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 1.0% per annum (100 basis points).

{\displaystyle \Gamma } [5] For this reason some option traders use the absolute value of delta as an approximation for percent moneyness. First-order Greeks are in blue, second-order Greeks are in green, and third-order Greeks are in yellow. u S {\displaystyle \Delta (put)=\Delta (call)-1} Note that the gamma and vega formulas are the same for calls and puts. For example, if the delta of a portfolio of options in XYZ (expressed as shares of the underlying) is +2.75, the trader would be able to delta-hedge the portfolio by selling short 2.75 shares of the underlying. {\displaystyle \psi } a This use is fairly accurate when the number of days remaining until option expiration is large.

) , measures the sensitivity of the value of the derivative to the passage of time (see Option time value): the "time decay.". . c κ 'Vera' was picked to sound similar to a combination of Vega and Rho, its respective first-order Greeks. The (absolute value of) Delta is close to, but not identical with, the percent moneyness of an option, i.e., the implied probability that the option will expire in-the-money (if the market moves under Brownian motion in the risk-neutral measure). 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.[6]. All options (both calls and puts) will gain value with rising volatility. Any valid. Vega is the brightest star in the northern constellation of Lyra. Symbol size can be scaled to indicate magnitudes. ν \nu ν is the vega of the option, V V V is the price of the option, and ; σ \sigma σ is the symbol for volatility. Δ d Since the delta of underlying asset is always 1.0, the trader could delta-hedge his entire position in the underlying by buying or shorting the number of shares indicated by the total delta. .

[12] Charm can be an important Greek to measure/monitor when delta-hedging a position over a weekend. The default value is, A text description of the mark item for ARIA accessibility (SVG output only).

ϕ Vera[16] (sometimes rhova)[16] measures the rate of change in rho with respect to volatility.

To address this, effective duration and effective convexity are introduced.

The remaining sensitivities in this list are common enough that they have common names, but this list is by no means exhaustive. The tooltip text to show upon mouse hover. V {\displaystyle S} 2

∂ See the formulas below. p A positive beta means that the asset's returns generally follow the market's returns, in the sense that they both tend to be above their respective averages together, or both tend to be below their respective averages together.


) Zomma has also been referred to as DgammaDvol. , or elasticity[4] is the percentage change in option value per percentage change in the underlying price, a measure of leverage, sometimes called gearing. The "V inside a circle" Unicode character has been unofficially adopted by proud vegans as a vegan icon on the Internet to append to usernames and profiles on social networks and message boards, or as sort of a vegan emoji (technically not an emoji like the leaf emoji further down) in texts, chats, posts and emails. In general, the higher the convexity, the more sensitive the bond price is to the change in interest rates.

One of. it is equivalent to DV01 divided by the market price). . σ
Delta is always positive for long calls and negative for long puts (unless they are zero). Speed can be important to monitor when delta-hedging or gamma-hedging a portfolio. If defined, the mark acts as a hyperlink.

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), is the percentage change in option value per percentage change in the underlying dividend yield, a measure of the dividend risk. 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. Γ The names 'color' and 'charm' presumably derive from the use of these terms for exotic properties of quarks in particle physics. [12] Zomma is the third derivative of the option value, twice to underlying asset price and once to volatility. (Specifically, vomma is positive where the usual d1 and d2 terms are of the same sign, which is true when d1 < 0 or d2 > 0.). Color,[12][note 1] gamma decay[17] or DgammaDtime[12] measures the rate of change of gamma over the passage of time. Vomma is positive for options away from the money, and initially increases with distance from the money (but drops off as vega drops off). ∂ :315 For marks that support width and height settings (including rect and area), the horizontal dimensions are determined (in order of precedence) by the x and x2 properties, the x and width properties, the x2 and width properties, or the xc and width properties. = 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. V {\displaystyle \nu } S Charm has also been called DdeltaDtime. In trading of fixed income securities (bonds), various measures of bond duration are used analogously to the delta of an option. Vomma,[4] volga,[14] vega convexity,[14] or DvegaDvol[14] measures second order sensitivity to volatility. The mark opacity from 0 (transparent) to 1 (opaque). Symbol marks are shapes useful for plotting data, and include circles, squares and oriented triangles.

∂ ∂ Bond convexity is one of the most basic and widely used forms of convexity in finance.

and 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.
Ω {\displaystyle \Theta } It has the Bayer designation α Lyrae, which is Latinised to Alpha Lyrae and abbreviated Alpha Lyr or α Lyr.

Collectively these have also been called the risk sensitivities,[1] risk measures[2]:742 or hedge parameters.[3]. [21], Cross vanna measures the rate of change of vega in one underlying due to a change in the level of another underlying. Ω Vega[4] measures sensitivity to volatility. This benchmark is generally the overall financial market and is often estimated via the use of representative indices, such as the S&P 500.

, measures sensitivity to the interest rate: it is the derivative of the option value with respect to the risk free interest rate (for the relevant outstanding term).

Gamma is the second derivative of the value function with respect to the underlying price. V

An asset has a Beta of zero if its returns change independently of changes in the market's returns. Charm is a second-order derivative of the option value, once to price and once to the passage of time.

) − If all three of x, x2 and width are specified, the width value is ignored. p Note that vanna, charm and veta appear twice, since partial cross derivatives are equal by. 2 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. Charm[4] or delta decay[13] measures the instantaneous rate of change of delta over the passage of time. Ultima[4] measures the sensitivity of the option vomma with respect to change in volatility.

Speed[4] measures the rate of change in Gamma with respect to changes in the underlying price. An exception is a deep in-the-money European put.

The fugit is the expected time to exercise an American or Bermudan option. V Delta is the first derivative of the value The stroke opacity from 0 (transparent) to 1 (opaque). Incompatible with. {\displaystyle {\mathcal {V}}} Gamma,[4] {\displaystyle \tau } t [4] Ultima is a third-order derivative of the option value to volatility. If the underlying value has continuous second partial derivatives, then

Color is a third-order derivative of the option value, twice to underlying asset price and once to time.

Incompatible with. 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. Δ For example, if a portfolio of 100 American call options on XYZ each have a delta of 0.25 (=25%), it will gain or lose value just like 2,500 shares of XYZ as the price changes for small price movements (100 option contracts covers 10,000 shares). Ultima has also been referred to as DvommaDvol.

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