Chapter 16, Problem 44C

Summary Introduction

(A)

To Compute:

What will be the payoff to the put, $P_u$, if the stock goes up?

Introduction:

The payoff to the put, $P_u$, if the stock goes up is given in the calculation below.

Explanation of Solution

  $\begin{array}{l}Datacanbesummarizedasfollows:\\ Exercisepriceofstock\text{ =}\$110\\ Riskfreerate,r_t-5\%\\ Factorbywhichstockincreases\left(u\right)=1.1\\ Factorbywhichstockincreases\left(d\right)=0.95\end{array}$

Valuation of one year European put option using the two-state approach is described as

follows:

The put price for increase factor of 1.1, $S_{01}is\text{ \$}121\left[\$110x1.1-\$121\right].$The put price for

decrease factor of 0.95, $S_{02}is\$104.50\left[\$110\times 0.95=\$104.50\right].$The strike or exercise price is

$110.

Calculate the probable value of put, in case stock price increases ( $P_u$), as follows

Conclusion

  ${\text{The put can fall to an expiration-date value of p}}_{\text{u1}}\text{=\$0}\text{and the put can rise to a}{\text{final value of P}}_{\text{u2=}}\text{\$5}\text{.50}$

Summary Introduction

(B)

To Compute:

What will be the payoff, $P_d$if the stock price falls?

Introduction:

The payoff, $P_d$if the stock price falls is given using the calculation below.

Explanation of Solution

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

  $H=\frac{P_{u1}-P_{u2}}{S_{o1}-S_{o2}}$

Substitute the value to calculate hedge ratio as follows:

  $\begin{array}{l}{P}_{u1}=\$0,P_{u2}=\$5.50,S_{o1}=\$121,and{S}_{02}=\$104.50\\ H=\frac{P_{u1}-P_{u2}}{S_{o1}-S_{o2}}\\ =\frac{\$0-\$5.50}{\$121-\$104.50}\\ =-\frac{1}{3}\end{array}$

Hence, the hedge ratio is $-\frac{1}{3}$

The final stock price of the portfolio that will be worth $121 at expiry regardless of the final stock price is as follows:

The portfolio must have a current market value equal to the present value of 4121 defined by the following equation:

  $\begin{array}{l}\$110+3P_u=\frac{\$121}{1.05}\\ P_u=\frac{\$115.238-\$110}{3}\\ P_u=\frac{\$5.238}{3}\\ P_u=\$1.746\\ Hence,thevalueofputincasestockincreaseis\$1.746\end{array}$

Calculate the value of put in case stock price decreases $\left(P_d\right)$as follows:

The put price for increase factor of 1.1, $S_{03}=\$90.25$The put price for

decrease factor of 0.95, $S_{02}=\$104.50\left[\$110\times 0.95=\$104.50\right].$The strike or exercise price is

$110

Calculation of the hedge ratio:

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

  $\begin{array}{l}{P}_{u1}=\$5.50,P_{u2}=\$19.75,S_{o2}=\$104.50,and{S}_{03}=\$90.25\\ H=\frac{P_{u2}-P_{u3}}{S_{o2}-S_{o3}}\\ =\frac{\$5.50-\$19.75}{\$104.50-\$90.25}\\ =\frac{-\$14.25}{\$14.25}\\ =-1.0\end{array}$

Therefore, the hedge ratio is -1.0

The final stock price of the portfolio that will be worth $110 at expiry is as follows:

The portfolio must have a current market value equal to the present value of $110,which is defined by the following equation:

Hence, the value of put in case stock price decrease is $9.762.

Calculate P using the values of $P_u$and $P_d$

The put can increase to a value of $P_d$=$9.762, at this point the stock price is $d{S}_o=\$95$from its

Initial value.

The put can fall to a value of $P_u=\$1.746$, At this point the stock price is $u{S}_o=\$110$from its

Initial value.

Calculate hedge ratio at this point as follows:

$\begin{array}{l}H=\frac{P_u-P_d}{u{S}_o-d{S}_o}\\ =\frac{\$1.746-\$9.762}{\$110-\$95}\\ =\frac{-\$8.016}{\$15}\\ =-0.5344\end{array}$

Conclusion

Thus, the payoff, $P_d$if the stock price falls is obtained successfully.

Summary Introduction

(C)

To Compute:

Value the put option using the risk-neutral shortcut described in the box. Confirm that your answer matches the value you get using the two-state approach.

Introduction:

Explanation of Solution

The portfolio will be worth $60.53 at expiry irrespective of the final stock price as follows:

The portfolio must have a market value equal to the present value of $60.53 which is defined by the following equation:

$\begin{array}{l}\$53.44+P=\frac{\$60.53}{1.05}\\ =\$57.648-\$53.44\\ =\$4.208\end{array}$

Hence, the value of put option binomial model option pricing is $4.208

Valuation of put option using risk neutral shortcut is described as follows:

Formula for risk neutral probability is as follows:

$\rho =\frac{1+r_f-d}{u-d}$

Here,

$\rho$is the probability

$r_f$is the risk free rate

u is factor by which stock increases

d is factor by which stock increases

Calculate the risk neutral probability that the stock price will increase as follows:

$\begin{array}{l}\rho =\frac{1+r_f-d}{u-d}\\ =\frac{1+0.05-0.95}{1.1-0.95}\\ =\frac{0.1}{0.15}\\ =0.6667\end{array}$

Calculate the expected cash flows at expiration and discount it by the risk free rate to find $P_u$

and $P_d$using the probability as follows:

Calculate $P_u$as follows:

  $\begin{array}{l}uE\left(CF\right)=0.6667\times 0+\left(1-0.6667\right)\times \$5.5\\ =0.6667\times 0+0.3333\times \$5.5\\ =\$1.8333\\ P_u=\frac{E\left(CF\right)}{1+r_f}\\ =\frac{\$1.8333}{1+0.05}\\ =\frac{\$1.8333}{1.05}\\ =\$1.7460\end{array}$

Calculate $P_d$as follows:

  $\begin{array}{l}dE\left(CF\right)=0.6667\times \$5.50+\left(1-0.6667\right)\times \$19.75\\ =0.6667\times \$5.50+0.3333\times \$19.75\\ =\$10.25\\ P_d=\frac{E\left(CF\right)}{1+r_f}\\ =\frac{\$10.25}{1+0.05}\\ =\frac{\$10.25}{1.05}\\ =\$9.7619\end{array}$

Calculate the expected cash flow in 6 months and discount the E(CF) by the 6 month risk free

Rate as follows:

  $\begin{array}{l}E\left(CF\right)=0.6667\times \$1.7460+\left(1-0.6667\right)\times \$9.7619\\ =0.6667\times \$1.7460+0.3333\times \$9.7619\\ =\$4.4180\\ P=\frac{E\left(CF\right)}{1+r_f}\\ =\frac{\$4.4180}{1+0.05}\\ =\frac{\$4.4180}{1.05}\\ =\$4.208\end{array}$

Hence, the value of put option using risk neutral shortcut is $4.208

Therefore, the value of put option through two state approach and risk neutral shortcut is same, that is, $4.208.

Conclusion

Therefore, the value of put option through two state approach and risk neutral shortcut is same, that is, $4.208.