FeynmanKac Class

FeynmanKac Class

Solves the generalized Feynman-Kac PDE.

Inheritance Hierarchy

SystemObject
Imsl.MathFeynmanKac

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

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[SerializableAttribute]
public class FeynmanKac

<SerializableAttribute>
Public Class FeynmanKac

[SerializableAttribute]
public ref class FeynmanKac

[<SerializableAttribute>]
type FeynmanKac =  class end

The FeynmanKac type exposes the following members.

Constructors

	Name	Description
	FeynmanKac	Constructs a PDE solver to solve the Feynman-Kac PDE.

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Methods

	Name	Description
	ComputeCoefficients	Determines the coefficients of the Hermite quintic splines that represent an approximate solution for the Feynman-Kac PDE.
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetAbsoluteErrorTolerances	Returns absolute error tolerances.
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetRelativeErrorTolerances	Returns relative error tolerances.
	GetSplineCoefficients	Returns the coefficients of the Hermite quintic splines that represent an approximate solution of the Feynman-Kac PDE.
	GetSplineCoefficientsPrime	Returns the first derivatives of the Hermite quintic spline coefficients that represent an approximate solution of the Feynman-Kac PDE.
	GetSplineValue	Evaluates for time value 0 or a time value in tGrid the derivative of the Hermite quintic spline interpolant at evaluation points within the range of xGrid.
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	SetAbsoluteErrorTolerances(Double)	Sets the absolute error tolerances.
	SetAbsoluteErrorTolerances(Double)	Sets the absolute error tolerances.
	SetForcingTerm	Sets the user-supplied method that computes approximations to the forcing term $\phi(x)$ and its derivative $\partial \phi/\partial y$ used in the FeynmanKac PDE.
	SetInitialData	Sets the user-supplied method for adjustment of initial data or as an opportunity for output during the integration steps.
	SetRelativeErrorTolerances(Double)	Sets the relative error tolerances.
	SetRelativeErrorTolerances(Double)	Sets the relative error tolerances.
	SetTimeDependence	Sets the time dependence of the coefficients, boundary conditions and function $\phi$ in the Feynman Kac equation.
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	GaussLegendreDegree	The number of quadrature points used in the Gauss-Legendre quadrature formula.
	InitialStepsize	The starting step size for the integration.
	MaximumBDFOrder	The maximum order of the backward differentiation formulas (BDF).
	MaximumStepsize	The maximum internal step size used by the integrator.
	MaxSteps	The maximum number of internal steps allowed.
	Method	The step control method used in the integration of the Feynman-Kac PDE.
	TimeBarrier	Sets or returns the barrier used for integration in the time direction.

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Remarks

Class FeynmanKac solves the generalized Feynman-Kac PDE on a rectangular grid with boundary conditions using a finite element Galerkin method. The generalized Feynman-Kac differential equation has the form

$f_t +\mu(x,t) f_x +\frac{\sigma^2(x,t)}{2}f_{xx}-\kappa(x,t)f=\phi(f,x,t),$

where the initial data satisfies

. The derivatives are $f_t=\frac{\partial{f}}{\partial{t}}$ , etc. Method ComputeCoefficients uses a finite element Galerkin method over the rectangle

$[x_{\min},x_{\max}] \times [\bar{T},T]$

to compute the approximate solution. The interval $[x_{\min},x_{\max}]$ is decomposed with a grid

$(x_{\min}=)x_1 \lt x_2 \lt \ldots \lt x_m(=x_{\max})\,.$

On each subinterval the solution is represented by

$f(x,t) = f_ib_0(z)+f_{i+1}b_0(1-z)+h_i f_i' b_1(z)-h_i f_{i+1}' b_1(1-z)+ h_i^2f_i''b_2(z)+h_i^2f_{i+1}''b_2(1-z).$

The values

$f_i,f_i',f_i'',f_{i+1},f_{i+1}',f_{i+1}''$

are time-dependent coefficients associated with each interval. The basis functions

are given for

$x \in [x_i, x_{i+1}], \, h_i=x_{i+1}-x_i, \, z=(x-x_i)/h_i, z \in [0,1],$

$b_2(z)=\frac{1}{2}(-z^5+3z^4-3z^3+z^2)=\frac{1}{2}(1-z)^3z^2.$

By adding the piece-wise definitions the unknown solution function may be arranged as the series

$f(x,t) = \sum_{i=1}^{3m}y_i\beta_i(x), \; x \in [x_{\min},x_{\max}],$

where the time-dependent coefficients are defined by re-labeling:

$y_{3i-2} = f_i,\, y_{3i-1} = f_i',\, y_{3i} = f_i'',\, i=1,\ldots,m.$

The Galerkin principle is then applied. Using the provided initial and boundary conditions leads to an Index 1 differential-algebraic equation for the coefficients $y_i, \, i=1,\ldots,3m$ .

This system is integrated using the variable order, variable step algorithm DDASLX noted in Hanson, R. and Krogh, F. (2008), Solving Constrained Differential-Algebraic Systems Using Projections. Solution values and their time derivatives at a grid preceding time T, expressed in units of time remaining, are given back by methods GetSplineCoefficients and GetSplineCoefficientsPrime, respectively. For further details of deriving and solving the system see Hanson, R. (2008), Integrating Feynman-Kac Equations Using Hermite Quintic Finite Elements. To evaluate f or its partials $f_x, f_{xx}, f_{xxx}, f_t, f_{tx}, f_{txx}, f_{txxx}$ at any time point in the grid, use method GetSplineValue.

One useful application of the FeynmanKac class is financial analytics. This is illustrated in Example 2, which solves a diffusion model for call options which, in the special case $\alpha = 2$ reduces to the Black-Scholes (BS) model. Another useful application for the FeynmanKac class is the calculation of the Greeks, i.e. various derivatives of Feynman-Kac solutions applicable to, e.g., the pricing of options and related financial derivatives. In Example 5, the FeynmanKac class is used to calculate eleven of the Greeks for the same diffusion model introduced in Example 2 in the special BS case. These Greeks are also calculated using the BS closed form Greek equations (see http://en.wikipedia.org/wiki/The_Greeks). The Feynman-Kac and BS solutions are output and compared. Example 5 illustrates that the FeynmanKac class can be used to explore the Greeks for a much wider class of financial models than can BS.

Reference

Imsl.Math Namespace

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