Options Greeks: Delta and Gamma

• "Greeks" measure an option's sensitivity to changes in underlying parameters.
• They are useful for hedging and risk management.
• First order Greeks include Delta, Vega, Theta, and Rho.
• Second order Greeks include Gamma, Vanna, and Volga.

Delta ($\Delta$)

• Delta represents the change in option price for a $1 change in the underlying asset price.
• Formula: $\Delta = \frac{\partial V}{\partial S}$, where V = option price and S = asset price
• For a call option, 0 < Delta < 1
• For a put option, -1 < Delta < 0
• Example: If a stock price increases by $1 and the call option price increases by $0.75, Delta = $0.75.

Gamma ($\Gamma$)

• Gamma represents the rate of change of delta with respect to changes in the underlying asset price.
• Formula: $\Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial^2 V}{\partial S^2}$
• Measures the stability of the option's delta.
• It is highest for at-the-money options.
• Example: If a call option has a gamma of 0.05, a $1 stock price increase raises the option's delta by 0.05.

Vega

• Vega represents the change in option price for a 1% change in implied volatility.
• Formula: $Vega = \frac{\partial V}{\partial \sigma}$, where V = option price and $\sigma$ = implied volatility.
• Volatility is specified in percentage points, so vega is defined as the change in option price for a 1% change in volatility.
• Example: For every 1% increase in implied volatility, the call option price increases by $0.60, Vega is $0.60.

Theta ($\Theta$)

• Theta represents the rate of change in option price with respect to time, also known as time decay.
• Formula: $\Theta = - \frac{\partial V}{\partial t}$, where V = option price and t = time to expiration.
• Expressed as change in price per day.
• Example: For every day that passes, the call option price decreases by $0.10, Theta is -$0.10.

Rho ($\rho$)

• Rho represents the change in option price for a 1% change in the risk-free interest rate.
• Formula: $\rho = \frac{\partial V}{\partial r}$, where V = option price and r = risk-free interest rate.
• The impact is generally small, especially for short-term options.
• Example: For every 1% increase in the risk-free interest rate, the call option price increases by $0.05, Rho is $0.05.

Second Order Greeks

• Gamma measures rate of change of delta.
• Vanna represents the change in delta with respect to changes in implied volatility, $Vanna = \frac{\partial \Delta}{\partial \sigma} = \frac{\partial Vega}{\partial S}$.
• Volga represents the change in vega with respect to changes in implied volatility, $Volga = \frac{\partial Vega}{\partial \sigma}$.

Implied Volatility

• Implied Volatility is volatility implied by the market price of an option.
• It is derived by "inverting" an option pricing model, e.g., Black-Scholes.
• Represents the market's expectation of future volatility.
• Example: A call option with a market price of $5, stock price of $100, strike price of $100, risk-free rate of 5%, and time to expiration of 1 year has an implied volatility of 20%.

Implied Volatility Surface

• These are implied volatilities for options with different strike prices and expiration dates.
• It typically forms a "skew" or "smile" shape.
• Provides insights into market sentiment and risk aversion.

Volatility Smile

• The volatility smile is a pattern where options with strike prices far from the current market price have higher implied volatilities than options with strike prices closer to the market price.

Summary

• Greeks measure an option's sensitivity to changes in underlying parameters.
• Delta, Gamma, Vega, Theta, and Rho are the most commonly used Greeks.
• Implied volatility represents the market's expectation of future volatility.
• The implied volatility surface provides insights into market sentiment and risk aversion.