compute the Gamma of a European-style option with given payoff

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Finance[BlackScholesGamma] - compute the Gamma of a European-style option with given payoff

	Calling Sequence
	BlackScholesGamma(, K, T, sigma, r, d, optiontype) BlackScholesGamma(, P, T, sigma, r, d)

Parameters

	-	algebraic expression; initial (current) value of the underlying asset
K	-	algebraic expression; strike price
T	-	algebraic expression; time to maturity
sigma	-	algebraic expression; volatility
r	-	algebraic expression; continuously compounded risk-free rate
d	-	algebraic expression; continuously compounded dividend yield
P	-	operator or procedure; payoff function
optiontype	-	call or put; option type

Description

•	The Gamma of an option or a portfolio of options is the sensitivity of the Delta to changes in the value of the underlying asset

GAMMA = diff(S, S[0], S[0])

•	The BlackScholesGamma command computes the Gamma of a European-style option with the specified payoff function.

•	The parameter is the initial (current) value of the underlying asset. The parameter T is the time to maturity in years.

•

The parameter K specifies the strike price if this is a vanilla put or call option. Any payoff function can be specified using the second calling sequence. In this case the parameter P must be given in the form of an operator, which accepts one parameter (spot price at maturity) and returns the corresponding payoff.

•	The sigma, r, and d parameters are the volatility, the risk-free rate, and the dividend yield of the underlying asset. These parameters can be given in either the algebraic form or the operator form. The parameter d is optional. By default, the dividend yield is taken to be 0.

Compatibility

•	The Finance[BlackScholesGamma] command was introduced in Maple 15.

•	For more information on Maple 15 changes, see Updates in Maple 15.

Examples

First you compute the Gamma of a European call option with strike price 100, which matures in 1 year. This will define the Gamma as a function of the risk-free rate, the dividend yield, and the volatility.

(1/200)*2^(1/2)*exp(-(1/8)*(4*d*sigma^2+4*r^2-8*r*d+4*r*sigma^2+4*d^2+sigma^4)/sigma^2)/(sigma*Pi^(1/2))

(1)

In this example you will use numeric values for the risk-free rate, the dividend yield, and the volatility.

(2)

You can also use the generic method in which the option is defined through its payoff function.

(1/200)*2^(1/2)*exp(-(1/8)*(4*d*sigma^2+4*r^2-8*r*d+4*r*sigma^2+4*d^2+sigma^4)/sigma^2)/(sigma*Pi^(1/2))

(3)

(4)

(5)

Here are similar examples for the European put option.

(6)

(7)

(8)

(9)

In this example, you will compute the Gamma of a strangle.

(10)

C := (1/200)*2^(1/2)*exp(-(1/8)*(4*d*sigma^2+4*r^2-8*r*d+4*r*sigma^2+4*d^2+sigma^4)/sigma^2)/(sigma*Pi^(1/2))

(11)

(12)

Check:

(13)

	See Also
	Finance[AmericanOption], Finance[BermudanOption], Finance[BlackScholesDelta], Finance[BlackScholesPrice], Finance[BlackScholesRho], Finance[BlackScholesTheta], Finance[BlackScholesVega], Finance[EuropeanOption], Finance[ImpliedVolatility], Finance[LatticePrice]