Week 5

Question 1

Delta neutral:

Answer 1

Delta neutral means the portfolio has no tendency to change value as the underlying portfolio value changes.

Question 2

Consider the following two statements:

1. Suppose two stocks have identical firm-specific risk. A put option on a higher-beta stock is worth more than one on a low-beta stock.
2. All else equal, a call option with a high exercise price has a higher hedge ratio than one with a low exercise price.

Which of the statements are true?

Answer 2

Only Statement 1 is correct.

Statement 1 is correct. Holding firm-specific risk constant, higher beta implies higher total stock volatility. Therefore, the value of the put option increases as beta increases.

Statement 2 is incorrect. A call option with a high exercise price has a lower hedge ratio. This call option is less in the money. Both d1 and N(d1) are lower when X is higher.

Question 3

Which put option is written on the stock with the lower stock price?

Put T X σ Price of Put
A 0.5 50 0.20 $ 10
B 0.5 50 0.25 $ 10

Answer 3

Put A must be written on the stock with the lower price (which is an attractive feature in a put contract). Otherwise, given the lower volatility of Stock A, Put A would sell for less than Put B.

Question 4

Vega is defined as:

Answer 4

Vega is defined as the sensitivity of an option’s price to changes in volatility.

Question 5

Which of the following is/are correct?

The Black-Scholes formula assumes that

1. the risk-free interest rate is constant over the life of the option.
2. the stock price volatility is constant over the life of the option.
3. the expected rate of return on the stock is constant over the life of the option.
4. there will be no sudden extreme jumps in stock prices.

Answer 5

1, 2, and 4.

The risk-free rate and stock price volatility are assumed to be constant, but the option value does not depend on the expected rate of return on the stock. The model also assumes that stock prices will not jump markedly.

Question 6

All the inputs in the Black-Scholes Option Pricing Model are directly observable except for..

1. the price of the underlying security.
2. the time to expiration.
3. the variance of returns of the underlying asset return.
4. the strike price.

Answer 6

The variance of the returns of the underlying asset is not directly observable, but must be estimated from historical data, from scenario analysis, or from the prices of other options.

Question 7

Give an example of an option for which we prefer using binomial trees
instead of Black-Scholes option pricing

Answer 7

The Black-Scholes option price does not consider the path of the stock price
between maturity and 𝑡 = 0, whereas binomial trees do consider this.

Example: Asian options depend on the average stock price in a predefined
period, for example the last three months before maturity.