Class FDMAsianSabrADI3D

java.lang.Object

net.finmath.finitedifference.solvers.adi.AbstractADI3D

net.finmath.finitedifference.solvers.adi.FDMAsianSabrADI3D

All Implemented Interfaces:

FDMSolver

public class FDMAsianSabrADI3D
extends AbstractADI3D

Specialized three-dimensional ADI solver for arithmetic Asian options under a lifted SABR state (S,\alpha,I).

The lifted formulation augments the SABR state variables spot S and volatility factor \alpha by the running integral

I_t = \int_0^t S_u \, du,

so that an arithmetic-average payoff can be represented as a terminal payoff in the three-dimensional state vector (S_t,\alpha_t,I_t). The dynamics are of the form

dS_t = \mu_S dt + diffusion terms,

d\alpha_t = \mu_{\alpha} dt + diffusion terms,

dI_t = S_t dt.

In time-to-maturity coordinates \tau = T - t, the backward pricing PDE is split as

u_{\tau} = A_0 u + A_1 u + A_2 u + A_3 u,

where

• A_0 contains discounting and mixed-derivative terms and is treated explicitly,
• A_1 contains the S-direction drift and diffusion and is treated implicitly,
• A_2 contains the \alpha-direction drift and diffusion and is treated implicitly,
• A_3 is the transport term S u_I and is treated by a specialized implicit upwind solve.

There is no diffusion in the integral direction and no mixed derivative involving I. Hence the I-direction is a pure transport direction. In \tau-time, the consistent upwind discretization is forward in I.

Relative to the generic AbstractADI3D implementation, this class specializes the third-direction transport operator and the corresponding implicit line solve while leaving the first and second implicit line solves aligned with the standard three-dimensional SABR discretization.

Author:

Alessandro Gnoatto

Nested Class Summary

Nested classes/interfaces inherited from class AbstractADI3D

AbstractADI3D.DoubleQuaternaryOperator, AbstractADI3D.DoubleTernaryOperator

Constructor Summary

Constructors

Constructor	Description
FDMAsianSabrADI3D(FiniteDifferenceEquityModel model, FiniteDifferenceEquityProduct product, SpaceTimeDiscretization spaceTimeDiscretization, Exercise exercise)	Creates the lifted three-dimensional ADI solver for arithmetic Asian pricing under SABR.

Method Summary

Modifier and Type	Method	Description
protected double[]	applyA3Explicit(double[] u, double time)	Applies the explicit transport operator A3 u = S u_I.
protected double[]	solveFirstDirectionLines(double[] rhs, double time, double dt)	Solves the implicit systems in the first spatial direction.
protected double[]	solveSecondDirectionLines(double[] rhs, double time, double dt)	Solves the implicit systems in the second spatial direction.
protected double[]	solveThirdDirectionLines(double[] rhs, double time, double dt)	Solves the implicit systems in the integral direction.

Methods inherited from class AbstractADI3D

add, applyA0Explicit, applyA1Explicit, applyA2Explicit, applyExerciseObstacleIfNeeded, applyFullExplicitOperator, applyInternalConstraints, applyOuterBoundaries, extractBoundaryValue, flatten, getExercise, getLowerBoundaryValueForFirstDirection, getLowerBoundaryValueForSecondDirection, getLowerBoundaryValueForThirdDirection, getModel, getN, getN0, getN1, getN2, getProduct, getSpaceTimeDiscretization, getTheta, getUpperBoundaryValueForFirstDirection, getUpperBoundaryValueForSecondDirection, getUpperBoundaryValueForThirdDirection, getValue, getValue, getValue, getValue, getValues, getValues, getValues, getValues, getX0Grid, getX1Grid, getX2Grid, isInternalConstraintActive, overwriteBoundaryRow, performDouglasHalfStep, performStableDouglasStep, sanitize, scale, subtract

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Methods inherited from interface FDMSolver

getValue, getValue, getValues, getValues

Constructor Details

FDMAsianSabrADI3D

public FDMAsianSabrADI3D(FiniteDifferenceEquityModel model,
 FiniteDifferenceEquityProduct product,
 SpaceTimeDiscretization spaceTimeDiscretization,
 Exercise exercise)

Creates the lifted three-dimensional ADI solver for arithmetic Asian pricing under SABR.

Parameters:

model - The finite-difference model.

product - The product to be valued.

spaceTimeDiscretization - The space-time discretization.

exercise - The exercise specification.

Method Details

applyA3Explicit

protected double[] applyA3Explicit(double[] u,
 double time)

Applies the explicit transport operator A3 u = S u_I.

The derivative in the integral direction is approximated by forward upwinding:

u_I(S_i,\alpha_j,I_k) \approx (u_{i,j,k+1} - u_{i,j,k}) / (I_{k+1} - I_k).

Overrides:

applyA3Explicit in class AbstractADI3D

Parameters:

u - Current solution vector.

time - Current running time.

Returns:

Explicit contribution of A3.

solveFirstDirectionLines

protected double[] solveFirstDirectionLines(double[] rhs,
 double time,
 double dt)

Solves the implicit systems in the first spatial direction.

Specified by:

solveFirstDirectionLines in class AbstractADI3D

Parameters:

rhs - Right-hand side vector.

time - Current running time.

dt - Time-step size.

Returns:

Updated solution vector.

solveSecondDirectionLines

protected double[] solveSecondDirectionLines(double[] rhs,
 double time,
 double dt)

Solves the implicit systems in the second spatial direction.

Specified by:

solveSecondDirectionLines in class AbstractADI3D

Parameters:

rhs - Right-hand side vector.

time - Current running time.

dt - Time-step size.

Returns:

Updated solution vector.

solveThirdDirectionLines

protected double[] solveThirdDirectionLines(double[] rhs,
 double time,
 double dt)

Solves the implicit systems in the integral direction.

For fixed (S,\alpha), the third-direction transport discretization is

(1 + \lambda_k) v_k - \lambda_k v_{k+1} = rhs_k,

where \lambda_k = \theta \Delta \tau S / (I_{k+1} - I_k).

Specified by:

solveThirdDirectionLines in class AbstractADI3D

Parameters:

rhs - Right-hand side vector.

time - Current running time.

dt - Time-step size.

Returns:

Updated solution vector.