# Option Greeks

What Are The Option Greeks?

The Mathematical characteristics of the Black-Scholes Model are named after the greek letters used to represent them in the equations. The 5 Option Greeks measure the sensitivity of the price  of stock options in relation to 4 different factors; Changes in the underlying stock price, interest rate, volatility and time decay.

Delta

The movement of the option position relative to the movement of the underlying stock  position. Measures the speed at which the option position is moving relative to the underlying stock position. Therefore, a Delta of 1 means the option position is moving 1 point for every point the stock moves. A Delta of –1 means the option position ismoving –1 point for every point the underlying stock moves. Delta is another way of expressing the probability of an option expiringin-the-money.

Formula for calculating option Delta: Formula Components

C = Value of the Call Option

S(t) = Current value of the underlying asset

N(d1) = Rate of change of the option price with respect to the price of the underlying asset

T = Option life as a percentage of the year

ln = Natural log

Rf = Risk free rate of return

Gamma

Gamma is mathematically the second derivative of Delta and can be viewed in two ways: the acceleration of the option position relative to the underlying stock price, or the odds of a change in the probability of the position expiring ITM (in other words, the odds of a change in Delta). Gamma is effectively an early warning to the fact that Delta could be about to change.Both calls and puts have positive Gammas. Typically, deep OTM and deep ITM options have near zero Gamma because the odds of a change in Delta are very low. Logically, Gamma tends to peak around the strike price. Gamma is important because it shows us how fast our position delta changes in relation to the market price of the underlying asset.

Formula for calculation option Gamma: Forumla Components

d1 = Refer to Delta Calculation

S = Current value of underlying asset

T = Option life as a percentage of a year

Theta

Theta stands for the option position’s sensitivity to time decay. Long options have negative Theta, meaning that everyday you own that option, time decay is eroding the Time Value portion of the option’s value. In other words, time decay is hurting the position of a Long option position. When you short options, Theta is positive, indicating that time decay is helping the option writer’s position. The closer to the expiration date, the higher the theta and the father away the expiration date, the lower the theta.

The below graphs show the effect of Theta on options during the last 30 days to expiration. ITM and ATM options decay fastest during the last 30 days to expiration. OTM options decay the least during the final 30 days.  Formula for Calculating Theta d1 = Refer to Delta Calculation

T = Option life as a percentage of year

C = Value of Call Option

St = Current price of underlying asset

X = Strike Price

Rf = Risk free rate of return

N(d2) = Probability of option being in the money

Vega

Vega stands for the option position’s sensitivity to volatility. Options tend toincrease in value when the underlying stock’s volatility increases. So, volatility helps the owner of an option and hurts the writer of an option. Vega is positivefor long option positions and negative for short option positions.

Formula for Calculating Vega: Forumula Components

d1 = Refer to Delta Caculation above

S = Current Value of underlying asset

T = Option life as percentage of year

C = Value of Call Option

Rho

Rho stands for the option position’s sensitivity to interest rates. A positive Rho means that higher interest rates are helping the position, and a negative Rho means that higher interest rates are hurting the position. Rho is the least important of all the Greeks as far as stock options are concerned.

Rho Charateristics – Options Rho come in positive or negative polarity. Long call options produces positive options Rho and Long put options produces negative options rho. This means that call options rise in value and put options drop in value with a rise in interest rates. Options Rho increases as time to expiration becomes longer. Options Rho is almost equal for all ITM and decreases for OTM options. d1 = Refer to Delta calculation

T = Option life as a percentage of year

C = Value of Call Option

X = Strike Price

N(d2) = Probabilty of option being in the money