FFT

Create FFT pricer object for Vanilla instrument using Merton, Heston, or Bates model

Description

Create and price a Vanilla instrument object with a Heston, Bates, or Merton model and an FFT pricing method using this workflow:

1. Use fininstrument to create a Vanilla instrument object.

2. Use finmodel to specify a Heston, Bates, or Merton model for the Vanilla instrument.

3. Use finpricer to specify an FFT pricer object for the Vanilla instrument.

For more information on this workflow, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

For more information on the available pricing methods for a Vanilla instrument, see Choose Instruments, Models, and Pricers.

Creation

Syntax

FFTPricerObj = finpricer(PricerType,'Model',model,'DiscountCurve',ratecurve_obj)

FFTPricerObj = finpricer(___,Name,Value)

Description

FFTPricerObj = finpricer(PricerType,'Model',model,'DiscountCurve',ratecurve_obj) creates an FFT pricer object by specifying PricerType and sets the properties for the required name-value pair arguments Model and DiscountCurve.

FFTPricerObj = finpricer(___,Name,Value) sets optional properties using additional name-value pairs in addition to the required arguments in the previous syntax. For example, FFTPricerObj = finpricer("FFT",'Model',FFTModel, 'DiscountCurve',ratecurve_obj,'SpotPrice',1000,'DividendValue',0.01,'VolRiskPremium',0.9) creates an FFT pricer object. You can specify multiple name-value pair arguments.

Input Arguments

PricerType — Pricer type

string with value "FFT" | character vector with value 'FFT'

Pricer type, specified as a string with the value of "FFT" or a character vector with the value of 'FFT'.

Data Types: char | string

FFT Name-Value Pair Arguments

Specify required and optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: FFTPricerObj = finpricer("FFT",'Model',FFTModel, 'DiscountCurve',ratecurve_obj,'SpotPrice',1000,'DividendValue',0.01,'VolRiskPremium',0.9)

Required FFT Name-Value Pair Arguments

'Model' — Model

model object

Model, specified as the comma-separated pair consisting of 'Model' and the name of the previously created Merton, Bates, or Heston model object using finmodel.

Data Types: object

'DiscountCurve' — ratecurve object for discounting cash flows

ratecurve object

This property is read-only.

ratecurve object for discounting cash flows, specified as the comma-separated pair consisting of 'DiscountCurve' and the name of the ratecurve object.

Note

Specify a flat ratecurve object for DiscountCurve. If you use a nonflat ratecurve object, the software uses the rate in the ratecurve object at Maturity and assumes that the value is constant for the life of the equity option.

Data Types: object

'SpotPrice' — Current price of underlying asset

nonnegative numeric

Current price of the underlying asset, specified as the comma-separated pair consisting of 'SpotPrice' and a scalar nonnegative numeric.

Data Types: double

Optional FFT Name-Value Pair Arguments

'DividendValue' — Dividend yield

0 (default) | scalar nonnegative numeric

Dividend yield, specified as the comma-separated pair consisting of 'DividendValue' and a scalar nonnegative numeric in decimals.

Data Types: double

'VolRiskPremium' — Volatility risk premium

0 (default) | numeric

Volatility risk premium, specified as the comma-separated pair consisting of 'VolRiskPremium' and a scalar numeric value.

Data Types: double

'LittleTrap' — Flag indicating Little Heston Trap formulation

true (default) | logical with value true or false

Flag indicating Little Heston Trap formulation by Albrecher et al., specified as the comma-separated pair consisting of 'LittleTrap' and a logical:

• true — Use the Albrecher et al. formulation.

  For more information on the LittleTrap, see [1] and also the Little Trap formulation is defined by C_j and D_j in Heston Stochastic Volatility Model and Bates Stochastic Volatility Jump Diffusion Model.

• false — Use the original Heston formation.

Note

LittleTrap is supported only for Heston and Bates models.

Data Types: logical

'NumFFT' — Number of grid points in characteristic function variable

4096 (default) | numeric

Number of grid points in the characteristic function variable and in each column of the log-strike grid, specified as the comma-separated pair consisting of 'NumFFT' and a scalar numeric value.

