net.finmath.fouriermethod.models

Class BatesModel

java.lang.Object

net.finmath.fouriermethod.models.BatesModel

All Implemented Interfaces:

CharacteristicFunctionModel, Model

public class BatesModel
extends Object
implements CharacteristicFunctionModel

Implements the characteristic function of a Bates model. The Bates model for an underlying \( S \) is given by \[ dS(t) = r^{\text{c}} S(t) dt + \sqrt{V(t)} S(t) dW_{1}(t) + S dJ, \quad S(0) = S_{0}, \] \[ dV(t) = \kappa ( \theta - V(t) ) dt + \xi \sqrt{V(t)} dW_{2}(t), \quad V(0) = \sigma^2, \] \[ dW_{1} dW_{2} = \rho dt \] \[ dN(t) = r^{\text{d}} N(t) dt, \quad N(0) = N_{0}, \] where \( W \) is Brownian motion and \( J \) is a jump process (compound Poisson process). The free parameters of this model are:

\( S_{0} \)

spot - initial value of S

\( r \)

the risk free rate

\( \sigma \)

the initial volatility level

\( \xi \)

the volatility of volatility

\( \theta \)

the mean reversion level of the stochastic volatility

\( \kappa \)

the mean reversion speed of the stochastic volatility

\( \rho \)

the correlation of the Brownian drivers

\( a \)

the jump size mean

\( b \)

the jump size standard deviation

The process \( J \) is given by \( J(t) = \sum_{i=1}^{N(t)} (Y_{i}-1) \), where \( \log(Y_{i}) \) are i.i.d. normals with mean \( a - \frac{1}{2} b^{2} \) and standard deviation \( b \). Here \( a \) is the jump size mean and \( b \) is the jump size std. dev. The model can be rewritten as \( S = \exp(X) \), where \[ dX = \mu dt + \sqrt{V(t)} dW + dJ^{X}, \quad X(0) = \log(S_{0}), \] with \[ J^{X}(t) = \sum_{i=1}^{N(t)} \log(Y_{i}) \] with \( \mu = r - \frac{1}{2} \sigma^2 - (exp(a)-1) \lambda \).

Version:

1.0

Author:

Christian Fries, Andy Graf, Lorenzo Toricelli

Constructor Summary

Constructors

Constructor and Description
BatesModel(double initialValue, double riskFreeRate, double[] volatility, double[] alpha, double[] beta, double[] sigma, double[] rho, double[] lambda, double k, double delta) Create a two factor Bates model.
BatesModel(double initialValue, double riskFreeRate, double[] volatility, double discountRate, double[] alpha, double[] beta, double[] sigma, double[] rho, double[] lambda, double k, double delta) Create a two factor Bates model.
BatesModel(double initialValue, double riskFreeRate, double volatility, double alpha, double beta, double sigma, double rho, double lambdaZero, double lambdaOne, double k, double delta) Create a one factor Bates model.

Method Summary

Modifier and Type	Method and Description
CharacteristicFunction	apply(double time) Returns the characteristic function of X(t), where X is this stochastic process.

Methods inherited from class java.lang.Object

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

Constructor Detail

BatesModel

public BatesModel(double initialValue,
                  double riskFreeRate,
                  double[] volatility,
                  double discountRate,
                  double[] alpha,
                  double[] beta,
                  double[] sigma,
                  double[] rho,
                  double[] lambda,
                  double k,
                  double delta)

Create a two factor Bates model.

Parameters:

initialValue - Initial value of S.

riskFreeRate - Risk free rate.

volatility - Square root of initial value of the stochastic variance process V.

discountRate - Rate used for the discount factor.

alpha - The parameter alpha/beta is the mean reversion level of the variance process V.

beta - Mean reversion speed of variance process V.

sigma - Volatility of volatility.

rho - Correlations of the Brownian drives (underlying, variance).

lambda - Coefficients of for the jump intensity.

k - Jump size mean.

delta - Jump size variance.

BatesModel

public BatesModel(double initialValue,
                  double riskFreeRate,
                  double[] volatility,
                  double[] alpha,
                  double[] beta,
                  double[] sigma,
                  double[] rho,
                  double[] lambda,
                  double k,
                  double delta)

Create a two factor Bates model.

Parameters:

initialValue - Initial value of S.

riskFreeRate - Risk free rate.

volatility - Square root of initial value of the stochastic variance process V.

alpha - The parameter alpha/beta is the mean reversion level of the variance process V.

beta - Mean reversion speed of variance process V.

sigma - Volatility of volatility.

rho - Correlations of the Brownian drives (underlying, variance).

lambda - Coefficients of for the jump intensity.

k - Jump size mean.

delta - Jump size variance.

BatesModel

public BatesModel(double initialValue,
                  double riskFreeRate,
                  double volatility,
                  double alpha,
                  double beta,
                  double sigma,
                  double rho,
                  double lambdaZero,
                  double lambdaOne,
                  double k,
                  double delta)

Create a one factor Bates model.

Parameters:

initialValue - Initial value of S.

riskFreeRate - Risk free rate.

volatility - Square root of initial value of the stochastic variance process V.

alpha - The parameter alpha/beta is the mean reversion level of the variance process V.

beta - Mean reversion speed of variance process V.

sigma - Volatility of volatility.

rho - Correlations of the Brownian drives (underlying, variance).

lambdaZero - Constant part of the jump intensity.

lambdaOne - Coefficients of the jump intensity, linear in variance.

k - Jump size mean.

delta - Jump size variance.

Method Detail

apply

public CharacteristicFunction apply(double time)

Description copied from interface: CharacteristicFunctionModel

Returns the characteristic function of X(t), where X is this stochastic process.

Specified by:

apply in interface CharacteristicFunctionModel

Parameters:

time - The time at which the stochastic process is observed.

Returns:

The characteristic function of X(t).