The Greeks

In mathematical finance, the Greeks are the quantities representing the
market sensitivities of options or other derivatives, with each measuring a
different aspect of the risk in an option position, and corresponding to the
set of parameters on which the value of an instrument or portfolio of
financial instruments is dependent. The name is used because most of the
parameters are denoted by Greek letters.

     The delta, δ, of an instrument is the derivative of the value
     function with respect to the underlying price,
     [\frac{\partial V}{\partial S}]; delta measures sensitivity to price.
     The gamma, γ is the second derivative of the value function with
     respect to the underlying, [\frac{\partial^2 V}{\partial S^2}]; gamma
     measures second order sensitivity to price.
     The vega, which is not a Greek letter, is the derivative of the option
     value with respect to the volatility of the underlying,
     [\frac{\partial V}{\partial \sigma}]; vega measures sensitivity to
     implied volatility.
     The theta, θ is the derivative of the option value with respect
     to the amount of time to expiry of the option,
     [\frac{\partial V}{\partial T}]; theta measures sensitivity to the
     passage of time (see Option time value).
     The rho, ρ is the derivative of the option value with respect to
     the risk free rate, [\frac{\partial V}{\partial r}]; rho measures
     sensitivity to the applicable interest rate.

     Less commonly used:
     The lambda, λ is the percentage change in option value per
     change in the underlying price, or [\frac{\partial V}{\partial S}/V].

The Greeks are vital tools in risk management. With the exception of the
theta, each of the above quantities represent a specific measure of risk in
owning an option. Thus a desirable property of a model of a financial market
is the ability to compute the Greeks. The Greeks in the Black-Scholes model
are very easy to calculate and this is one reason for the model's continued
popularity in the market.

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