Data Types: double

'CharacteristicFcnStep' — Characteristic function variable grid spacing

0.01 (default) | numeric

Characteristic function variable grid spacing, specified as the comma-separated pair consisting of 'CharacteristicFcnStep' and a scalar numeric value.

Data Types: double

'LogStrikeStep' — Log-strike grid spacing

2*pi/NumFFT/CharacteristicFcnStep (default) | numeric

Log-strike grid spacing, specified as the comma-separated pair consisting of 'LogStrikeStep' and a scalar numeric value.

Note

If (LogStrikeStep*CharacteristicFcnStep) is 2*pi/NumFFT, FFT is used. Otherwise, FRFT is used.

Data Types: double

'DampingFactor' — Damping factor for Carr-Madan formulation

1.5 (default) | numeric

Damping factor for the Carr-Madan formulation, specified as the comma-separated pair consisting of 'DampingFactor' and a scalar numeric value.

Data Types: double

'Quadrature' — Type of quadrature

"simpson" (default) | string with value "simpson" or "trapezoidal" | character vector with value 'simpson' or 'trapezoidal'

Type of quadrature, specified as the comma-separated pair consisting of 'Quadrature' and a scalar string or character vector.

Data Types: char | string

Properties

Model — Model

model object

Model, returned as a model object.

Data Types: object

SpotPrice — Current price of underlying asset

nonnegative numeric

Current price of the underlying asset, returned as a scalar nonnegative numeric.

Data Types: double

DividendValue — Dividend yield

0 (default) | scalar nonnegative numeric

Dividend yield, returned as a scalar nonnegative numeric in decimals.

Data Types: double

VolRiskPremium — Volatility risk premium

0 (default) | numeric

Volatility risk premium, returned as a scalar numeric value.

Data Types: double

LittleTrap — Flag indicating Little Heston Trap formulation

true (default) | logical with value true or false

Flag indicating Little Heston Trap formulation by Albrecher et al., returned as a logical.

Data Types: logical

NumFFT — Number of grid points in the characteristic function variable

4096 (default) | numeric

Number of grid points in the characteristic function variable and in each column of the log-strike grid, returned as a scalar numeric value.

Data Types: double

CharacteristicFcnStep — Characteristic function variable grid spacing

0.01 (default) | numeric

Characteristic function variable grid spacing, returned as a scalar numeric value.

Data Types: double

LogStrikeStep — Log-strike grid spacing

2*pi/NumFFT/CharacteristicFcnStep (default) | numeric

Log-strike grid spacing, returned as a scalar numeric value.

Data Types: double

DampingFactor — Damping factor for Carr-Madan formulation

1.5 (default) | numeric

Damping factor for the Carr-Madan formulation, returned as a scalar numeric value.

Data Types: double

Quadrature — Type of quadrature

"simpson" (default) | string with value "simpson" or "trapezoidal"

Type of quadrature, returned as a string.

Data Types: string

Object Functions

price

Compute price for equity instrument with FFT pricer

Examples

Use FFT Pricer and Heston Model to Price Vanilla Instrument

This example shows the workflow to price a Vanilla instrument when you use a Heston model and an FFT pricing method.

Create Vanilla Instrument Object

Use fininstrument to create a Vanilla instrument object.

VanillaOpt = fininstrument("Vanilla",'ExerciseDate',datetime(2022,9,15),'Strike',105,'ExerciseStyle',"european",'Name',"vanilla_option")

VanillaOpt = 
  Vanilla with properties:

       OptionType: "call"
    ExerciseStyle: "european"
     ExerciseDate: 15-Sep-2022
           Strike: 105
             Name: "vanilla_option"

Create Heston Model Object

Use finmodel to create a Heston model object.

HestonModel = finmodel("Heston",'V0',0.032,'ThetaV',0.1,'Kappa',0.003,'SigmaV',0.2,'RhoSV',0.9)

HestonModel = 
  Heston with properties:

        V0: 0.0320
    ThetaV: 0.1000
     Kappa: 0.0030
    SigmaV: 0.2000
     RhoSV: 0.9000

Create ratecurve Object

Create a flat ratecurve object using ratecurve.

