Option Greeks Based on Black-Scholes Option Pricing Model

Option Greeks measure the sensitivity of the option from its parameters. In this section, we will explore each of the Greeks and we will begin with Delta.

Delta measures rate of change of the option's value with respect to the stock pricing; it is the first differential of option price with respect to the price of the underlying asset. Delta also changes gradually over time even if there is no price movement of the underlying asset. The change in delta for a given change in the asset price is known as Gamma. Gamma is the second derivative of the option value with respect to the price of the underlying asset. Variation in Delta requires that a hedged position be rebalanced if it is to remain delta neutral after the price of the underlying asset has changed. How much adjustment needed depends on how much the Delta changes, that is, on Gamma.

Theta refers to the rate of time decay for an option. It is the first differential of the option value with respect to time. Holding all other things constant, an option loses value as it approaching to the expiration day. Theta measures the cost of holding an option long, and the reward fo writing it.

Vega measures the relationship between the volatility of the underlying asset and the option value. It is the first differential of the option price with respect to the volatility (standard deviation). The more volatility the underlying asset is, the more valuable the option becomes since the chance for the option to be deep-in-the-money is greater.

Rho measures the sensitivity of the option value to the interest rate. It is the derivative of the option value with respect to the interest rate. The higher the interest rate, the greater the time value of the option. Hence, Rho is positive for calls and negative for puts. For both calls and puts, the longer the time to expiration, the larger is the effect of the interest rate on the option value.

Below are the formulas for the greeks.

Delta measures rate of change of the option's value with respect to the stock pricing; it is the first differential of option price with respect to the price of the underlying asset. Delta also changes gradually over time even if there is no price movement of the underlying asset. The change in delta for a given change in the asset price is known as Gamma. Gamma is the second derivative of the option value with respect to the price of the underlying asset. Variation in Delta requires that a hedged position be rebalanced if it is to remain delta neutral after the price of the underlying asset has changed. How much adjustment needed depends on how much the Delta changes, that is, on Gamma.

Theta refers to the rate of time decay for an option. It is the first differential of the option value with respect to time. Holding all other things constant, an option loses value as it approaching to the expiration day. Theta measures the cost of holding an option long, and the reward fo writing it.

Vega measures the relationship between the volatility of the underlying asset and the option value. It is the first differential of the option price with respect to the volatility (standard deviation). The more volatility the underlying asset is, the more valuable the option becomes since the chance for the option to be deep-in-the-money is greater.

Rho measures the sensitivity of the option value to the interest rate. It is the derivative of the option value with respect to the interest rate. The higher the interest rate, the greater the time value of the option. Hence, Rho is positive for calls and negative for puts. For both calls and puts, the longer the time to expiration, the larger is the effect of the interest rate on the option value.

Below are the formulas for the greeks.

The input and output are shown as followed. Note that the sign for each of the greeks indicates the direction that the option price will go when the corresponding parameter changes.

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