Question

Stock Z is trading at $50 today. In one year, the value will go either up to $62.50 or down to $40. A call option on Z with exactly one year to expiration has a strike price of $55. Inflation is high, so the interest rate is 10% per year.

1. Find the value of the call option using binomial approach.

2. What is the hedge ratio of the call option?

3. From the put-call parity, what should be the price of the identical put option?

Answer 1

Strike price = $55
Current Market price = $50
Risk-free rate = 10%
Rate for 1 year (r) = 10%*1 = 10%
Continuous compounded rate (e^r) formula =(r)^1/ 1 + (r)^2 / (2*1) + (r)^3 / (3*2*1) + (r)^4 / (4*3*2*1)
(10%)^1 /1 + ((10%)^2 / (2*1) )+ ((10%)^3 / (3*2*1)) + ((10%)^4 / (4^3*2*1))
0.1051674479
Possible upside price (UP)=$62.50
Possible downside price (DP)= $40
Value of call option = Stock price at Expiration -Strike price(subject to 0, value cannot be negative)
VC at UP = 62.50-55 = 7.5
VC at DP = 40-55 = 0
Call detlta = (VC at upper - VC at lower)/(Upper price - downside price)
(7.5-0)/(62.50-40)
0.3333333333
Call option price as per binomial model= (Current Market price - (downside price/(1+e^i)))* Call delta
(50-(40/(1+0.1051674479)))*0.3333333333
4.60
So call option price is $4.60 according to binomial model.

Put option has same terms as call option. So we will Calculate put option price by Put call parity equation

As per put call parity = Current market price + Put option value = (Excercise price/(1+e^i))+call option value

50+VP = (55/(1+0.1051674479))+4.60

50 + VP = 54.36621426

VP = 54.36621426-50

VP = 4.366214255

So put option price is 4.37

a. Call option priece is $4.60

b. hedge ratio is 0.3333

c. put option price is 4.37