Question

Calculate the value of a call option on the stock in the previous problem with an exercise price of 110. Verify that the put-call parity relationship is satisfied by your answers to both Problems 8 and 9. (Do not use continuous compounding to calculate the present value of X in this example, because the interest rate is quoted as an effective per-period rate.)

Short answer

a. 0.4

b. $145.455

c. As below

Step-by-step solution

Given information

When S_T = $130, the C = 0

When S_T = $80, the C = $20

Where, S_T is stock price

And C is call price.

Calculation of the hedge ratio

Formula for hedge ratio (H) = Cu – Cd / uS₀– dS₀

= 20 – 0 / 130 – 80

= 0.4

Where, u stands for up

And d stands for down

C is the call option

And S is the stock price

Calculation of the present value of the portfolio 

Riskless Portfolio	ST= 80	ST= 130
2 shares	160	260
5 puts	0	-100
Total	160	160

Present value = $160 / 1.10 = $145.455

Explanation on the value of the put

Portfolio cost = 2S – 5C = $145.455

= 2 x 100 – 5C = $145.455

= 200 – 5C = $145.455

= -5C = $145.455 – 200

= -5C = $54.545

= C = $10.9

Verification of put call parity relationship

P = C – S₀+ PV(X)

$10.91 = $10.91 - $100 + ($110 / 1.10)

$10.91 = $10.91 -$100 + $100

$10.91 = $10.91