Chapter 21, Problem 1PS

Binomial model* Over the coming year, Ragwort’s stock price will halve to $50 from its current level of $100 or it will rise to $200. The one-year interest rate is 10%.

1. a. What is the delta of a one-year call option on Ragwort stock with an exercise price of $100?
2. b. Use the replicating-portfolio method to value this call.
3. c. In a risk-neutral world, what is the probability that Ragwort stock will rise in price?
4. d. Use the risk-neutral method to check your valuation of the Ragwort option.
5. e. If someone told you that in reality there is a 60% chance that Ragwort’s stock price will rise to $200, would you change your view about the value of the option? Explain.

Summary Introduction

To compute: The delta of one year call option on R stock with a strike price of $100.

Explanation of Solution

The formula to calculate delta is:

$\text{Delta}=\frac{\text{Spread of possible option price}}{\text{spread of possible share price}}$

The calculation of delta is as follows:

$\begin{array}{c}\text{Delta}=\left(\frac{\$100-0}{\$200-50}\right)\\ =.6667\end{array}$

Summary Introduction

To discuss: Apply the replicating portfolio technique to value this call.

Explanation of Solution

The replicating portfolio technique of valuing call is as follows.

Summary Introduction

To discuss: The probability of increasing stock R price in a risk neutral world.

Explanation of Solution

The probability of increasing stock R calculated as follows:

$\begin{array}{l}\text{Expected value of calloption}=\left(\text{Probability of rise}\times \text{call option value in rise}\right)\\ +\left[\left(1-\text{Probability of rise}\right)\times \text{Call option value in fall}\right]\end{array}$

The computation as follows:

$\begin{array}{c}\text{Expected value of call option}=\left(p\times 1.00\right)+\left(1–p\right)\left(–.50\right)\\ =.10\\ p=.40\end{array}$

Foot note: The probability is calculated on the basis of expected return.

Summary Introduction

To compute: The value of stock R using the risk neutral method.

Explanation of Solution

The option value is calculated using the following formula:

$\begin{array}{c}\text{Value of call}=\frac{\text{Expected future value}}{\text{1+Interest rate}}\\ =\frac{.40}{1.10}\\ =0.3636\text{ OR \$36}\text{.36}\end{array}$

Hence, the value of call is $36.36

Summary Introduction

To discuss: Whether person X change his option regarding the value of option.

Explanation of Solution

Person X does not change his opinion regarding the value of option. The chance of price increase is most likely higher than the risk- neutral probability, but it does not aid to value the option.