EBK INVESTMENTS
EBK INVESTMENTS
11th Edition
ISBN: 9781259357480
Author: Bodie
Publisher: MCGRAW HILL BOOK COMPANY
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Chapter 21, Problem 36PS
Summary Introduction

To calculate the value of call option using the given prices

Expert Solution & Answer
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Answer to Problem 36PS

Hence, the value of call option is $13.6364.

Explanation of Solution

Given Information:

You are attempting to value a call option with an exercise price of $100 and one year to expiration. The underlying stock pays no dividends, its current price is $100 , and you believe it has a 50% chance of increasing to $120 and a 50% chance of decreasing to $80 . The risk-free rate of interest is 10%.

Data can be summarized as follows:

Exercise price of call option (X)=$100

Time to expiration (T)=1 year

Dividend payout = 0

Stock price (S0)=$100

Probability of stock price increasing to $120(uS0)=0.5

Probability of stock decreasing to $80(dS0)=0.5

Risk-free rate of interest (r)=10%

Methodology to arrive at the option value is as follows:

1. Calculate the probable call values

2. Then based on the call values arrived in both the cases, i.e. in case of stock price increase

and decrease, the hedge ratio needs to be calculated

3. Calculate riskless portfolio.

4. Then the present value of stock price highly probable at the end of year can be calculated.

5. Then set the value of hedged position equal to the present value of certain pay-off.

6. Finally, value of the call option can be calculated.

Step 1 Probable Call value for stock price $120=uS0X

CU=$120$100=$20

Probable Call value for stock price $80=dS0X

Cd=$80$100=0

Therefore, Cu=$20 is in-the-money and Cd=$0 is out-of-the-money.

Calculate the hedge ratio.

The hedge ratio, H can be calculated by using the following formula:

H=(CuCd)(uS0dS0)

Here,

Value of the call option in case of in-the-money scenario is Cu

Value of the call option in case of out-of-the-money scenario is Cd

Stock price in case stock price increases is uS0

Stock price in case stock price decreases is dS0

Factor by which stock increases is u

Factor by which stock decreases is  d

Substitute the values in the formula as follows:

Cu=$20 , Cd=$0 , uS0=$120 and dS0=$80

H=(CuCd)(uS0dS0)=($20$0)($120$80)=$20$40=0.5

Hence, the hedge ratio is 0.5.

Create a riskless portfolio that consists of two written calls and one share of stock. The riskless

Portfolio's cost is S02C =1002C.

The following table shows the payoff of the portfolio to be risk free.

    Portfolio   S=$80   S=$120
    Buy 1 share   $80   $120
    Write 2 calls   $0   $40
    Total   $80   $80

Present value of portfolio is as follows:

Present value of stock price =Payoff(1+r)1

=$80( 1+0.10)1=$801.10=$72.7273

Hence, the present value of portfolio is $72.7273.

Step 5:

Set the value of the hedged position equal to the present value of the payoff as follows:

  $1002C = $72.7273

Step 6:

Calculate the value of call option (C) as follows:

  $1002C = $72.73C=10072.72732=$13.6364

Hence, the value of call option is $13.6364.

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