create new Black-Scholes process

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Finance[BlackScholesProcess] - create new Black-Scholes process

	Calling Sequence
	BlackScholesProcess(, sigma, r, d) BlackScholesProcess(, sigma, r, d, t, S)

Parameters

	-	non-negative constant; initial value
r	-	non-negative constant, procedure or yield term structure; risk-free rate
sigma	-	non-negative constant, procedure or a local volatility structure; volatility
d	-	non-negative constant, procedure or yield term structure; dividend yield
t	-	name; time variable
S	-	name; state variable

Description

•	The BlackScholesProcess command creates a new Black-Scholes process. This is a process governed by the stochastic differential equation (SDE)

dS(t)/S(t) = (r(t)-d(t)-(1/2)*sigma(S(t), t)^2)*dt+sigma(S(t), t)*dW(t)

where

–	is the risk-free rate,

–	is the local volatility,

–	is the dividend yield,

and

–	is the standard Wiener process.

•	The parameter defines the initial value of the underlying stochastic process. It must be a real constant.

•

The parameter r is the risk-free rate. The parameter d is the continuous dividend yield. Time-dependent risk-free rate and dividend yield can be given either as an algebraic expression, a Maple procedure, or a yield term structure. If r or d is given as an algebraic expression, then the fifth parameter t must be passed to specify which variable in r should be used as the time variable. Maple procedure defining a time-dependent drift must accept one parameter (the time) and return the corresponding value for the drift.

•

The sigma parameter is the local volatility. It can be constant or it can be given as a function of time and the value of the state variable. In the second case it can be specified as an algebraic expression, a Maple procedure or a local volatility term structure. If sigma is specified in the algebraic form, the parameters t and S must be given to specify which variable in sigma represents the time variable and which variable represents the value of the underlying.

Compatibility

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

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

Examples

First define a Black-Scholes process with constant parameters.

(1)

(2)

(3)

(4)

You can compute the expected payoff of a European call option with strike 100 maturing in 1 year.

(5)

(6)

(7)

You can then compare the result to the theoretical price.

(8)

This is incorporating local volatility term structure.

Again, you can compute the expected payoff of a European call option with strike 100 maturing in 1 year.

(9)

Then you can compute the implied volatility.

(10)

In this example we implied volatility surface obtained using a piecewise interpolation of known prices.

(11)

(12)

(13)

(14)

(15)

(16)

(17)

	See Also
	Finance[BlackScholesPrice], Finance[BrownianMotion], Finance[Diffusion], Finance[Drift], Finance[ExpectedValue], Finance[ForwardCurve], Finance[GeometricBrownianMotion], Finance[ImpliedVolatility], Finance[ItoProcess], Finance[LocalVolatility], Finance[LocalVolatilitySurface], Finance[MertonJumpDiffusion], Finance[PathPlot], Finance[SamplePath], Finance[SampleValues], Finance[StochasticProcesses]