Ch 17 - The Greek Letters Flashcards
The Greeks assist traders in managing their risks
What are the 5 inputs in the Black-Scholes-Merton model?
asset price exercise price asset price volatility time to expiration risk free rate
What is the relationship between option price and its underlying price for calls and puts?
Call: Positive curve (directly related) Increasing gradient
Put: Negative curve (inversely related) increasing gradient
What is the relationship between option price and its exercise price for calls and puts?
Call: Negative curve (inversely related) increasing gradient
Put: Positive curve (directly related) Increasing gradient
What is the relationship between option price and the risk-free rate for calls and puts?
Call: Slightly positive line (directly related)
Put: Slightly negative line (inversely related)
What is the relationship between option price and the time to expiration for calls and puts?
Call & Put: Negative curve (decreasing gradient)
What is the relationship between option price and the stock price volatility for calls and puts?
Call: Slightly positive curve
Put: Slightly positive curve
What does Delta measure?
The sensitivity of the option price to a change in the price of the underlying
What does Gamma measure?
How well the delta sensitivity measure will approximate the option price’s response to a change in the price of the underlying
What does Theta measure?
The rate at which the time value decays as the option approaches expiration
What does Vega measure?
The sensitivity of the option price to volatility
What does Rho measure?
The sensitivity of the option price to the risk free rate
Give the definition of Delta
The rate of change of the option price with respect to the price of the underlying asset
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?
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
In the BSM model, what does N(d1) approximate?
delta of a call option for SMALL CHANGES
in S
In the BSM model, what does N(d1) -1 approximate?
The delta of a put option for SMALL CHANGES in
S
Give the general formula for calculating delta.
Change in option price / change in underlying price
The delta of a European call on a non-dividend-paying stock is?
N(d1) for small changes in S
The delta of a European put on the stock is?
[N(d1) - 1] for small changes in S
How is a delta used to calculate the change in option price given a change in underlying price? Give the general formula
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
If there is a change in the share price what must be done with delta and why? What is this iro hedging?
Delta has changed
Must be recalculated
Position in underlying changed accordingly
Dynamic hedging
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?
Therefore the dealer needs to buy a number
of the underlying shares to hedge this risk, 673 units of the underlying at 52.75
How does delta hedging work?
By creating a long position in the option synthetically, it neutralises the short position
How does delta hedging a short call position generally work when the stock price increases and where it goes down?
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!)
Give the definition of Gamma.
Gamma (Γ) is the rate of change of delta (Δ) with respect to the price of the underlying asset
What happens if gamma is small?
What happens if gamma is large?
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
When is gamma larger and when is it largest?
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
What happens to gamma if the option is deep in or deep out of the money?
Approaches zero –> delta not significantly affected by stock price changes
What can gamma be viewed as?
How poorly a dynamic hedge will perform when it is not rebalanced
How does gamma vary as time to maturity increases? What about short life ATM options?
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
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.
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.
Give the definition of Theta.
Theta (Θ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time
What is Time Decay?
As time passes and a call option approaches
maturity, its value declines (all else being equal)
What is Theta normally (sign wise)?
Theta is generally less than zero – as time passes and the option approaches maturity its value decreases
All else being equal, if an option is going to expire worthless, what has happened to its value?
It will have lost its entire value to time decay
How does the variation of theta of a European call look like graphically?
Similar to an unside-down lognormal graph
Give the definition of Vega.
Vega (ν) is the rate of change of the value of
a derivatives portfolio with respect to volatility
If Vega is high in absolute terms, what does this mean for the portfolio? Why is this the case?
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
Through what mechanism is vega and gamma adjusted?
By taking a position in an option or other derivative
Define Rho.
Rho is the rate of change of the value of a
derivative with respect to the interest rate
Do European options (both call and puts) change much from changes in the RFR? How do they change?
No.
Call options increase in value as the risk free rate increases
Put options decrease in value as the risk free rate increases
The delta of a European call is what for the long and short positions?
Long: N(d1)
Short: N(d1) - 1
For small changes in S
The delta of a European put is what for the long and short positions?
Long: N(d1) - 1
Short: N(d1)
For small changes in S
