The Greeks Flashcards
What is a market maker?
A company that quotes both a buy and a sell price in a financial instrument held in inventory, hoping to make a profit on the bid-ask spread. The market maker is willing to buy at P^bid and sell at P^ask. The bid-ask spread is P^ask-P^bid and is positive.
Delta
Delta (Δ) measures the sensitivity of the derivative price to changes in the price of the underlying. It can be interpreted as (1) change in option price for a small change in the price of the underlying, (2) graphically the slope of the option value profile, and (3) the number of shares to hold to replicate the option. For a call and put option, we have
Δ_c=∂C/∂S=N(d_1 ),where 0≤Δ_c≤1
Δ_p=∂C/∂S=N(d_1 )-1,where -1≤Δ_c≤0
What is delta hedging?
Using delta hedging for short position in a European call option involves maintaining a long position of N(d_1) shares for each option sold and vice versa for a long position. For a European put option, delta is negative, so a long put option should be hedged with a long position in the underlying stock, and a short option should be hedged with a short position in the underlying stock.
The delta of a stock is Δ_s=1, while the delta of a forward is: Δ_f=1
When the delta of the trader’s overall position is zero, it is referred to as delta neutral, i.e., the delta of the stock position offsets the delta of the option position. Since the delta of an option does not remain constant, the trader’s position remains delta hedged (or delta neutral) only for a short period of time. The hedge must be rebalanced periodically. When the hedge is adjusted on a regular basis, it is referred to as dynamic hedging, which can be contracted with static hedging where the hedge is set up initially and never adjusted.
Gamma
The gamma (Γ) of a derivative is the rate of change of the derivative’s delta with respect to the price of the underlying. In other words, the sensitivity of the derivative’s delta is measured by gamma, i.e.,
Γ=(∂^2 C)/(∂S_t^2 )=(N^’ (d_1 ))/(S_t σ√(T-t)) N^’ (d_1 )=1/√2π e^(-(d_1^2)/2)
second partial derivative of the option price with respect to the stock price.
What if Gamma is small and what if its large?
The gamma of a stock is Γ_S=0. If gamma is small, delta changes slowly, so the delta hedge must be changed infrequently, and vice versa. For a long position, gamma is always positive. Gamma is the same for put and call options. Gamma varies across different stock prices and time to maturity.
What is Gamma neutrality?
A portfolio is gamma neutral if Γ_V=0. A position in the underlying asset has zero gamma and cannot be used to change the gamma of a portfolio. Instead, we require a position in an instrument (derivative) that is not linearly dependent on the underlying asset.
A portfolio with Δ_V=0, Γ_V=-3000. We want to make it both delta and gamma neutral. We can trade a call option with Δ_c=0.62 and Γ_c=1.5. How?
We can trade a call option with Δ_c=0.62 and Γ_c=1.5. The portfolio will be gamma neutral if we buy 3000÷1.5=2000 calls. The delta of the new portfolio is 2000*0.62=1240. The portfolio is gamma and delta neutral if we sell 1240 stocks, as each stock has a gamma of 0 and a delta of 1.
Theta
Theta (Θ) is the rate of change of the value of the derivative with respect to time, i.e., the sensitivity of the option price to the passage of time. Theta is referred to as the time decay of the portfolio.
Θ_c=∂C/∂t=-(σSN^’ (d_1 ))/(2√(T-t))-rKe^(-r(T-t)) N(d_2 )
Θ_p=∂P/∂t=-(S_t N^’ (d_1 )σ)/(2√(T-t))+rKe^(-r(T-t)) N(〖-d〗_2)
Is Theta usually positive or negative?
Theta is usually negative for an option, because as time passes with all else remaining the same, the option tends to become less valuable. Both calls and puts suffer time decay through erosion of insurance value. Calls lose value because the benefit of time value of strike decreases. Puts however gain value because the time value of strike increases, so theta may be positive for deep ITM puts.
Vega
Vega (ν) is the sensitivity of the option price to volatility. Because volatility changes over time, the value of a derivative is liable to change because of movements in volatility as well as due to changes in the asset price and the passage of time. We have that
V_C=V_P=∂C/∂σ=∂P/∂σ=S√(T-t) N^’ (d_1 )
What value does vega take?
Vega is usually reported on a per % basis, i.e., true vega divided by 100. The vega of a long posi-tion in a European or American option is always positive. If vega is highly positive or negative, the portfolio’s value is very sensitive to small changes in volatility. If it is close to zero, volatility changes have relatively little impact on the value of the portfolio.
Rho
the rate of change of the value of the portfolio with respect to the interest rate, i.e.,
ρ_C=∂C/∂r=KTe^(-rT) N(d_2 ),ρ_P=∂P/∂r=-KTe^(-rT) N(-d_2 )
What is the delta of a straddle?
The delta of a straddle is Δ_straddle=Δ_C+Δ_P.
The Greek of a portfolio is given by a weighted sum of the individual Greeks with weights given by the number of options or shares held.
To what changes in prices is a Δ-hedge better?
Better for smaller compared to larger price changes.
Reasons:
Delta hedging is more effective for smaller price changes because it is based on delta, which accurately estimates the option’s price change only for slight movements in the underlying asset. For larger price changes:
Non-linearity: Delta, being a linear approximation, fails to capture the full effect of significant price movements due to the option’s non-linear pricing characteristics.
Gamma Impact: The option’s gamma (the rate of change of delta) becomes more influential with larger movements, meaning that the initial hedge quickly becomes inaccurate as the underlying price shifts substantially.
Rebalancing Needs: Maintaining an effective hedge against large price swings would require frequent rebalancing, which is costly and impractical due to higher transaction costs and operational demands.
Thus, delta hedging is better suited to environments with smaller, incremental price changes where the adjustments remain minimal and more predictable.
Where can we see the highest gamma?
Gamma is highest around the place where the option is at the money.
Gamma is highest for an at-the-money (ATM) call option because it is at this point that the delta of the option is most sensitive to changes in the price of the underlying asset. Here’s why:
- Delta Behavior: Delta measures how much the price of an option is expected to move per a one-unit change in the price of the underlying asset. For call options, delta ranges from 0 to 1. It is around 0.5 when the option is ATM, meaning the option’s price is very responsive to changes in the underlying asset price at this point.
- Sensitivity and Inflection Point: When an option is ATM, any small change in the underlying price can significantly impact whether the option will end in or out of the money at expiration. This makes the ATM point a critical inflection point where the rate of change of delta (gamma) is maximized.
- Curvature of Option Value: The value of a call option as a function of the underlying price has the greatest curvature when the option is ATM. This curvature translates into higher gamma, as gamma is a measure of this curvature or the rate of change of delta.
Thus, gamma peaks when the call option is ATM because this is where the option’s sensitivity to changes in the underlying price (as measured by delta) changes most rapidly.