Settle = datetime(2018,9,15);
Maturity = datetime(2023,9,15);
Rate = 0.035;
myRC = ratecurve('zero',Settle,Maturity,Rate,'Basis',12)

myRC = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 12
                Dates: 15-Sep-2023
                Rates: 0.0350
               Settle: 15-Sep-2018
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create FFT Pricer Object

Use finpricer to create an FFT pricer object and use the ratecurve object for the 'DiscountCurve' name-value pair argument.

outPricer = finpricer("fft",'DiscountCurve',myRC,'Model',HestonModel,'SpotPrice',100,'CharacteristicFcnStep', 0.2,'NumFFT',2^13)

outPricer = 
  FFT with properties:

                    Model: [1x1 finmodel.Heston]
            DiscountCurve: [1x1 ratecurve]
                SpotPrice: 100
             DividendType: "continuous"
            DividendValue: 0
                   NumFFT: 8192
    CharacteristicFcnStep: 0.2000
            LogStrikeStep: 0.0038
        CharacteristicFcn: @characteristicFcnHeston
            DampingFactor: 1.5000
               Quadrature: "simpson"
           VolRiskPremium: 0
               LittleTrap: 1

Price Vanilla Instrument

Use price to compute the price and sensitivities for the Vanilla instrument.

[Price, outPR] = price(outPricer,VanillaOpt,["all"])

Price = 14.7545

outPR = 
  priceresult with properties:

       Results: [1x7 table]
    PricerData: []

outPR.Results

ans=1×7 table
    Price      Delta      Gamma       Theta       Rho       Vega     VegaLT
    ______    _______    ________    ________    ______    ______    ______

    14.754    0.44868    0.021649    -0.20891    120.45    88.192    1.3248

More About

Vanilla Option

A vanilla option is a category of options that includes only the most standard components.

A vanilla option has an expiration date and straightforward strike price. American-style options and European-style options are both categorized as vanilla options.

The payoff for a vanilla option is as follows:

• For a call: $$\max (St-K,0)$$

• For a put: $$\max (K-St,0)$$

Here:

St is the price of the underlying asset at time t.

K is the strike price.

For more information, see Vanilla Option.

Heston Stochastic Volatility Model

The Heston model is an extension of the Black-Scholes model, where the volatility (the square root of the variance) is no longer assumed to be constant, and the variance now follows a stochastic (CIR) process. This process allows modeling the implied volatility smiles observed in the market.

The stochastic differential equation is

$$\begin{array}{l}d{S}_t=(r-q)S_{t}dt+\sqrt{v_t}{S}_{t}d{W}_t\\ d{v}_t=\kappa (\theta -v_t)dt+{\sigma }_v\sqrt{v_t}d{W}_t^v\\ \text{E}\left[d{W}_{t}d{W}_t^v\right]=pdt\end{array}$$

Here:

r is the continuous risk-free rate.

q is the continuous dividend yield.

S_t is the asset price at time t.

v_t is the asset price variance at time t.

v₀ is the initial variance of the asset price at t = 0 for (v₀ > 0).

θ is the long-term variance level for (θ > 0).

κ is the mean reversion speed for the variance for (κ > 0).

σ_v is the volatility of the variance for (σ_v > 0).

p is the correlation between the Weiner processes W_t and W^v_t for (-1 ≤ p ≤ 1).

The characteristic function $$f_{Hesto{n}_j}(\varphi )$$ for j = 1 (asset price measure) and j = 2 (risk-neutral measure) is

$$\begin{array}{l}{f}_{Hesto{n}_j}(\varphi )=\exp (C_j+D_j{v}_0+i\varphi \ln S_t)\\ C_j=(r-q)i\varphi \tau +\frac{\kappa \theta }{{\sigma }_v^2}\left[\left(b_j-p{\sigma }_{v}i\varphi +d_j\right)\tau -2\ln \left(\frac{1-g_j{e}^{d_j\tau }}{1-g_j}\right)\right]\\ D_j=\frac{b_j-p{\sigma }_{v}i\varphi +d_j}{{\sigma }_v^2}\left(\frac{1-e^{d_j\tau }}{1-g_j{e}^{d_j\tau }}\right)\\ g_j=\frac{b_j-p{\sigma }_{v}i\varphi +d_j}{b_j-p{\sigma }_{v}i\varphi -d_j}\\ d_j=\sqrt{{\left(b_j-p{\sigma }_{v}i\varphi \right)}^2-{\sigma }_v^2\left(2u_{j}i\varphi -{\varphi }^2\right)}\\ \text{where for }j=1,2:\\ u_1=\frac{1}{2},u_2=-\frac{1}{2},b_1=\kappa +{\lambda }_{VolRisk}-p{\sigma }_v,b_2=\kappa +{\lambda }_{VolRisk}\end{array}$$

