Greeks and its Derivatives: Charm & Color

In mathematical finance, the quantities that represent the sensitivities of derivatives such as options to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent are referred to as the Greeks. This name is given to them due to the reason that most common of these sensitivities are usually denoted by Greek letters. Collectively these sensitivities have also been referred to as the risk sensitivities, risk measures or hedge parameters.

risk management

Use of the Greeks

There are more common sensitivities to the four primary inputs into the Black-Scholes model (spot price of the underlying security, time remaining until option expiration, volatility and the rate of return of a risk-free investment) and to the option’s value, such as delta, gamma, vega and vomma. Greeks which are a first-order derivative are indicated as blue, second-order derivatives are in green, and third-order derivatives are indicated by yellow color. Here you have to note that vanna is used, intentionally, in two places due to the reason that these two sensitivities are mathematically equivalent.

In risk management the Greeks are vital tools. The sensitivity of the value is measured by each Greek to a small change in a given underlying parameter, it is done so that component risks may be treated in isolation, and the portfolio could be rebalanced accordingly to achieve a desired exposure.

It is relatively easy to calculate for the Greeks in the Black-Scholes model,which is a desirable property of financial models, and these are very useful for derivatives traders, particularly those who are looking forward to hedge their portfolios from adverse changes in market conditions. Due to this reason, those Greeks which are specifically useful for hedging delta, gamma and vega are well-defined for measuring changes in Price, Time and Volatility. Although it is true that rho is a primary input into the Black-Scholes model, the overall impact is insignificant on the value of an option corresponding to changes in the risk-free interest rate and therefore higher-order derivatives that involve the risk-free interest rate are not common.

charm

Among Geeks the most common are the first order derivatives: Delta, Theta, Vega and Rho as well as Gamma, which is a second-order derivative of the value function. Below I have explained two Geeks:

Charm
Color

Charm

$Charm = frac{partial Delta}{partial tau} = -frac{partial Theta}{partial S} = frac{partial^2 V}{partial S partial tau}$

The instantaneous rate of change of delta over the passage of time is measured by Charm that is also referred to as delta decay.Charm has also been known as DdeltaDtime. When delta-hedging a position over a weekend then charm can be an important greek to measure/monitor. Charm is a second order derivative of the option value, once to price and once to time. Then it is also the (minus) derivative of theta with respect to the underlying’s price.

Practical Use

The mathematical result of the formula for charm is shown in delta/year. In order to arrive at the delta decay per day, it is often useful to divide this by the number of days per year. When the number of days remaining until option expiration is large then this use is fairly accurate. Charm itself may change quickly when an option nears expiration, and they render full day estimates of delta decay inaccurate.

Color

$Color = frac{partial Gamma}{partial tau} = frac{partial^3 V}{partial S^2 partial tau}$

Color which is also known as gamma decay or DgammaDtime measures the rate of change of gamma over the passage of time. Color is a third-order derivative of the option value, twice to underlying asset price and once to time. When maintaining a gamma-hedged portfolio color can be an important sensitivity to monitor as it can help the trader to expect the effectiveness of the hedge as time passes.

Practical Use

The mathematical result of the formula for color is expressed in gamma/year. To arrive at the change in gamma per day, it is often useful to divide this by the number of days per year. When the number of days remaining until option expiration is large then this use is fairly accurate. color itself may change quickly when an option nears expiration, they render full day estimates of gamma change inaccurate.

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