In mathematical finance , the Greeks are the quantities representing the sensitivity of the price 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. The name is used because the most common of these sensitivities are denoted by Greek letters as are some other finance measures.

Collectively these have also been called the risk sensitivities , [1] risk measures [2]: The Greeks are vital tools in risk management. Each Greek measures the sensitivity of the value of a portfolio to a small change in a given underlying parameter, so that component risks may be treated in isolation, and the portfolio rebalanced accordingly to achieve a desired exposure; see for example delta hedging. The Greeks in the Black—Scholes model are relatively easy to calculate, a desirable property of financial models , and are very useful for derivatives traders, especially those who seek to hedge their portfolios from adverse changes in market conditions.

For this reason, those Greeks which are particularly useful for hedging—such as delta, theta, and vega—are well-defined for measuring changes in Price, Time and Volatility.

Although rho is a primary input into the Black—Scholes model, the overall impact on the value of an option corresponding to changes in the risk-free interest rate is generally insignificant and therefore higher-order derivatives involving the risk-free interest rate are not common. The most common of the Greeks are the first order derivatives: The remaining sensitivities in this list are common enough that they have common names, but this list is by no means exhaustive.

The use of Greek letter names is presumably by extension from the common finance terms alpha and beta , and the use of sigma the standard deviation of logarithmic returns and tau time to expiry in the Black—Scholes option pricing model. Several names such as 'vega' and 'zomma' are invented, but sound similar to Greek letters.

The names 'color' and 'charm' presumably derive from the use of these terms for exotic properties of quarks in particle physics. For a vanilla option, delta will be a number between 0. The difference between the delta of a call and the delta of a put at the same strike is close to but not in general equal to one, but instead is equal to the inverse of the discount factor. These numbers are commonly presented as a percentage of the total number of shares represented by the option contract s.

This is convenient because the option will instantaneously behave like the number of shares indicated by the delta. For example, if a portfolio of American call options on XYZ each have a delta of 0. The sign and percentage are often dropped — the sign is implicit in the option type negative for put, positive for call and the percentage is understood. Delta is always positive for long calls and negative for long puts unless they are zero.

The total delta of a complex portfolio of positions on the same underlying asset can be calculated by simply taking the sum of the deltas for each individual position — delta of a portfolio is linear in the constituents. Since the delta of underlying asset is always 1. This portfolio will then retain its total value regardless of which direction the price of XYZ moves. Albeit for only small movements of the underlying, a short amount of time and not-withstanding changes in other market conditions such as volatility and the rate of return for a risk-free investment.

The absolute value of Delta is close to, but not identical with, the percent moneyness of an option, i. For example, if an out-of-the-money call option has a delta of 0. At-the-money puts and calls have a delta of approximately 0. The actual probability of an option finishing in the money is its dual delta , which is the first derivative of option price with respect to strike.

Given a European call and put option for the same underlying, strike price and time to maturity, and with no dividend yield, the sum of the absolute values of the delta of each option will be 1 — more precisely, the delta of the call positive minus the delta of the put negative equals 1. This is due to put—call parity: If the value of delta for an option is known, one can calculate the value of the delta of the option of the same strike price, underlying and maturity but opposite right by subtracting 1 from a known call delta or adding 1 to a known put delta.

For example, if the delta of a call is 0. Vega [4] measures sensitivity to volatility. Vega is the derivative of the option value with respect to the volatility of the underlying asset. Vega is not the name of any Greek letter. Presumably the name vega was adopted because the Greek letter nu looked like a Latin vee , and vega was derived from vee by analogy with how beta , eta , and theta are pronounced in American English.

Another possibility is that it is named after Joseph De La Vega, famous for Confusion of Confusions , a book about stock markets and which discusses trading operations that were complex, involving both options and forward trades. All options both calls and puts will gain value with rising volatility. Vega can be an important Greek to monitor for an option trader, especially in volatile markets, since the value of some option strategies can be particularly sensitive to changes in volatility.

The value of an option straddle , for example, is extremely dependent on changes to volatility. The mathematical result of the formula for theta see below is expressed in value per year. By convention, it is usual to divide the result by the number of days in a year, to arrive at the amount an option's price will drop, in relation to the underlying stock's price. Theta is almost always negative for long calls and puts, and positive for short or written calls and puts. An exception is a deep in-the-money European put.

The total theta for a portfolio of options can be determined by summing the thetas for each individual position. The value of an option can be analysed into two parts: The time value is the value of having the option of waiting longer before deciding to exercise. Even a deeply out of the money put will be worth something, as there is some chance the stock price will fall below the strike before the expiry date.

However, as time approaches maturity, there is less chance of this happening, so the time value of an option is decreasing with time.

