Corporate Finance, Module 23: “Advanced Option Valuation”

Volatility Questions

(The attached PDF file has better formatting.)

{This posting contains more information than is needed for the corporate finance on-line course.}


Question 23.1: Volatility

The stock of XYZ Company has an expected annual return of 10%. For the binomial tree pricing method, the upward movement u is modeled as eσ√h and the downward movement d is modeled as 1/u. If the downward movement of the stock is –15% in the next six months and the annual risk-free rate is 6%, which of the following is closest to the standard deviation of the stock’s annual return?

A.    (–ln(0.85) / 0.50.5)0.5
B.    –ln(0.85) × √0.5
C.    ln(0.85) / √0.5
D.    –ln(0.85) / √0.5
E.    ln(0.85) × 0.5

Answer 23.1: D

The downside change is

e–σ√t = 0.85
–σ √t = ln(0.85)
–σ √0.5 = ln(0.85)
σ = –ln(0.85) / √0.5

This is a standard convention for choosing the upward and downward movements.

Question 23.2: Volatility

As the stock price volatility increases, which of the following is true?

A.    The call option value and put option value both increase.
B.    The call option value and put option value both decrease.
C.    The call option value increases and the put option value decreases.
D.    The call option value decreases and the put option value increases.
E.    None of A, B, C, or D is true.

Answer 23.2: A

Options are a one-sided investment: one gains if the stock moves one way, but one does not lose (except for the original premium, which is a sunk cost) if the stock moves the opposite way. More uncertainty in the stock price movement increases the value of the option, whether it is a call option or a put option.

Question 23.3: Volatility

European call and put options are trading on a stock. The stock price is $80, the strike price is $80, the time to maturity is one year, and the risk-free interest rate is 6% with continuous compounding. As the stock price volatility increases without bound (i.e., goes to infinity), what happens to the value of the options?

A.    Call → $80; put → $80e–6%
B.    Call → $80; put → $80 × (1 – e–6%)
C.    Call → $80 × (1 – e–6%); put → $80e–6%
D.    Call → $80 × (1 – e–6%); put → $80 × (1 – e–6%)
E.    None of A, B, C, or D is true.

Answer 23.3: A

The call value → $80 and the put value → $80e–6%.

In the Black-Scholes formula, as σ → ∞, d1 → ∞ and d2 → –∞, so N(d1) → 1 and N(d2) → 0.

●    The call option value → S0 × 1 – PV(X) × 0 = S0.
●    The put option value → PV(X) × (1 – 0) – S0 × (1 – 1) = PV(X).

The maximum value of a call option is the current price of the stock, and the maximum value of a put option is the present value of the strike price. As the volatility increases without bound, the option approaches its maximum value.

Question 23.4: Volatility

One year European call and put options are traded on a stock whose stock price volatility is 30%. An investor is using a binomial tree pricing method with N periods to estimate the value of the options, with an upward stock price movement of e0.25 × √N and a downward price movement that is the reciprocal of the upward price movement. Which of the following is true?

A.    The investor has over-estimated the value of both the call and put options.
B.    The investor has under-estimated the value of both the call and put options.
C.    The investor has over-estimated the value of the call option and under-estimated the value of the put option.
D.    The investor has under-estimated the value of the call option and over-estimated the value of the put option.
E.    None of A, B, C, or D is true.

Answer 23.4: B

The volatility used in the binomial tree is too low, so the investor has under-estimated the value of both the call and put options.