Ch 17 - The Greek Letters

Question 1

What are the 5 inputs in the Black-Scholes-Merton model?

Answer 1

asset price
exercise price
asset price volatility
time to expiration
risk free rate

Question 2

What is the relationship between option price and its underlying price for calls and puts?

Answer 2

Call: Positive curve (directly related) Increasing gradient
Put: Negative curve (inversely related) increasing gradient

Question 3

What is the relationship between option price and its exercise price for calls and puts?

Answer 3

Call: Negative curve (inversely related) increasing gradient
Put: Positive curve (directly related) Increasing gradient

Question 4

What is the relationship between option price and the risk-free rate for calls and puts?

Answer 4

Call: Slightly positive line (directly related)
Put: Slightly negative line (inversely related)

Question 5

What is the relationship between option price and the time to expiration for calls and puts?

Answer 5

Call & Put: Negative curve (decreasing gradient)

Question 6

What is the relationship between option price and the stock price volatility for calls and puts?

Answer 6

Call: Slightly positive curve
Put: Slightly positive curve

Question 7

What does Delta measure?

Answer 7

The sensitivity of the option price to a change in the price of the underlying

Question 8

What does Gamma measure?

Answer 8

How well the delta sensitivity measure will approximate the option price’s response to a change in the price of the underlying

Question 9

What does Theta measure?

Answer 9

The rate at which the time value decays as the option approaches expiration

Question 10

What does Vega measure?

Answer 10

The sensitivity of the option price to volatility

Question 11

What does Rho measure?

Answer 11

The sensitivity of the option price to the risk free rate

Question 12

Give the definition of Delta

Answer 12

The rate of change of the option price with respect to the price of the underlying asset

Question 13

Give the type of gradient of a call option’s delta and of a put option’s delta. What does this mean in each case?

Answer 13

Delta of call is positive - as the underlying price increases so too does the call

Delta of put is negative - as the underlying price increases, the put decreases

Question 14

In the BSM model, what does N(d1) approximate?

Answer 14

delta of a call option for SMALL CHANGES

in S

Question 15

In the BSM model, what does N(d1) -1 approximate?

Answer 15

The delta of a put option for SMALL CHANGES in

S

Question 16

Give the general formula for calculating delta.

Answer 16

Change in option price / change in underlying price

Question 17

The delta of a European call on a non-dividend-paying stock is?

Answer 17

N(d1) for small changes in S

Question 18

The delta of a European put on the stock is?

Answer 18

[N(d1) - 1] for small changes in S

Question 19

How is a delta used to calculate the change in option price given a change in underlying price? Give the general formula

Answer 19

Given that Delta = (change in option price/change in
underlying price)

Then we should expect that:
Change in option price = delta x change in underlying price

For EXTRA EXPLANATION ONLY:
If delta = 0.6733 then a $1 change in the underlying
would result in a:
􀁺 0.6733 change in the call option price
􀁺 -0.3267 change in the put option price
Therefore new call and put option prices can be
calculated:
􀁺 Call option: 8.6186 + 0.6733 = 9.2919
􀁺 Put option: 4.0717 – 0.3267 = 3.7450

Question 20

If there is a change in the share price what must be done with delta and why? What is this iro hedging?

Answer 20

Delta has changed
Must be recalculated
Position in underlying changed accordingly

Dynamic hedging

Question 21

Suppose a dealer sells (shorts) 1,000 call options for 8.619 to a customer
How can the dealer hedge this risk if the delta is 0.6733?

Answer 21

Therefore the dealer needs to buy a number

of the underlying shares to hedge this risk, 673 units of the underlying at 52.75

Question 22

How does delta hedging work?

Answer 22

By creating a long position in the option synthetically, it neutralises the short position

Question 23

How does delta hedging a short call position generally work when the stock price increases and where it goes down?

Answer 23

Delta hedging a short call position generally involves selling stock just after the price has one down, and buying stock just after the price has gone up (buy-high; sell-low trading strategy!)

Question 24

Give the definition of Gamma.

Answer 24

Gamma (Γ) is the rate of change of delta (Δ) with respect to the price of the underlying asset

Question 25

What happens if gamma is small?

What happens if gamma is large?

Answer 25

If gamma is small, delta changes slowly
If gamma is big, delta is very sensitives to changes in the price of the underlying.

When gamma is large, the delta changes rapidly and cannot provide a good approximation of how much an option moves for each unit of movement in the underlying

Question 26

When is gamma larger and when is it largest?

Answer 26

Gamma is large when there is more uncertainty about whether the option will expire in or out of the money

Gamma is largest when a call or put is at the money and close to expiration - means delta is very sensitive to changes in underlying’s price

Question 27

What happens to gamma if the option is deep in or deep out of the money?

Answer 27

Approaches zero –> delta not significantly affected by stock price changes

Question 28

What can gamma be viewed as?

Answer 28

How poorly a dynamic hedge will perform when it is not rebalanced

Question 29

How does gamma vary as time to maturity increases? What about short life ATM options?

Answer 29

For an at-the-money option, gamma increases as TTM decreases

Short life at the money options have very high gammas –> holder’s position highly sensitive to jumps in stock price

Question 30

Consider a call option on a non-dividend
paying stock where the stock price is $49,
the strike price is $50, the risk-free rate is
5%, time to maturity is 20 weeks (=0.3846
years) and the volatility is 20%.

Calculate gamma and interpret it.

Answer 30

Gamma=0.39828/ 49(0.2) sqrt(0.3846)

When the stock price changes by ΔS, the delta of the option changes by 0.066ΔS.

Therefore if the stock price change by $1 then delta changes by 0.066.

Question 31

Give the definition of Theta.

Answer 31

Theta (Θ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time

Question 32

What is Time Decay?

Answer 32

As time passes and a call option approaches

maturity, its value declines (all else being equal)

Question 33

What is Theta normally (sign wise)?

Answer 33

Theta is generally less than zero – as time passes and the option approaches maturity its value decreases

Question 34

All else being equal, if an option is going to expire worthless, what has happened to its value?

Answer 34

It will have lost its entire value to time decay

Question 35

How does the variation of theta of a European call look like graphically?

Answer 35

Similar to an unside-down lognormal graph

Question 36

Give the definition of Vega.

Answer 36

Vega (ν) is the rate of change of the value of

a derivatives portfolio with respect to volatility

Question 37

If Vega is high in absolute terms, what does this mean for the portfolio? Why is this the case?

Answer 37

Portfolio value will be very sensitive to small changes
in volatility
Both calls and puts are more valuable the higher the volatility (all else being equal) so Vega for calls and puts is positive

Question 38

Through what mechanism is vega and gamma adjusted?

Answer 38

By taking a position in an option or other derivative

Question 39

Define Rho.

Answer 39

Rho is the rate of change of the value of a

derivative with respect to the interest rate

Question 40

Do European options (both call and puts) change much from changes in the RFR? How do they change?

Answer 40

No.
Call options increase in value as the risk free rate increases
Put options decrease in value as the risk free rate increases

Question 41

The delta of a European call is what for the long and short positions?

Answer 41

Long: N(d1)
Short: N(d1) - 1
For small changes in S

Question 42

The delta of a European put is what for the long and short positions?

Answer 42

Long: N(d1) - 1
Short: N(d1)
For small changes in S