Thus if you are long an option you are short theta: Except under extreme circumstances, the value of an option is less sensitive to changes in the risk free interest rate than to changes in other parameters. For this reason, rho is the least used of the first-order Greeks. Rho is typically expressed as the amount of money, per share of the underlying, that the value of the option will gain or lose as the risk free interest rate rises or falls by 1.

Gamma is the second derivative of the value function with respect to the underlying price. Most long options have positive gamma and most short options have negative gamma. Long options have a positive relationship with gamma because as price increases, Gamma increases as well, causing Delta to approach 1 from 0 long call option and 0 from -1 long put option.

The inverse is true for short options. Gamma is important because it corrects for the convexity of value. When a trader seeks to establish an effective delta-hedge for a portfolio, the trader may also seek to neutralize the portfolio's gamma, as this will ensure that the hedge will be effective over a wider range of underlying price movements. Vanna , [4] also referred to as DvegaDspot [12] and DdeltaDvol , [12] is a second order derivative of the option value, once to the underlying spot price and once to volatility.

It is mathematically equivalent to DdeltaDvol , the sensitivity of the option delta with respect to change in volatility; or alternatively, the partial of vega with respect to the underlying instrument's price. Vanna can be a useful sensitivity to monitor when maintaining a delta- or vega-hedged portfolio as vanna will help the trader to anticipate changes to the effectiveness of a delta-hedge as volatility changes or the effectiveness of a vega-hedge against change in the underlying spot price.

Charm [4] or delta decay [13] measures the instantaneous rate of change of delta over the passage of time. Charm has also been called DdeltaDtime. Charm is a second-order derivative of the option value, once to price and once to the passage of time. It is also then the derivative of theta with respect to the underlying's price. It is often useful to divide this by the number of days per year to arrive at the delta decay per day.

This use is fairly accurate when the number of days remaining until option expiration is large. When an option nears expiration, charm itself may change quickly, rendering full day estimates of delta decay inaccurate. Vomma , [4] volga , [14] vega convexity , [14] or DvegaDvol [14] measures second order sensitivity to volatility. Vomma is the second derivative of the option value with respect to the volatility, or, stated another way, vomma measures the rate of change to vega as volatility changes.

With positive vomma, a position will become long vega as implied volatility increases and short vega as it decreases, which can be scalped in a way analogous to long gamma.

And an initially vega-neutral, long-vomma position can be constructed from ratios of options at different strikes. Vomma is positive for options away from the money, and initially increases with distance from the money but drops off as vega drops off. Veta [15] or DvegaDtime [14] measures the rate of change in the vega with respect to the passage of time.

Veta is the second derivative of the value function; once to volatility and once to time. It is common practice to divide the mathematical result of veta by times the number of days per year to reduce the value to the percentage change in vega per one day. Vera [16] sometimes rhova [16] measures the rate of change in rho with respect to volatility. Vera is the second derivative of the value function; once to volatility and once to interest rate. Vera can be used to assess the impact of volatility change on rho-hedging.

Speed [4] measures the rate of change in Gamma with respect to changes in the underlying price. This is also sometimes referred to as the gamma of the gamma [2]: Speed can be important to monitor when delta-hedging or gamma-hedging a portfolio. Zomma [4] measures the rate of change of gamma with respect to changes in volatility.

Zomma has also been referred to as DgammaDvol. Zomma can be a useful sensitivity to monitor when maintaining a gamma-hedged portfolio as zomma will help the trader to anticipate changes to the effectiveness of the hedge as volatility changes.

Color , [12] [note 1] gamma decay [17] or DgammaDtime [12] 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. Color can be an important sensitivity to monitor when maintaining a gamma-hedged portfolio as it can help the trader to anticipate the effectiveness of the hedge as time passes.

It is often useful to divide this by the number of days per year to arrive at the change in gamma per day. When an option nears expiration, color itself may change quickly, rendering full day estimates of gamma change inaccurate. Ultima [4] measures the sensitivity of the option vomma with respect to change in volatility.

Ultima has also been referred to as DvommaDvol. If the value of a derivative is dependent on two or more underlyings , its Greeks are extended to include the cross-effects between the underlyings. Correlation delta measures the sensitivity of the derivative's value to a change in the correlation between the underlyings. Cross gamma measures the rate of change of delta in one underlying to a change in the level of another underlying. Cross vanna measures the rate of change of vega in one underlying due to a change in the level of another underlying.

Equivalently, it measures the rate of change of delta in the second underlying due to a change in the volatility of the first underlying. Cross volga measures the rate of change of vega in one underlying to a change in the volatility of another underlying.

More...