Delta
Δ = δV/δS
V - option price
S - price of the underlying asset
The Delta measures the change in the option price with respect to small movement in the price of underlying asset, assuming that the other variables from the Black Scholes model remain unchanged. The delta sign indicates that the position is bullish or bearish. A positive delta means that the position will profit when the price of the underlying asset rises. A negative delta indicates that the option portfolio position will be profitable if the asset price falls.
Mathematically, the delta represents the 1st derivative of the option price with respect to the price of the underlying asset. The delta of long call option has a positive value because the call option price rises and falls together with the price of the asset. The long put has a negative delta. Among traders, the delta is used for hedging purposes.
Delta (call)
Delta (put)
For example, delta of call option equal to 0.8 (80%) means that if the price of the underlying asset rises (falls) by 1%, then the price of the option call will rise (fall) by 0.8%. Therefore, in order to get rid of the sensitivity to the price movement of the underlying at the time, 8 units of the underlying asset must be sold for every 10 call options bought. Thus, small movements in the price of the asset will not change the total value of the portfolio consisting of 10 call options and a short position of 8 units of the asset.
The value of the long put delta, as opposed to a call option, is negative because the price of the put falls (rises) as the price of the underlying asset increases (falls). The put with Delta -0.8 will lose 0.8% if the price of the underlying rises by 1%.
As the expiry date approaches, the delta of ITM call option increases to 1, while the delta of ITM put will approach -1.
Gamma
Γ = δΔ/δS = δ^{2}V/δS^{2}
Gamma measures the impact of the price of the underlying asset on the option delta and indicates how much the delta will change if the price of the asset moves by 1%. Mathematically, the gamma represents the 1st derivative of the delta with respect to the change in underlying asset price and the 2nd derivative of the option against the price of the underlying. Purchase of call and put options leads to a long (or positive) gamma, while selling options leads to a negative gamma position.
Gamma (long call and long put)
The positive gamma allows trader to hedge the delta position at a profit regardless of the direction of the underlying asset price movement, because the delta position will increase as the asset price rises and decrease as the price falls.
A negative gamma, which is formed when an option is sold, on the contrary, leads to an increase in the delta when the price of the underlying asset falls and reduces the delta when the price of the underlying asset rises. The gamma is most important when the option is at-the-money.
Positive gamma is always accompanied by a negative theta. Options with the highest gamma, i.e. short-term call and put options that are at-the-money, are used to trade realized volatility against implied volatility.
If the gamma is small, the delta will change slowly and the rebalancing of the delta-neutral portfolio is rare. However, if the option gamma is very high (positive or negative), then the delta is highly sensitive to the price movements of the underlying asset. Therefore, leaving a delta neutral portfolio with a large gamma without additional hedging becomes risky.
Theta
Θ = δV/δt
Theta measures the sensitivity of an option price to the time, all other parameters being equal, and usually has a negative value if the trader is long options, and a positive value if the trader has a short option position.
Theta (long call and long put)
Theta is expressed in dollars and shows how much the value of the option will decrease per day if other option parameters remain the unchanged. As an example, let's consider an option that costs $3, it has a theta equal to $0.15. If the price of the underlying asset, volatility, interest rates and dividends have not changed in one day, then after one day the option will cost $2.85.
Theta and gamma of options are closely related. Since a positive gamma allows the optionholder to earn on any movement in the price of the underlying asset, regardless of direction, the gamma must have a price. This is the price of theta. The higher the gamma of an option, the larger the theta.
Vega
υ = δV/δσ
Vega indicates the sensitivity of the option price to changes in implied volatility. A positive vega is obtained by buying options (call and put) and means that if the implied volatility increases by one point, the value of the option will increase by the value of the vega. Selling options is accompanied by a negative vega.
Vega (long call and long put)
As a rule, a fall in the price of the underlying asset leads to an increase in the implied volatility of options, and if the price of the asset rises, the implied volatility falls. This is due to the fact that asset prices (e.g., shares) usually fall very rapidly and rise slowly. Implied volatility can be traded using options with the highest vega - ATM options with a long expiry time.
In practice, the volatility of an underlying asset is an unknown value that option traders are trying to predict. Let's assume that according to our calculations the volatility should be 20%. If the implied volatility increases by 1%, the option price will increase by the value of the vega. Vega is measured in monetary units (US dollars, euros, etc.). Call and put options with the same strike have the same vega value.
Rho
ρ = δV/δr
Rho measures the sensitivity of the option price with respect to 1% change in interest rates. Thus, mathematically rho is the first derivative of the option price with respect to interest rate. Buying a call option is accompanied by a positive rho, i.e. an increase in the interest rate will increase the value of the call option. While buying a put option means a negative rho, and rising rates will reduce the value of the put option.
Rho (call)
Rho (put)
For example, the current risk-free interest rate is 3% and the rho value for a call option is 7.53. This means that a 1% increase in the risk-free interest rate (from 3% to 4%) will lead to an increase in the price of the call option by about 0.01 x 7.53 = $ 0.0753.Mathematically, rho is the 1st derivative of the option price at the interest rate.
Summary
Strategy |
Delta |
Gamma |
Thetta |
Vega |
Rho |
Long call option |
+ |
+ |
To pay |
+ |
+ |
Short call option |
- |
- |
receive |
- |
- |
Long put option |
- |
+ |
To pay |
+ |
- |
Short put option |
+ |
- |
receive |
- |
+ |
The table shows the Greeks signs of the 4 basic option strategies. For example, a long position in a call option has a positive delta, i.e. the value of the call option will increase with the price of the underlying asset. The gamma (convection) of a position also has a positive value, i.e. the profit from the price movement of the underlying asset increases exponentially while the loss decreases as the asset price falls. The theta position is negative; this means that the time value will fall day by day, all other things being equal. Vega and rho have positive values as the value of a call option will increase as the volatility and interest rates increase.