Question

Question 1 - 35 Points Consider a European put option on a non-dividend-paying stock where the stock price is $15, the strike price is $13, the risk-free rate is 3% per annum, the volatility is 30% per annum and the time to maturity is 9 months. Consider a three-step troc. (Hint: dt = 3 months). (a) Compute u and d. (b) Compute the European put price using a three-step binomial tree. (c) If the option in (b) is American instead of European, would it ever be optimal to exercise it earlier than maturity at any of the nodes of the tree? In which node is the option early exercised? Compute the American put price. (d) Calculate the delta of the European put option at each time step in (b). (e) Using the result obtained in (b) and the put-call parity, compute the price of an European call with same underlying, strike and maturity of the put option. (f) If the call option in (e) is American instead of European, would it ever be optimal to exercise it earlier than maturity at any of the nodes of the tree? Explain in detail and give the price of this Amer- ical call option.

Answer 1

As HOMEWORKLIB's policy, we are allowed to answer only 4 subparts. I have answered a, b,c and e here.

Given information

St = $45

X = $43

Rf = 3%

Volatility Sigma = 30%

Time to maturity = 9 months

Part a) Compute u and d

The formula to calculate u and d are given below:

The formula to calculate this is given below:

1 + upside change = u =

1+ downside change = d = 1/u

where e is the base of natural logarithm = 2.718

h = time interval = 3 months = 3/12 years (since we have a three step binomial for a period of 9 months h = 9/3 = 3 months)

r = risk free rate

q = dividend rate

Substitute the given values:

u = = 1.1618

d = 1/u = 1/ 1.1618 = 0.8607

Part b:

First calculate p using the formula

probability of upside p =

p = = 0.4876

1-p = 0.5124

Create the 3 step binomial tree using the calculated values as follows:

Put option is a bearish strategy. If the stock price moves up higher than the exercise price, then the payoff is zero, whereas if the stock price falls below the exercise price then there is a profit of (X*e^-rt-St) where r is risk free rate and t is the time left for maturity. In step 1, t = 6 months = 6/12 =0.5 yr, whereas in step 2, t = 3 months = 3/12 = 0.25 yr

Option payoff at each step = max{0, (X*e^-rt - St) }. The calculation for the option payoff is shown for the step 1 below. the remaining is calculated on similar lines.

For the calculation of option payoff for step 1, t = 6 months whereas for step 2, t = 3 months and step 3, t = 0 months. This is used in the calculation below.

	Time to expiry	0.5			0.25			0
	Step 1	option payoff		Step 2	option payoff		Step 2	option payoff
upside (St*u)	$52.28	$0.00 max(0,43*e-3%*0.5 - 52.28) = max (0,-9.92) = 0	upside	$60.74	$0.00	upside	$70.57	$0.00
						downside	$52.28	$0.00
			downside	$45.00	$0.00	upside	$52.28	$0.00
						downside	$38.73	$4.27
downside (St*d)	$38.73	$3.63 max(0,43*e-3%*0.5 - 38.73) = max (0,3.63) = 3.63	upside	$45.00	$0.00	upside	$52.28	$0.00
						downside	$38.73	$4.27
			downside	$33.34	$9.34	upside	$38.73	$4.27
						downside	$28.69	$14.31

Now we calculate the present value of the option payoff at each step after multiplying it by the probability of that event occurring.

Step 1 present value = p*0 + (1-p)*3.63*e^-rt = 0 + 0.5124*3.63*e^3%*0.25 = 0 + 1.8452 = 1.8452 (time period used is 3 months as step 1 occurs after 3 months)

For step 2 events to occur, we have to navigate through step 1, hence we need to multiply the probability to reach step 1 and then multiply the probability again to reach step 2. For example to reach the first upside in step 2, we have to reach the upside of step1 which occurs at a probability of p and then reach the upside of step 2 which occurs at a probability of p. hence in effect the probability for step 2 upside to occur is p*p. We apply similar logic to all the events.

Step 2 present value = p*p*0 + p*(1-p)*0 + (1-p)*p*0 + (1-p)*(1-p)*9.34*e^-rt = 0+0+0+0.5124*0.5124*9.34*e^-3%*0.5 = 2.4165

Step 3 present value = p*p*p*0 + p*p*(1-p)*0 + p*(1-p)*p*0 + p*(1-p)*(1-p)*4.27*e^-3%*0.75 + (1-p)*p*p*0 + (1-p)*p*(1-p)*4.27*e^-3%*0.75 + (1-p)*(1-p)*p*4.27*e^-3%*0.75 + (1-p)*(1-p)*(1-p)*14.31*e^-3%*0.75

= 0.5343 + 0.5343 + 0.5343 + 1.8822 = 2.9508

European put option premium = Step 1 present value + Step 2 present value + Step 3 present value

= 1.8452 + 2.4165 + 2.9508 = 7.2125

European put option premium = $7.2125

Part c:

Here we have an American option instead of European and the stock doesn't pay dividend. Since there is no dividend the formula to calculate probability remains the same with q=0.

We will make one changes to the calculations above.

1. We will not discount the strike price X in Step 1, 2 and 3 to calculate the option payoff.

The revised binomial tree is copied below:

	Time to expiry	0.5			0.25			0
	Step 1	option payoff		Step 2	option payoff		Step 2	option payoff
upside (St*u)	$52.28	$0.00 Max(0,X-St) = Max(0, 43-52.28)= Max(0,-9.28)	upside	$60.74	$0.00	upside	$70.57	$0.00
						downside	$52.28	$0.00
			downside	$45.00	$0.00	upside	$52.28	$0.00
						downside	$38.73	$4.27
downside (St*d)	$38.73	$4.27	upside	$45.00	$0.00	upside	$52.28	$0.00
						downside	$38.73	$4.27
			downside	$33.34	$9.66	upside	$38.73	$4.27
						downside	$28.69	$14.31

The highlighted cells in green have the option payoff significantly greater (even after discounting the payoff with the interest rate) than the option payoff from the European put option calculated above. Hence it is optimal to exercise the put option at this steps.

Part e:

Put call parity formula

p + S0 = c + Xe^-rT .\

The put price calculated above = 7.2125

S0= 45

substitute the values in the formula:

7.2125 + 45 = c + 43 * e^-3%*9/12

Solving for c = 52.2125 - 42.0433 = 10.1692

Hence call option price = $10.1692