Derivatives: Option Greeks, graphical explanation, part 2

Key words: Options, Greeks, Quantlib,Delta, Gamma, Vega, Theta, Time decay, Maturity, Moneyness, ATM, ITM, OTM
Option Greeks

The previous part describes Delta versus underlying price as well as Delta versus time to maturity relationships.
Here I describe the rest of Greeks for plain vanilla option and provide Quantlib code for my calculations.
I assume Strike=62.50, today =01 Sep 2012, dividends=0, risk free rate = 0.275%.
All other parameters are given in the code I provide below.

Gamma measures the rate of change of delta as the underlying price moves. As for me, gamma is an acceleration of delta.
Therefore, Gamma is the second derivative of the option price with respect to the underlying price.

As for delta, moneyness, maturity and volatility have an effect on gamma.
For simplicity reasons, I will show gamma-moneyness and gamma-maturity dependencies here, but using the code provided above it is easy to make modification to plot gamma-volatility relationship.

Gamma versus price

The graphs above show:

• Bought options have always positive gamma, sold options have always negative gamma

• The most extreme gamma is ATM, this is why delta changes too fast while ATM. OTM and ITM options have less convexity

• Short term options have higher gamma than long term options.

• Gamma is symmetrical, for example, 10% OTM gamma is equal to 10% ITM gamma.

Vega measures the rate of change of an option with respect to 1% changes in its implied volatility. Vega affects only time value and not the intrinsic value.

Vega versus price

The graphs above show:

• Bought options have always positive vega, sold options have always negative vega

• The most extreme vega is ATM

• Long term options have higher vega than short term options.

• Vega is symmetrical, for example, 10% OTM gamma is equal to 10% ITM gamma.

Theta or time decay measures the decline of an option with respect to a unit change in the time to expiration.
Theta affects only time value and not the intrinsic value. Often the one day theta based on a seven day week is used.

Theta versus price

The graphs above show:

• Options are wasting assets, time decay decreases the option value as time approaches to expiration

• Buyer has negative theta while seller profits on it

• Theta accelerates as expiration approaches

• ATM options have more significant theta than ITM and OTM.

Major conclusions:

• Short term options are mostly sensitive to Gamma while long term options are mostly sensitive to Vega.

• Pay attention to moneyness as greeks behave differently across strikes.

• Greeks and prices can behave irrationally at expiry, i.e. accelerating time decay, volatility collapse, gamma tends to infinity etc.

Using the code 1,2,3,4 attached, it is possible to plot other relationships, for example, Rho and volatility.


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