Here:

ϕ is the characteristic function variable.

ƛ_VolRisk is the volatility risk premium.

τ is the time to maturity (τ = T - t).

i is the unit imaginary number (i² = -1).

The definitions for C_j and D_j in the Little Heston Trap by Albrecher et al. (2007) are

$$\begin{array}{l}{C}_j=(r-q)i\varphi \tau +\frac{\kappa \theta }{{\sigma }_v{}^2}\left[\left(b_j-p{\sigma }_{v}i\varphi -d_j\right)\tau -2\ln \left(\frac{1-{\epsilon }_j{e}^{-d_j\tau }}{1-{\epsilon }_j}\right)\right]\\ Dj=\frac{b_j-p{\sigma }_{v}i\varphi -d_j}{{\sigma }_v^2}\left(\frac{1-e^{-d_j\tau }}{1-{\epsilon }_j{e}^{-d_j\tau }}\right)\\ {\epsilon }_j=\frac{b_j-p{\sigma }_{v}i\varphi -d_j}{b_j-p{\sigma }_{v}i\varphi +d_j}\end{array}$$

Bates Stochastic Volatility Jump Diffusion Model

The Bates model (Bates 1996) is an extension of the Heston model where, in addition to stochastic volatility, the jump diffusion parameters similar to Merton (1976) are also added to model sudden asset price movements.

The stochastic differential equation is

$$\begin{array}{l}d{S}_t=(r-q-{\lambda }_p{\mu }_J)S_{t}dt+\sqrt{v_t}{S}_{t}d{W}_t+J{S}_{t}d{P}_t\\ d{v}_t=\kappa (\theta -v_t)dt+{\sigma }_v\sqrt{v_t}d{W}_t\\ \text{E}\left[d{W}_{t}d{W}_t^v\right]=pdt\\ \text{prob(}d{P}_t=1)={\lambda }_{p}dt\end{array}$$

Here:

r is the continuous risk-free rate.

q is the continuous dividend yield.

S_t is the asset price at time t.

v_t is the asset price variance at time t.

J is the random percentage jump size conditional on the jump occurring, where ln(1+J) is normally distributed with mean $$\ln (1+{\mu }_J)-\frac{{\delta }^2}{2}$$ and the standard deviation δ, and (1+J) has a lognormal distribution:

$$\frac{1}{(1+J)\delta \sqrt{2\pi }}\exp \left\{{\frac{-\left[\ln (1+J)-\left(\ln (1+{\mu }_J\right)-\frac{{\delta }^2}{2}\right]}{2{\delta }^2}}^2\right\}$$

Here:

v₀ is the initial variance of the asset price at t = 0 (v₀> 0).

θ is the long-term variance level for (θ > 0).

κ is the mean reversion speed for (κ > 0).

σ_v is the volatility of variance for (σ_v > 0).

p is the correlation between the Weiner processes W_t and $$W_t^v$$ for (-1 ≤ p ≤ 1).

μ_J is the mean of J for (μ_J > -1).

δ is the standard deviation of ln(1+J) for (δ ≥ 0).

$${\lambda }_p$$ is the annual frequency (intensity) of Poisson process P_t for ($${\lambda }_p$$ ≥ 0).

