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Chapter 5: Q4B (page 552)

We showed in the text that the value of a call option increases with the volatility of the stock. Is this also true of put option values? Use the put-call parity relationship as well as a numerical example to prove your answer.

Short Answer

Expert verified

Put in the low volatility example has half the expected value of the other.

Step by step solution

01

Given information

Parity relationship formula: C = P + S0 - PV (X) - PV (Dividends)

This implies that with an increase in C due to volatility (with value S and risk free rate), P must increase to balance the equation.

02

Validation through examples

A put with exercise price of 100 can have a value from 90, 100, 110

In this case, the payoff to the put would be 10, 0, and 0 respectively.

Similarly for the stock price of 100 (for less volatility), the payoff would be 5, 0 and 0 respectively.

From the above examples, it is clear that the put in the low volatility example has half the expected value of the other.

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Most popular questions from this chapter

Consider an increase in the volatility of the stock in the previous problem. Suppose that if the stock increases in price, it will increase to \(130, and that if it falls, it will fall to \)70. Show that the value of the call option is higher than the value derived using the original assumptions.

Rich McDonald, CFA, is evaluating his investment alternatives in Ytel Incorporated by analyzing a Ytel convertible bond and Ytel common equity. Characteristics of the two securities are given in the following exhibit:

a. Calculate, based on the exhibit, the

i. Current market conversion price for the Ytel convertible bond.

ii. Expected one-year rate of return for the Ytel convertible bond.

iii. Expected one-year rate of return for the Ytel common equity.

One year has passed and Ytel’s common equity price has increased to $51 per share. Also, over the year, the yield to maturity on Ytel’s nonconvertible bonds of the same maturity increased, while credit spreads remained unchanged.

b. Name the two components of the convertible bond’s value. Indicate whether the value of each component should decrease, stay the same, or increase in response to the:

i. Increase in Ytel’s common equity price.

ii. Increase in bond yield.

A bearish spread is the purchase of a call with exercise price X 2 and the sale of a call with exercise price X 1, with X 2 greater than X 1. Graph the payoff to this strategy and compare it to Figure 15.10 .

A put option on a stock with a current price of \(33 has an exercise price of \)35. The price of the corresponding call option is $2.25. According to put-call parity, if the effective annual risk-free rate of interest is 4% and there are three months until expiration, what should be the value of the put?

Suppose the value of the S&P 500 Stock Index is currently $1,200. If the one-year T-bill rate is 3% and the expected dividend yield on the S&P 500 is 2%, what should the one-year maturity futures price be? What if the T-bill rate is less than the dividend yield, for example, 1%?
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