Class FDMAsianADI2D

java.lang.Object

net.finmath.finitedifference.solvers.adi.AbstractADI2D

net.finmath.finitedifference.solvers.adi.FDMAsianADI2D

All Implemented Interfaces:

FDMSolver

public class FDMAsianADI2D
extends AbstractADI2D

Specialized ADI solver for arithmetic Asian options in the lifted state (S,I).

The lifted formulation augments the spot process S 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 two-dimensional state variables (S_t,I_t). The state dynamics are

dS_t = \mu_S dt + \sigma(S_t,t) dW_t,

dI_t = S_t dt.

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

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

with

A_0 u = -r u,

A_1 u = \mu_S u_S + \frac{1}{2} a_{SS} u_{SS},

A_2 u = S u_I.

Hence there is no mixed derivative and no diffusion in the integral direction. The second state direction is a pure transport direction. In \tau-time, transport is propagated toward increasing I, so the consistent upwind discretization is forward in the I direction.

Relative to the generic AbstractADI2D implementation, this class therefore:

• uses only discounting in the explicit A0 part,
• uses a forward-upwind discretization for A2,
• solves the second-direction implicit systems with the corresponding transport stencil.

Author:

Alessandro Gnoatto

Nested Class Summary

Nested classes/interfaces inherited from class AbstractADI2D

AbstractADI2D.DoubleTernaryOperator

Constructor Summary

Constructors

Constructor	Description
FDMAsianADI2D(FiniteDifferenceEquityModel model, FiniteDifferenceEquityProduct product, SpaceTimeDiscretization spaceTimeDiscretization, Exercise exercise)	Creates the lifted two-dimensional ADI solver for arithmetic Asian products.

Method Summary

Modifier and Type	Method	Description
protected double[]	applyA0Explicit(double[] u, double time)	Applies the explicit operator part A0.
protected double[]	applyA2Explicit(double[] u, double time)	Applies the explicit transport operator A2 u = S u_I.
protected double[]	solveSecondDirectionLines(double[] rhs, double time, double dt)	Solves the implicit systems in the integral direction.

Methods inherited from class AbstractADI2D

add, applyA1Explicit, applyContinuousObstacleIfNeeded, applyExerciseObstacleIfNeeded, applyFullExplicitOperator, applyInternalConstraints, applyOuterBoundaries, applyProductEventConditionIfNeeded, extractBoundaryValue, flatten, getExercise, getLowerBoundaryValueForFirstDirection, getLowerBoundaryValueForSecondDirection, getModel, getN, getN0, getN1, getProduct, getSpaceTimeDiscretization, getStencilBuilder, getTheta, getUpperBoundaryValueForFirstDirection, getUpperBoundaryValueForSecondDirection, getValue, getValue, getValue, getValues, getValues, getValues, getValuesWithContinuousObstacle, getValueWithContinuousObstacle, getX0Grid, getX1Grid, isInternalConstraintActive, overwriteBoundaryRow, performDouglasHalfStep, performStableDouglasStep, sanitize, scale, solveFirstDirectionLines, 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

FDMAsianADI2D

public FDMAsianADI2D(FiniteDifferenceEquityModel model,
 FiniteDifferenceEquityProduct product,
 SpaceTimeDiscretization spaceTimeDiscretization,
 Exercise exercise)

Creates the lifted two-dimensional ADI solver for arithmetic Asian products.

Parameters:

model - The finite-difference model.

product - The product to be valued.

spaceTimeDiscretization - The space-time discretization.

exercise - The exercise specification.

Method Details

applyA0Explicit

protected double[] applyA0Explicit(double[] u,
 double time)

Applies the explicit operator part A0.

For the lifted arithmetic Asian PDE, A0 contains only the discounting term -r u.

Overrides:

applyA0Explicit in class AbstractADI2D

Parameters:

u - Current solution vector.

time - Current running time.

Returns:

Explicit contribution of A0.

applyA2Explicit

protected double[] applyA2Explicit(double[] u,
 double time)

Applies the explicit transport operator A2 u = S u_I.

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

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

This is the appropriate upwind choice in time-to-maturity coordinates for the transport equation induced by dI_t = S_t dt.

Overrides:

applyA2Explicit in class AbstractADI2D

Parameters:

u - Current solution vector.

time - Current running time.

Returns:

Explicit contribution of A2.

solveSecondDirectionLines

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

Solves the implicit systems in the integral direction.

For each fixed spot index, the second-direction system corresponds to the transport discretization

(1 + \lambda_j) v_j - \lambda_j v_{j+1} = rhs_j,

where \lambda_j = \theta \Delta \tau S / (I_{j+1} - I_j).

The upper integral boundary is the inflow side and is imposed if it is Dirichlet. The lower integral boundary is overwritten only when an explicit Dirichlet condition is provided.

Overrides:

solveSecondDirectionLines in class AbstractADI2D

Parameters:

rhs - Right-hand side vector.

time - Current running time.

dt - Time-step size.

Returns:

Updated solution vector.