Class QdPlusAmericanEngine

Namespace

NanoQL.PricingEngines

Assembly

NanoQL.dll

The American engine based on the QD+ approximation to the exercise boundary (M. Li, "Analytical Approximations for the Critical Stock Prices of American Options: A Performance Comparison", 2009): the boundary at Chebyshev nodes in √τ, the early-exercise premium integrated over it by tanh-sinh quadrature.

public class QdPlusAmericanEngine : QdPutCallParityEngine

Inheritance

object

QdPutCallParityEngine

QdPlusAmericanEngine

Inherited Members

QdPutCallParityEngine.Calculate(VanillaOption, EvaluationContext)

QdPutCallParityEngine.Process

object.GetType()

object.MemberwiseClone()

object.ToString()

object.Equals(object)

object.Equals(object, object)

object.ReferenceEquals(object, object)

object.GetHashCode()

Remarks

Its main purpose is to provide a good initial guess to the exercise boundary for the superior fixed-point engine, QdFpAmericanEngine. The engine reports the value.

Immutable: each τ's boundary is solved by a working evaluator built for that solve, and each price builds its own boundary, as QuantLib's does. The static Calculate builds an engine with the given settings and prices through it. An extension point, as in QuantLib.

Constructors

QdPlusAmericanEngine(GeneralizedBlackScholesProcess, int, QdPlusAmericanEngineSolverType, double, int?)

Initializes a new instance of the QdPlusAmericanEngine class.

public QdPlusAmericanEngine(GeneralizedBlackScholesProcess process, int interpolationPoints = 8, QdPlusAmericanEngineSolverType solverType = QdPlusAmericanEngineSolverType.Halley, double eps = 1E-06, int? maxIter = null)

Parameters

process GeneralizedBlackScholesProcess

The Black-Scholes process.

interpolationPoints int

The number of Chebyshev nodes of the boundary.

solverType QdPlusAmericanEngineSolverType

The boundary's root finder.

eps double

The boundary's and the premium's accuracy.

maxIter int?

The most evaluations of a boundary solve, or null for QuantLib's default: 100 for Brent's, Newton's and Ridder's methods, 10 for the Halley iterations.

Exceptions

ArgumentNullException

process is null.

ArgumentOutOfRangeException

interpolationPoints or maxIter is negative, or the solver type isn't a member of its type.

Methods

Calculate(VanillaOption, EvaluationContext, GeneralizedBlackScholesProcess, int, QdPlusAmericanEngineSolverType, double, int?)

Prices an American vanilla option by the QD+ approximation.

public static OneAssetOptionResults Calculate(VanillaOption option, EvaluationContext context, GeneralizedBlackScholesProcess process, int interpolationPoints = 8, QdPlusAmericanEngineSolverType solverType = QdPlusAmericanEngineSolverType.Halley, double eps = 1E-06, int? maxIter = null)

Parameters

option VanillaOption

The option.

context EvaluationContext

The evaluation context.

process GeneralizedBlackScholesProcess

The Black-Scholes process.

interpolationPoints int

The number of Chebyshev nodes of the boundary.

solverType QdPlusAmericanEngineSolverType

The boundary's root finder.

eps double

The boundary's and the premium's accuracy.

maxIter int?

The most evaluations of a boundary solve, or null for QuantLib's default.

Returns

OneAssetOptionResults

The option's results: the value.

Exceptions

ArgumentNullException

An argument is null.

NanoQLException

The engine refuses the option.

CalculatePut(double, double, double, double, double, double)

The value of an American put.

protected override double CalculatePut(double s, double k, double r, double q, double vol, double t)

Parameters

s double

The spot.

k double

The strike.

r double

The risk-free rate.

q double

The dividend yield.

vol double

The volatility.

t double

The time to expiry.

Returns

double

The value.

GetPutExerciseBoundary(double, double, double, double, double, double)

The put's exercise boundary as a Chebyshev interpolation, on the second kind's nodes in z = 2√(τ/T) − 1, of ln(b(τ)/x_max)²: each node's boundary solved by PutExerciseBoundaryAtTau(double, double, double, double, double, double, double).

public ChebyshevInterpolation GetPutExerciseBoundary(double s, double k, double r, double q, double vol, double t)

Parameters

s double

The spot.

k double

The strike.

r double

The risk-free rate.

q double

The dividend yield.

vol double

The volatility.

t double

The time to expiry.

Returns

ChebyshevInterpolation

The interpolation.

Exceptions

NanoQLException

A node's solve refuses.

PutExerciseBoundaryAtTau(double, double, double, double, double, double, double)

The put's exercise boundary at a time to expiry, and the number of evaluations its solve took.

public (int Evaluations, double Boundary) PutExerciseBoundaryAtTau(double s, double k, double r, double q, double vol, double t, double tau)

Parameters

s double

The spot.

k double

The strike.

r double

The risk-free rate.

q double

The dividend yield.

vol double

The volatility.

t double

The option's time to expiry.

tau double

The time to expiry the boundary is at.

Returns

(int Evaluations, double Boundary)

The number of evaluations and the boundary; 0 and XMax(double, double, double) below a τ of ε.

Exceptions

NanoQLException

The solver refuses.

XMax(double, double, double)

The exercise boundary's limit at expiry: Table 2 of L. Andersen and M. Lake (2021), "Fast American Option Pricing: The Double-Boundary Case".

public static double XMax(double k, double r, double q)

Parameters

k double

The strike.

r double

The risk-free rate.

q double

The dividend yield.

Returns

double

The limit; 0 in the European cases.

Exceptions

NanoQLException

A rate is NaN ("internal error").