Chapter 21, Problem 44PS

Summary Introduction

(A)

To Compute:

If the desired put option were traded. How much would it cost to purchase?

Introduction:

The value of put option is calculated using the formula given below.

Explanation of Solution

a) Computation of value of put:

Here, the investor would hold a put position. The current price is $100. This price could either

increase or decrease by 10%

  $\begin{array}{l}Exercisepriceofputoption\left(X\right)-\$100\\ Timetoexpiration\left(T\right)-1year\\ Dividend=0\\ Stockprice\left(S_0\right)\$100\\ Probabilityofstockpriceincreasingto\text{ \$}110\left(P_1\right)0.5\\ Probabilityofstockpricedecreasingto\text{ \$}90\left(P_2\right)0.5\end{array}$

Factor by which stock increases is u

Factor by which stock decreases is d

$T-billsrate\left(r\right)-5\%$

Calculation of option value

The two possible stock prices are

$S_1\text{=\$}110and{S}_2\text{=}\$90$

since, the exercise price is $100, and the corresponding two possible put values are

$P_d-\$0\left(out-of-the-money\right)and{P}_u=10\left(in-the-money\right).$

Calculation of the hedge ratio (H)

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

$H=\frac{\left(P_d-P_u\right)}{u{S}_0-d{S}_0}$

Here,

The value of the call option in case of in-the-money scenario is $P_u$

The value of the call option in case of out-of-the-money scenario is $P_d$

Factor by which stock increases u.

Factor by which stock decreases is d.

Stock price in case stock price increases is $u{S}_0$

Stock price in case stock price decreases is $d{S}_0$

Substitute:

  $\begin{array}{l}{P}_u=\$10,P_d=\$0,u{S}_0=\$110,d{S}_0=\$90,u=1.1,andd=.9:\\ H=\frac{\left(P_d-P_u\right)}{u{S}_0-d{S}_0}\\ =\frac{\left(0-10\right)}{\left(110-90\right)}\\ =\frac{-10}{20}\\ =-0.5\end{array}$

Hence, the hedge ratio is -.5

The portfolio combining of two puts and one share brings an assured payoff $ 110 making it a riskless portfolio.

Calculation of the present value of certain stock price of $110 with a one-year interest rate of 5%:

  $\begin{array}{l}\mathrm{Pr}esent\text{}\text{}valueofstockprice=\frac{S_1}{{\left(1+r\right)}^1}\\ Substitute:\\ S_1=\$110andr=5\%\end{array}$

  $\begin{array}{l}\mathrm{Pr}esent\text{}\text{}valueofstockprice=\frac{S_1}{{\left(1+r\right)}^1}\\ =\frac{\$110}{1+.05}\\ =\frac{\$110}{1.05}\\ =\$104.76\end{array}$

Hence, the present value of certain stock price of $110 with a one-year interest rate of 5% $104.76

Comparing the protective put strategy and present value of stock we get,

One stock and two put options

  $So+2P\text{ =\$}104.76$

Solve for the value of put (P):

  $\begin{array}{l}So+2P\text{ =\$}104.76\\ Substitute:\\ S_0\text{=}\$100\left(currentprice\right)\\ S_0+2P=\$104.76\\ P=\frac{\$104.76-\$100}{2}\\ =\frac{\$4.76}{2}\\ =\$2.38\end{array}$

Hence, the value of put option is $2.38.

Conclusion

Hence, the value of put option is $2.38.

Summary Introduction

(B)

To Compute:

What would have been the cost of the protective put portfolio? Introduction: The cost of protective put portfolio that comes out is calculated based on the given formula below.

Explanation of Solution

(b) Computation of cost of protective put portfolio:

The cost of the protective put portfolio is the cost of one share plus the cost of one put. It

can be calculated by using the following formula:

  $\begin{array}{l}Costofprotectiveput\text{ =}{S}_0+P\\ Substitute:\\ S_0\$100andP=\$2.38\\ Costofprotectiveput\text{=}{S}_0+P\\ =\$100+S2.38\\ =\$102.38\end{array}$

Hence, the cost of protective put portfolio comes out to be $102.38.

Conclusion

Hence, the cost of protective put portfolio comes out to be $102.38.

Summary Introduction

(C)

To Compute:

What portfolio position in stock and T-bills ill ensure you a payoff equal to the payoff that would be provided by a protective put with X-$100? Show that the payoff to this portfolio and

the cost of establishing the portfolio match those of the desired protective put.

Introduction:

The strategy is calculated in the table to determine the pay-off using the below calculation.

Explanation of Solution

To ensure equal pay-off in stock and T-bills and protective put is as follows

The hedge ratio is 0.50. The investor will invest 0.50 in stocks which costs $50 and

remaining in T-bills to earn 5% interest.

The above strategy will give same pay-off as the protective put option.

Conclusion

The above strategy will give same pay-off as the protective put option.