Greeks (finance)

The Greeks is the term used in mathematical finance for the quantities representing the market sensitivities of options or other derivatives - with each measuring a different aspect of the risk in an option position. These denote 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, $\delta$, of an instrument is the derivative of the value function with respect to the underlying price, ${\frac {\partial V}{\partial S}}$.

The gamma, $\gamma$ is the second derivative of the value function with respect to the underlying, ${\frac {\partial ^{2}V}{\partial S^{2}}}$

The vega, which is not a Greek letter, is the derivative of the option value with respect to the volatility of the underlying, ${\frac {\partial V}{\partial \sigma }}$.

The theta, $\theta$ is the derivative of the option value with respect to the amount of time to expiry of the option, ${\frac {\partial V}{\partial T}}$.

The rho, $\rho$ is the derivative of the option value with respect to the risk free rate, ${\frac {\partial V}{\partial r}}$.

Less commonly used:

The lambda, $\lambda$ 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 measure of risk in owning an option. Thus a desirable property of a model of a financial market is the ability to ability to compute Greeks. The Greeks in the Black-Scholes model are very easy to calculate and this is one reason for the models' continued popularity in the market.

External links

The Greeks: formulae

• options on non-dividend paying stocks: http://www.riskglossary.com/articles/black_scholes_1973.htm
• options on stock indexes: http://www.riskglossary.com/articles/merton_1973.htm
• options on forwards (the Black model): http://www.riskglossary.com/articles/black_1976.htm
• foreign exchange options: http://www.riskglossary.com/articles/garman_kohlhagen_1983.htm

Discussion

• The Greeks, Riskglossary: http://www.riskglossary.com/articles/greeks.htm
• Delta: http://www.quantnotes.com/fundamentals/options/thegreeks-delta.htm and http://www.riskglossary.com/articles/delta_and_gamma.htm
• Gamma: http://www.quantnotes.com/fundamentals/options/thegreeks-gamma.htm and http://www.riskglossary.com/articles/delta_and_gamma.htm
• Vega: http://www.riskglossary.com/articles/vega.htm
• Theta: http://www.quantnotes.com/fundamentals/options/thegreeks-theta.htm and http://www.riskglossary.com/articles/theta.htm
• Rho: http://www.riskglossary.com/articles/rho.htm