Question

a) The price company A’s stock is $50 and the price of its 3-month European call option on the stock with a strike price of $52 is $2. Draw the payoff graph for the option buyer.

b)The risk-free rate is 4% (compounded quarterly). The 3-month European put option with a strike price of $52 is sold for $3. The stock pays a quarterly dividend of $0.5. Given the call option information in a), describe the arbitrage strategy and calculate the profit.

c)Company B’s stock price is currently $20. It is known that at the end of 3 months it will be either $23 or $18. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a 6-month European call option with a strike price of $21? Show your work

Answer 1

The payoff of a long call option is given by = max( St-K, 0)

where max function returns the maximum of the two values separated by comma  

St is the stock price at maturity and K is the strike price

The payoff table for various possible values of St as well as the payoff diagram is given below

St	Payoff	Premium	Total payoff
48	0	2	-2
49	0	2	-2
50	0	2	-2
51	0	2	-2
52	0	2	-2
53	1	2	-1
54	2	2	0
55	3	2	1
56	4	2	2
57	5	2	3
58	6	2	4

b)

From the put call parity equation  

c+ K/(1+r)^t = p+S

where c and p are call and put option premiums respectively,=$2 and $3 respectively

K is the strike price of the options =$52

, r is the periodic interest rate = 4% per annum or 1% per quarter

and t is the no. of periods = 3 months or 1 period

, S is the adjusted spot price net of present value of dividends =(50-0.5/1.01)= 49.505

So, LHS = 2+52/1.01 = 53.48515

LHS = 49.505+3 =52.505

As p+S is cheaper and c+K/(1+r)^t is costlier , the arbitrage steps are as follows

1. Today Sell the call option for $2, borrow $51 for 3 months and from the $53, buy the stock and put option for $50 and $3

2. After 3 months, the payable amount= $51*1.01 = $51.51

Get the dividend of $0.5

If Stock price < $52, call option will be worthless and put option will be exercised , sell the stock using put option at $52 and pay the amount of $51.51 , arbitrage profit = $52 +$0.5 - $51.51 = $0.99

If Stock price > $52, call option will be exercised and put option will be worhtless , sell the stock using call option at $52 and pay the amount of $51.51 , arbitrage profit = $52 +$0.5 - $51.51 = $0.99

If Stock price = $52, both options are worhtless , sell the stock in market at $52 and pay the amount of $51.51 , arbitrage profit = $52 +$0.5 - $51.51 = $0.99

So, in any future situation, profit of $0.99 can be made

c) The stock price can follow the following route (2 period binomial model)

TODAY	AFTER 3 months	AFTER 6 months
		26.45	Value =5.45
	23
20		20.7	Value =0
	18
		16.2	Value =0

u =$23/$20 =1.15 , d= $18/$20 = 0.9

p = (exp(rt)-d)/(u-d) = (exp(0.05*3/12)-0.9)/(1.15-0.9) = 0.4503

Value of option when Stock price = $23 (after 3 months)

= (p*value of option when stock price is $26.45 + (1-p) *value of option when stock price is $20.7)* exp(-rt)

=(0.4503*5.45+0.5497*0)*exp(-0.05*3/12)

=$2.423724

Value of option when Stock price = $18 (after 3 months)

= (p*value of option when stock price is $20.7 + (1-p) *value of option when stock price is $16.2)* exp(-rt)

=(0.4503*0+0.5497*0)*exp(-0.05*3/12)

=$0

Value of option when Stock price = $20 (today)

= (p*value of option when stock price is $23+ (1-p) *value of option when stock price is $18)* exp(-rt)

=(0.4503*2.423724+0.5497*0)*exp(-0.05*3/12)

=$1.077878 or $1.08

Value of six month European call option with strike $21 is $1.08 today