Question

QST’s current share price is $200and it pays no dividends. In each of the next two years, the price goes up by 20%or go down by 25%. The annual,constant, risk-free rate is 6%.

(a) Using the binomial model (replication or risk-neutral valuation), what is the price of a two-year European call option on QST’s stock with a strike price of $190?

(b) Using the binomial model (replication or risk-neutral valuation), what is the price of a two-year European put option on QST’s stock with a strike price of $190?

(c) Verify that your answers to (a) and (b) indeed satisfy the put-call parity.

Answer 1

The binomial model is a very popular method used to calculate or price the value of an option. It is based on the idea that there are 2 possible outcomes for every iteration, a move-up or move down that forms a binomial tree. In this model, we have to calculate the probability of an up-move and a down-move and then compute the value of the option.

We are given an up-move probability of 20% and a down-move probability of 25%. So we can calculate the price in year 1 and 2 by forming a binomial tree,

240*1.20 = 288

200*1.20 = 240

200 240*.75 = 180 or 150*1.20 = 180

200*.75 = 150

150*.75 = 112.5

The value of the option at expiration in each state is equal to the stock price minus the exercise price ( or 0, if that difference is negative).

C++ = max( 0 , 288 - 190 ) = 98

C-+ = max( 0 , 180 - 190 ) = 0

C+- = max( 0 , 180 - 190 ) = 0

C-- = max( 0 , 112.5 - 190 ) = 0

Value of call option in up-state is,

Risk free rate (rf) = 6%

C+ =

To compute the probability of up-move and down-move we have a formula as follows,

Probability of Up-move =

= 1 +.06 - .75 / 1.20 - .75

= .31 / .45 = 69% approx

Probability of down-move = 1 - Prob of up-move = 1 - .69 = .31 or 31%

Putting these in the formula of valuation of call option,

C+ = ( (.69 * 98) + (.31 * 0) ) / 1 + Rf = (67.62 + 0) / 1.06 = 63.8

Similarly for C-,

C- = ( (.69 * 0) + (.31 * 0) ) / 1 + Rf = (0 + 0) / 1.06 = 0

Now we know the value of up-state and down state one period from now, to get the value of option today we apply the methodology one more time,

C =   

= (.69 * 63.8) + (.31 * 0) / 1 + Rf

= 44.02 / 1.06 = 41.53

Thus, value of call is 41.53

b)

Similarly for calculating the value of put option,

P++ = max( 0 , 190 - 288 ) = 0

P-+ = max( 0 , 190 - 180 ) = 10

P+- = max( 0 , 190 - 180 ) = 10

P-- = max( 0 , 190 - 112.5 ) = 77.5

Value of Put option at time T1,

P+ =

P+ = ( (.69 * 0) + (.31 * 10) ) / 1 + Rf = (0 +3.1 ) / 1.06 = 2.92

P- = ( (.69 * 10) + (.31 * 77.5) ) / 1 + Rf = (6.9 +24 ) / 1.06 = 29.17

P =   

=   (.69 * 2.92) + (.31 * 29.17) / 1 + Rf = (2 + 9) / 1.06 = 10.4

The value of put is 10.4

c)

According to Put-Call Parity,

S = Spot price of stock = 200

P = value of put = 10.4

C = value of call = 41.53

PV(X) = Present value of exercise value = 190/1.06*1.06 = 169

Putting these value in the Put-Call parity equation,

200 + 10.4 = 41.53 + 169 = 210.5 ( rounding for slight difference)

So the value of the Call and put option satisfies the Put-Call parity condition.

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