Question

A stock price is currently $40. Over each of the next two three-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding.

1. What is the value of a six-month European put option with a strike price of $42?
2. What is the value of a six-month American put option with a strike price of $42?
3. What is the value of a six-month American put option with a strike price of $45? What do you conclude about whether or not it is optimal to exercise this American option immediately (What would be the value of this American option if it were to be exercised immediately)

Show your working by displaying the stock values and the values of the European and American options at each node of the tree

Answer 1

a) The binomial tree and the value of the European put option with strike $42 at maturity are as shown below

		E 48.4	Value at E=0
	B 44
A   40		F 39.6	Value at F = 2.4
	C 36
		G 32.4	Value at G= 9.6

d= 0.9 (downward movement of 10%)

u =1.1 (upward movement of 10%)

p = (e^(0.12*3/12) -0.9)/(1.1-0.9) = 0.6523

So, considering tree at Node B, Value of option at node B

= (p*value of option at node E+ (1-p)*value of option at node F)*e^(-0.12*3/12)

=(0.6523*0+0.3477*2.4)* e^(-0.03) = $0.81

considering tree at Node C, Value of option at node C

= (p*value of option at node F+ (1-p)*value of option at node G)*e^(-0.12*3/12)

=(0.6523*2.4+0.3477*9.6)* e^(-0.03) = $4.7585

Considering tree at Node A, Value of option at node A

= (p*value of option at node B+ (1-p)*value of option at node B)*e^(-0.12*3/12)

=(0.6523*0.81+0.3477*4.7585)* e^(-0.03) = $2.12

So, value of a six-month European put option with a strike price of $42 is $2.12

b) If the Option is an American option, as above, it can be exercised early at node B or C

Value of option at node B if exercised = max(42-44,0) = 0

If not exercised, value = $0.81

So, the value of $0.81 (higher value) will prevail as the holder will choose not to exercise

Value of option at node C if exercised = max(42-36,0) = $6

If not exercised, value = $4.7585

So, the value of $6 (higher value) will prevail as the holder will choose to exercise the option at node C

Thus, Value of option at node A

= (p*value of option at node B+ (1-p)*value of option at node B)*e^(-0.12*3/12)

=(0.6523*0.81+0.3477*6)* e^(-0.03) = $2.54

So, value of a six-month American put option with a strike price of $42 is $2.54

(as it is not optimal to exercise it immediately at Node A)

c)

The binomial tree and the value of the American put option with strike $45 at maturity are as shown below

		E 48.4	Value at E=0
	B 44
A   40		F 39.6	Value at F = 5.4
	C 36
		G 32.4	Value at G= 12.6

So, considering tree at Node B, Value of option at node B

= (p*value of option at node E+ (1-p)*value of option at node F)*e^(-0.12*3/12)

=(0.6523*0+0.3477*5.4)* e^(-0.03) = $1.82

Value of option at node B if exercised = max (45-44,0) = $1

So, the value of $1.82 (higher value) will prevail as the holder will choose not to exercise

considering tree at Node C, Value of option at node C

= (p*value of option at node F+ (1-p)*value of option at node G)*e^(-0.12*3/12)

=(0.6523*5.4+0.3477*12.6)* e^(-0.03) = $7.67

Value of option at node C if exercised = max (45-36,0) = $9

So, the value of $9 (higher value) will prevail as the holder will choose to exercise the option at node C

Considering tree at Node A, Value of option at node A

= (p*value of option at node B+ (1-p)*value of option at node B)*e^(-0.12*3/12)

=(0.6523*1.82+0.3477*9)* e^(-0.03) = $4.19

Value of option at node A if exercised = max (45-40,0) = $5

So, the value of $5 (higher value) will prevail as the holder will choose to exercise the option at node A

So, value of a six-month European put option with a strike price of $45 is $5

It is optimal to exercise this option immediately upon purchase

Value of the option if immediately exercised =$5 as above