2.2: Valuation of Options

Question 1

How does the binomial option pricing model price options?

Answer 1

By making the simplifying assumption that at the end of the next period, the stock price has only two possible values. Payoffs can be replicated exactly by constructing a portfolio of a risk-free bond and the underlying stock.

Question 2

What is ∆ in the binomial option pricing model?

Answer 2

The number of shares of stock we purchase

Question 3

What is B in the binomial option pricing model?

Answer 3

The initial investment in bonds

Question 4

How do we calculate the value of an option in a multi period binomial tree?

Answer 4

Start at the end and work backwards.

Question 5

Between what values are ∆?

Answer 5

• Call: between 0 and 1.
• Put: between 0 and -1.

Question 6

How do we make the binomial model more realistic?

Answer 6

Binary movements are not realistic over long time periods. Hence we increase the number of time periods and decrease the length of each period. When letting the length shrink to 0 and the number of periods grow infinitely large, we arrive at Black-Scholes.

Question 7

What options’ values can be calculated with the Black-Scholes?

Answer 7

European puts and calls. (and American calls on non-dividend-paying stocks)

Question 8

What input parameters do we need to price a call with Black-Scholes?

Answer 8

1. Stock price
2. Strike price
3. Exercise date
4. Risk-free interest rate
5. Volatility of the stock (only one that needs to be forecasted)

Question 9

What is N(d) in Black-Scholes?

Answer 9

The cumulative normal distribution; the probability that a normally distributed variable is less than d.

Question 10

How do we adjust the Black-Scholes to account for dividends prior to the expiration of a call?

Answer 10

Insert S(x) in the place of S.

S(x) = S - PV(div)

Question 11

What are the two most used strategies to estimate the volatility parameter for the Black-Scholes?

Answer 11

1. Use historical data.
2. Implied volatility: Use current market price of traded options to back out the volatility that is consistent with these prices based on Black-Scholes.

Question 12

What are the betas on calls and puts in Black-Scholes compared to the beta of the underlying stock?

Answer 12

• Call on positive beta stock are more risky (borrow=leveraged), and hence have higher betas
• Put options on positive beta stock will have negative beta

Question 13

How do we measure the risk of an option?

Answer 13

By computing the option beta, which is easiest done by computing the beta of the replicating portfolio (weighted average of components)

Question 14

How does the beta of an option change as the option goes closer in-the-money?

Answer 14

The magnitude of beta falls. On SML: Out-of-the-money call have highest expected return, puts the lowest

Question 15

What is the value of a call at expiration?

Answer 15

C = max(S-K, 0)

Question 16

What is the value of a put at expiration?

Answer 16

P = max(K-S, 0)

Question 17

How do we adjust the put-call-parity for dividends?

Answer 17

Subtract the present value of the dividends:
-PV(div)

Question 18

What is the formula for the option price of a call in the binomial model?

Answer 18

C = S∆ + B

Question 19

What is the formula for ∆ in the binomial model?

Answer 19

∆ = (Cu - Cd) / (Su - Sd)

Question 20

What is the formula for B in the binomial model?

Answer 20

B = (Cd - Sd∆) / (1+rf)

Question 21

What does the option delta measure?

Answer 21

The number of shares of the underlying stock. It approximately measures the value change of the option in response to a $1 change in the value of the underlying stock, for small changes in it