The Black Scholes equation is a partial differential equation, which describes the price of the derivative (option or warrant) over time. In the financial world, a derivative is a financial instrument, whose value depends on the value of other, more basic, underlying variables. Very often the variables underlying derivatives are the prices of traded assets. A stock option, for example, is a derivative whose value is dependent on the price of a stock.
The extended Black Scholes models are jumpdiffusion models, stochastic volatility models, local volatility models, regimeswitching models, garch models. The Greeks are the sensitivities of the option prices to the various parameters.
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FirstOrder Greeks 
Delta (Δ)
Delta (Δ)
The delta of an option or a portfolio of options is the sensitivity of the option or portfolio to the change in the price of the underlying security. It is the rate of change of value with respect to the asset The delta of a portfolio of options is just the sum of the deltas of all the individual positions. Delta hedging means holding one of the option and short a quantity ∆ of the underlying. Delta can be expressed as a function of S and t. This function varies as S and t vary. This means that the number of assets held must be continuously changed to maintain a delta neutral position, this procedure is called dynamic hedging. Changing the number of assets held requires the continual purchase and/or sale of the stock. This is called rehedging or rebalancing the portfolio. 
Vega (ν)
Vega (ν)
Vega measures the calculated option value's sensitivity to small changes in volatility. As with gamma hedging, one can vega hedge to reduce sensitivity to the volatility. This is a major step towards eliminating some model risk, since it reduces dependence on a quantity that is not known very accurately. 
Theta (Θ)
Theta (Θ)
Theta measures the calcualted option value's sensitivity to changes in time till maturity. The Theta is related to the option value, the delta and the gamma by the BlackScholes equation. In a deltahedged portfolio the theta contributes to ensuring that the portfolio earns the riskfree rate. 
Rho (ρ)
Rho (ρ)
Rho, ρ is the sensitivity of the option value to the interest rate
used in BlackScholes

Lambda (λ)
Lambda (λ)
Lambda (elasticity) is the percentage change in option value per percentage change in the underlying security price, a measure of leverage, sometimes called gearing. 
SecondOrder Greeks 
Gamma (Γ)
Gamma (Γ)
Gamma is the sensitivity of the delta to the underlying it is a measure of by how much or how often a position must be rehedged in order to maintain a deltaneutral position. Since gamma is a measure of sensitivity of the hedge ratio ∆ to the movement in the underlying, the hedging requirement can be decreased by a gammaneutral strategy. This means buying or selling more options, not just the underlying. Because the gamma of the underlying (its second derivative) is zero, gamma cannot be added to position just with the underlying. We can have as many options in our position as we want, we choose the quantities of each such that both delta and gamma are zero. 
Vanna 
Vomma 
Charm 
Veta 
Vera 
ThirdOrder Greeks 
Color 
Speed 
Ultima 
Zomma 
Dual Delta 
Dual Gamma 
Call Price 
Put Price 
Call Delta  Put Delta 
Vega  
Call Theta  Put Theta 
Call Rho  Put Rho 
Lambda  
Gamma  
Vanna  
Vomma  
Call Charm  Call Charm 
Veta  
Vera  
Coloro  
Speedo  
Ultima  
Zomma  
Call Double Delta  Put Double Delta 
Double Gamma 
φ  The standard normal probability density function.  
Φ  The standard normal cumulative distribution function.  
S  Security Price  
X  Strike Price  
r  RiskFree Rate  
q  Annual Dividend Yield  
τ  Time to Maturity τ = Tt  
σ  Volatility  
d1  
d2  
Calls  Puts  
Option Value (V)  
Delta  
Vega  
Theta  
Rho  
Lambda  Delta* (S/V)  * (S/V)  * (S/V) 
Gamma  
Vanna  
Charm  
Speed  
Zomma  
color  
Veta  
Vomma  
Ultima  
Totto  dVomma/dT  
Dual Delta  
Dual Gamma 