net.finmath.montecarlo.interestrate.products

Class ForwardRateVolatilitySurfaceCurvature

java.lang.Object

net.finmath.montecarlo.AbstractMonteCarloProduct

net.finmath.montecarlo.interestrate.products.AbstractLIBORMonteCarloProduct

net.finmath.montecarlo.interestrate.products.ForwardRateVolatilitySurfaceCurvature

All Implemented Interfaces:

Product, TermStructureMonteCarloProduct, MonteCarloProduct

public class ForwardRateVolatilitySurfaceCurvature
extends AbstractLIBORMonteCarloProduct

This class implements the calculation of the curvature of the volatility surface of the forward rates.
The value returned by the getValue(double, LIBORModelMonteCarloSimulationModel) method is calculated as follows: For each forward rate's instantaneous volatility σ(t) we calculate \[ \sqrt{ \frac{1}{t_{n} - t_{1}} \sum_{i=1}^{n-1} ( f(t_{i}) )^{2} \cdot (t_{i+1} - t_{i}) } \] (this is the root mean square / L2 norm of f) where \[ f(t_{i}) = \frac{ x_{i+2} - 2 x_{i+1} + x_{i} }{ (t_{i+2} - t_{i+1}) (t_{i+1} - t_{i}) } \] and where x_i = σ²(t_i) is the instantaneous variance of a specific forward rate. The value returned is then calculated as the average of all those curvatures for all forward rates. Note: A tolerance level can be specified. See the documentation of the constructor ForwardRateVolatilitySurfaceCurvature(double).

While this is not a common financial product, this class can be helpful in calibration procedures, e.g. to put an additional constrain on the smoothness / curvature of the model surface.
A rough interpretation of this product is a follows: If the value 0.25 is returned, then - on average - the variance of the forward rate oscillates by 0.25 per year, i.e. it moves up by 0.25 and back. This corresponds to a movement of 0.5 and back per year for the volatility. Given that you have 100 other calibrations products, and the weight of all (including this one) is 1.0, then 0.1% deviation of the other calibration products is considered as severe as 0.1 movement per year in the variance (0.3 movement in volatility). Note: You can control the relation of these by weights.

Version:

1.0

Author:

Christian Fries

Date:

12.04.2014.

Constructor Summary

Constructors

Constructor and Description
ForwardRateVolatilitySurfaceCurvature() Create the calculation of the curvature of the volatility surface of the forward rates
ForwardRateVolatilitySurfaceCurvature(double tolerance) Create the calculation of the curvature of the volatility surface of the forward rates.

Method Summary

Modifier and Type	Method and Description
RandomVariable	getValue(double evaluationTime, LIBORModelMonteCarloSimulationModel model) This method returns the value random variable of the product within the specified model, evaluated at a given evalutationTime.
RandomVariable	getValues(double evaluationTime, LIBORMarketModel model) Calculates the squared curvature of the LIBOR instantaneous variance.

Methods inherited from class net.finmath.montecarlo.interestrate.products.AbstractLIBORMonteCarloProduct

getFactorDrift, getValue, getValueForModifiedData, getValues

Methods inherited from class net.finmath.montecarlo.AbstractMonteCarloProduct

getCurrency, getValue, getValue, getValues, getValues, getValues, getValuesForModifiedData, getValuesForModifiedData, getValuesForModifiedData, getValuesForModifiedData, toString

Methods inherited from class java.lang.Object

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

Methods inherited from interface net.finmath.montecarlo.MonteCarloProduct

getCurrency, getValue, getValue, getValues, getValues, getValues, getValuesForModifiedData, getValuesForModifiedData, getValuesForModifiedData, getValuesForModifiedData

Constructor Detail

ForwardRateVolatilitySurfaceCurvature

public ForwardRateVolatilitySurfaceCurvature()

Create the calculation of the curvature of the volatility surface of the forward rates

ForwardRateVolatilitySurfaceCurvature

public ForwardRateVolatilitySurfaceCurvature(double tolerance)

Create the calculation of the curvature of the volatility surface of the forward rates. A tolerance level may be specified. In that case, the curvature calculated by the getValue method is approximately
max(curvature - tolerance, 0).
A rough interpretation of the tolerance is as follows: With a tolerance = 0.04, then

• the variance can oscillate once from 0.0 to 0.04 and back within a year without generating a penalty term (i.e., the volatility is allowed to oscillate once from 0.0 to 0.2 and back within a year), or
• the variance can oscillate twice from 0.0 to 0.02 and back within a year without generating a penalty term (i.e., the volatility is allowed to oscillate twice from 0.0 to 0.14 (sqrt(0.02)) and back).

Parameters:

tolerance - The tolerance level.

Method Detail

getValue

public RandomVariable getValue(double evaluationTime,
                               LIBORModelMonteCarloSimulationModel model)

Description copied from interface: TermStructureMonteCarloProduct

This method returns the value random variable of the product within the specified model, evaluated at a given evalutationTime. Note: For a lattice this is often the value conditional to evalutationTime, for a Monte-Carlo simulation this is the (sum of) value discounted to evaluation time. Cashflows prior evaluationTime are not considered.

Specified by:

getValue in interface TermStructureMonteCarloProduct

Specified by:

getValue in class AbstractLIBORMonteCarloProduct

Parameters:

evaluationTime - The time on which this products value should be observed.

model - The model used to price the product.

Returns:

The random variable representing the value of the product discounted to evaluation time

getValues

public RandomVariable getValues(double evaluationTime,
                                LIBORMarketModel model)

Calculates the squared curvature of the LIBOR instantaneous variance.

Parameters:

evaluationTime - Time at which the product is evaluated.

model - A model implementing the LIBORModelMonteCarloSimulationModel

Returns:

The squared curvature of the LIBOR instantaneous variance (reduced a possible tolerance). The return value is ≥ 0.