 6%

Let us calculate the stock price for next 2 periods -

T0	T1	T2
		49.4384
	46.64	up 6%
	up 6%	43.8416
44		down 6%
		43.8416
	41.36	up 6%
	down 6%	38.8784
		down 6%

The Probability of going up or down is 50%

Hence probability of Prices in period 2 (T2) is 25% each

Price T2	Probability
49.4384	=50%*50%	=25%
43.8416	=50%*50%	=25%
43.8416	=50%*50%	=25%
38.8784	=50%*50%	=25%

(a) Price of Call option -  

Expected Price	Probability	Price after 2 periods	Strike Price	Call Option Value	Call option Value * Probability
49.4384	25%	12.3596	45	4.4384	1.1096
43.8416	25%	10.9604	45	0	0
43.8416	25%	10.9604	45	0	0
38.8784	25%	9.7196	45	0	0
Total	100%	44			1.1096

where, Price after 2 periods= (Expected Price * Probability)

Call option value= max(Expected price-Strike price,0)

Hence, PV = 1.1096/1.02² = $1.067

(b) Price of Put option -

Expected Price	Probability	Price after 2 period	Strike Price	Put Option Value	Put option Value * Probability
49.4384	25%	12.3596	45	0	0
43.8416	25%	10.9604	45	1.1584	0.2896
43.8416	25%	10.9604	45	1.1584	0.2896
38.8784	25%	9.7196	45	6.1216	1.5304
Total	100%	44			2.1096

where, Price after 2 periods= (Expected Price * Probability)

Put option value= max(Strike price-Expected price,0)

Hence, PV = 2.1096/1.02² = $2.028

(c) Let the number of stocks be x

Hence, x*44 = 10000*1.067

=> x = 242.5