Question

The following is a Binomial Option Pricing Model question. There will be 7 questions asked about it. Since the order of questions chosen is random, I suggest you solve the following all at once and choose your answer to each part as it comes up.

You will be asked the following questions:

1. What are the values of the calls at maturity, t=2?

2. What are the values of the calls at t =1?

3. What is the initial (t = 0) fair market price of the call?

4. What is the initial (t = 0) hedge ratio?

5. What are the hedge ratios at t = 1?

6. If one call was written initially, what is the value of the hedged portfolio one period later (t = 1)?

7. If the stock moves down in period 1 how would you adjust your t = 0 hedge by trading only stock?

We have a 2-state, 2-period world (i.e. t = 0, 1, 2). The current stock price is 100 and the risk-free rate each period is 5%. Each period the stock can either go up by 10% or down by 10%. A European call option on this stock with an exercise price of 90 expires at the end of the second period.

If one call was written initially, what is the value of the hedged portfolio one period later (t = 1)? (closest answer)

		62.65
		89.65
		96.65
		84.65
		73.65

Answer 1

The various possible stock prices and the value of call option at t=2 are as given below

Prices of Stock			Values of call
		121	31
	110	99	9
100	90	81	0
t=0	t=1	t=2

1. The values of the call option at maturity i.e. t=2 are

$31 (when stock price is $121)

$9 (when stock price is $99)

$0 (when stock price is $81)

2. Risk neutral probability p = (1.05- 0.9)/(1.1-0.9) = 0.75

So, the value of call (when stock price is $110 at t=1)

= (p*value of call when stock price is $121 +(1-p)*value of call when stock price is $99)/1.05

=(0.75*31+0.25*9)/1.05

=$24.28571

Similarly

the value of call (when stock price is $90 at t=1)

= (p*value of call when stock price is $99 +(1-p)*value of call when stock price is $81)/1.05

=(0.75*9+0.25*0)/1.05

=$6.428571

3. At t=0, the initial fair price of the call

= (p*value of call when stock price is $110 +(1-p)*value of call when stock price is $90)/1.05

=(0.75*24.28571+0.25*6.428571)/1.05

=$18.87755

or $18.88

4. The hedge ratio is the given by Delta of the option

Delta of the option initially at t=0

= (value of option when the stock price is $110 - value of the option when the stock price is $90)/($110-$90)

=(24.28571-6.428571)/20

=0.892857

So, for each option shorted we have to hold 0.892857 shares to create a riskless portfolio