## Black Scholes Calculator

#### Option Greeks

Option Greeks are option sensitivity measures. The Greek is used in the name because these are denoted by Greek letters.

Option price is a function of many variables such as time to maturity, underlying volatility, spot price of underlying asset, strike price and interest rate, option trader needs to know how the changes in these variables affect the option price or option premium. The Option Greeks are essentials for an option trader as they help option trader plan his trades and understand & estimate the extent of risk while trading options.

Here we explain Option Greeks in simple terms to measure the sensitivity of option price to changes in these variables. To understand the impact of change in any factor it is (as always) important to keep other factors constant so we can understand the impact of change of that particular factor.

#### Option Delta

Option delta represents the sensitivity of option price to small movements in the price of underlying asset. For instance, if a call option has a delta of 0.8, this means that if the underlying price increases by $1, the option price will increase by $0.80. Similarly, when we say a put option has a delta of say -0.8, this means that if there is an increase of $1 in the underlying price, the option price will decrease by $0.80. For OTM options, the absolute value of delta will be lower compared to ITM options. For ITM call option near expiry, its delta will approach 1 or 100%; Similarly, as an in-the-money put option nears expiration, its delta will approach -1 or -100%. Likewise, as an out-of-the-money option nears expiry its delta approaches 0.

#### Delta is the most important of all the option greeks. Delta is usually expressed in percentage or decimal number. The legitimate values of delta are as follows:

- Between 0 and 1 for call options
- and between -1 and 0 for put options*
- *In case of put options, option price and the underlying price move inversely i.e., put option price will increase as the underlying price decreases and vice versa.
- Therefore put option delta is always negative while call options have positive delta.
- At-the-money options have a delta of about 0.50 or 50% (in case of calls) or -0.50 or -50% (in case of puts)

#### Option Gamma:

Gamma measures the sensitivity of option delta with respect to changes in the underlying prices. It is first level derivative of Delta. Option traders need to know this because option delta does not remain constant in reality and it changes as the underlying price changes. Therefore option traders need to worry about delta sensitivity and accordingly measure gamma in order to understand and estimate the risk they are exposed to while trading options.

Deep in-the-money options and deep out-of-the-money options have relatively lower gamma. However, at-the-money options have higher gamma and trades need to be watchful when dealing with these options.

#### Option Vega:

Vega (also known as kappa or zeta) measures the option price sensitivity to the changes in the underlying volatility. It represents change in the price of an option to 1% change in the underlying volatility. For example, if vega of an option is 1.5, it means that if the volatility of the underlying were to increase by 1%, then the option price will increase by $1.50.

Again Vega is not constant and it changes when there are large price movements in the underlying. Also, Vega decreases as the option gets closer to expiration date.

#### Option Theta:

Theta measures the change in the option value relative to the change in the time to maturity of the option. All other option parameters remaining constant, the option value will constantly erode with every passing day since the time value of the option diminishes as it approaches option expiration. This is also called as the time decay of option.

Theta is always negative since if other things remaining same, option value declines as it gets closer to expiration due to diminishing time value. To understand option Theta with illustration, if an option has Theta value of -0.30, it indicates that the option price will decrease by $0.30 the next day if the price of the underlying next day remains at same price as today’s.

#### Option Rho:

Rho measures the sensitivity of option value with the changes in the risk-free interest rate. This is positive for call options (since higher the interests, the higher the call option premium) and negative for put options since higher the interest the lower the put option premium. For example, if Rho of a call option is 0.5, it indicates that if risk-free interest rate increase by 1% then the option price will increase by $0.5. Similarly, if Rho of a put option is -0.5, it means that the option price will decrease by $0.5 for a 1% increase in risk-free interest rate.

Deep ITM options have higher Rho since these options are most likely to be exercised and therefore the value will move in line with changes in the forward prices of the underlying asset. However, relatively speaking, when compared with other option Greeks, the impact of Rho on option price is least significant.