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Encyclopedia > The Greeks

In mathematical finance, the Greeks are the quantities representing the market sensitivities of options or other derivatives. Each "Greek" measures a different aspect of the risk in an option position, and corresponds to a parameter 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. Mathematical finance is the branch of applied mathematics concerned with the financial markets. ... In finance, an option is a contract whereby one party (the holder or buyer) has the right but not the obligation to exercise a feature of the contract (the option) on or before a future date (the exercise date or expiry). ... A derivative is a financial contract whose payoffs over time are derived from the performance of assets (such as commodities, shares or bonds), interest rates, exchange rates, or indices (such as a stock market index, consumer price index (CPI) or an index of weather conditions). ... The factual accuracy of this article is disputed. ... In finance, a portfolio is a collection of investments held by an institution or a private individual. ... Financial instruments package financial capital in readily tradeable forms - they do not exist outside the context of the financial markets. ... Due to technical limitations, some web browsers may not display some special characters in this article. ...

Contents


Use of the Greeks

The Greeks are vital tools in risk management. Each Greek (with the exception of theta - see below) represents a specific measure of risk in owning an option, and option portfolios can be adjusted accordingly ("hedged") to achieve a desired exposure; see for example Delta hedging. Financial risk management is the practice of creating value in a firm by using financial instruments to manage exposure to risk. ... Risk is the potential impact (positive or negative) to an asset or some characteristic of value that may arise from some present process or from some future event. ... In finance, a hedge is an investment that is taken out specifically to reduce or cancel out the risk in another investment. ... Delta hedging is the process of setting or keeping the delta of a portfolio of financial instruments zero, or as close to zero as possible - where delta is the sensitivity of the value of a derivative to changes in the price of its underlying instrument; see Hedge (finance). ...


As a result, a desirable property of a model of a financial market is that it allows for easy computation of 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. A diagram of the IS/LM model In economics, a model is a theoretical construct that represents economic processes by a set of variables and a set of logical and quantitative relationships between them. ... In economics, a financial market is a mechanism which allows people to trade money for securities or commodities such as gold or other precious metals. ... A calculation is a deliberate process for transforming one or more inputs into one or more results. ... The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ...


The Greeks

  • The delta measures sensitivity to price. The Δ, of an instrument is the derivative of the value function with respect to the underlying price, frac{partial V}{partial S}.
  • The gamma measures second order sensitivity to price. The Γ is the second derivative of the value function with respect to the underlying price, frac{partial^2 V}{partial S^2}.
  • The vega, which is not a Greek letter, measures sensitivity to implied volatility. The vega is the derivative of the option value with respect to the volatility of the underlying, frac{partial V}{partial sigma}; the term kappa, κ, is sometimes used instead of vega.
  • The theta measures sensitivity to the passage of time (see Option time value). Θ is minus the derivative of the option value with respect to the amount of time to expiry of the option, Theta = -frac{partial V}{partial T}.
  • The rho measures sensitivity to the applicable interest rate. The ρ is the derivative of the option value with respect to the risk free rate, frac{partial V}{partial r}.
  • 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}timesfrac{1}{V}.
    • The vega gamma or volga measures second order sensitivity to implied volatility. This is the second derivative of the option value with respect to the volatility of the underlying, frac{partial^2 V}{partial sigma^2}.
    • The vanna measures cross-sensitivity of option value with respect to change in underlier price and underlier volatility, frac{partial^2 V}{partial S partial sigma}, which can also be interpreted as the sensitivity of delta to a unit change in volatility.
    • The delta decay measures the time decay of delta, frac{partial Delta}{partial T}. This can be important when hedging a position over a weekend.

In mathematics, the derivative is defined as the instantaneous rate of change of a function. ... In financial mathematics, the implied volatility of a financial instrument is the volatility implied by the market price of a derivative based on a theoretical pricing model. ... Volatility is the standard deviation of the change in value of a financial instrument with a specific time horizon. ... Option Value In finance, the value of an option consists of two components, its intrinsic value and its time value. ... A percentage is a way of expressing a proportion, a ratio or a fraction as a whole number, by using 100 as the denominator. ... In financial mathematics, the implied volatility of a financial instrument is the volatility implied by the market price of a derivative based on a theoretical pricing model. ...

Black-Scholes

The Greeks under the Black-Scholes model are calculated as follows; where, φ is the normal probability density function. Note that the gamma and vega formulas are the same for calls and puts. The Black-Scholes model, often simply called Black-Scholes, is a model of the varying price over time of financial instruments, and in particular stocks. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... A call option is a financial contract between two parties, the buyer and the seller of this type of option. ... A put option (sometimes simply called a put) is a financial contract between two parties, the buyer and the seller of the option. ...

Calls Puts
delta N(d_1) , N(d_1) - 1 ,
gamma frac{phi(d_1)}{Ssigmasqrt{T}} ,
vega S phi(d_1) sqrt{T} ,
theta - frac{S phi(d_1) sigma}{2 sqrt{T}} - rKe^{-rT}N(d_2) , - frac{S phi(d_1) sigma}{2 sqrt{T}} + rKe^{-rT}N(-d_2) ,
rho KTe^{-rT}N(d_2), -KTe^{-rT}N(-d_2),

where

d_1 = frac{ln(S/K) + (r + sigma^2/2)T}{|sigma|sqrt{T}}
d_2 = d_1 - |sigma|sqrt{T}.

External links

  • Greeks for specific option models
    • options on non-dividend paying stocks, riskglossary.com
    • options on stock indexes, riskglossary.com
    • options on forwards (the Black model), riskglossary.com
    • foreign exchange options, riskglossary.com
  • Discussion
    • The Greeks: riskglossary.com or optiontutor or investopedia.com or optiontradingtips.com
    • Delta: quantnotes.com or riskglossary.com
    • Gamma: quantnotes.com or riskglossary.com
    • Vega: riskglossary.com
    • Theta: quantnotes.com or riskglossary.com
    • Rho: riskglossary.com
  • Calculation
    • Free Option Pricing spreadsheet to calculate the Greeks, optiontradingtips.com

See also


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