The characteristic function $$f_{Bate{s}_j(\varphi )}$$ for j = 1 (asset price mean measure) and j = 2 (risk-neutral measure) is

$$\begin{array}{l}{f}_{Bates(\varphi )}=\exp (C_j+D_j{v}_0+i\varphi \ln S_t)\exp {\left({\lambda }_p\tau (1+{\mu }_J\right)}^{m_j+\frac{1}{2}}\left[{(1+{\mu }_j)}^{i\varphi }{e}^{{\delta }^2(m_{j}i\varphi +\frac{{(i\varphi )}^2}{2})}-1\right]-{\lambda }_p\tau {\mu }_{J}i\varphi )\\ m_j=\left\{\begin{array}{l}{m}_1=\frac{1}{2}\\ m_2=-\frac{1}{2}\end{array}\right\}\\ C_j=(r-q)i\varphi \tau +\frac{\kappa \theta }{{\sigma }_v{}^2}\left[\left(b_j-p{\sigma }_{v}i\varphi +d_j\right)\tau -2\ln \left(\frac{1-g_j{e}^{d_j\tau }}{1-g_j}\right)\right]\\ Dj=\frac{b_j-p{\sigma }_{v}i\varphi +d_j}{{\sigma }_v^2}\left(\frac{1-e^{d_j\tau }}{1-g_j{e}^{d_j\tau }}\right)\\ g_j=\frac{b_j-p{\sigma }_{v}i\varphi +d_j}{b_j-p{\sigma }_{v}i\varphi -d_j}\\ d_j=\sqrt{{\left(b_j-p{\sigma }_{v}i\varphi \right)}^2-{\sigma }_v^2\left(2u_{j}i\varphi -{\varphi }^2\right)}\\ \text{where for }j=1,2:\\ u_1=\frac{1}{2},u_2=-\frac{1}{2},b_1=\kappa +{\lambda }_{VolRisk}-p{\sigma }_v,b_2=\kappa +{\lambda }_{VolRisk}\end{array}$$

Here:

ϕ is the characteristic function variable.

ƛ_VolRisk is the volatility risk premium.

τ is the time to maturity for (τ = T - t).

i is the unit imaginary number for (i²= -1).

The definitions for C_j and D_j in the Little Heston Trap by Albrecher et al. (2007) are

$$\begin{array}{l}{C}_j=(r-q)i\varphi \tau +\frac{\kappa \theta }{{\sigma }_v{}^2}\left[\left(b_j-p{\sigma }_{v}i\varphi -d_j\right)\tau -2\ln \left(\frac{1-{\epsilon }_j{e}^{-d_j\tau }}{1-{\epsilon }_j}\right)\right]\\ Dj=\frac{b_j-p{\sigma }_{v}i\varphi -d_j}{{\sigma }_v^2}\left(\frac{1-e^{-d_j\tau }}{1-{\epsilon }_j{e}^{-d_j\tau }}\right)\\ {\epsilon }_j=\frac{b_j-p{\sigma }_{v}i\varphi -d_j}{b_j-p{\sigma }_{v}i\varphi +d_j}\end{array}$$

Merton Jump Diffusion Model

The Merton jump diffusion model (Merton 1976) is an extension of the Black-Scholes model, where sudden asset price movements (both up and down) are modeled by adding the jump diffusion parameters with the Poisson process.

The stochastic differential equation is

$$\begin{array}{l}d{S}_t=(r-q-{\lambda }_p{\mu }_j)S_{t}dt+\sigma S_{t}d{W}_t+J{S}_{t}d{P}_t\\ \text{prob(}d{P}_t=1)={\lambda }_{p}dt\end{array}$$

Here:

r is the continuous risk-free rate.

q is the continuous dividend yield.

W_t is the Weiner process.

J is the random percentage jump size conditional on the jump occurring, where ln(1+J) is normally distributed with mean $$\ln (1+{\mu }_J)-\frac{{\delta }^2}{2}$$ and the standard deviation δ, and (1+J) has a lognormal distribution:

$$\frac{1}{(1+J)\delta \sqrt{2\pi }}\exp \left\{{\frac{-\left[\ln (1+J)-\left(\ln (1+{\mu }_J\right)-\frac{{\delta }^2}{2}\right]}{2{\delta }^2}}^2\right\}$$

Here:

μ_J is the mean of J for (μ_J > -1).

δ is the standard deviation of ln(1+J) for (δ≥ 0).

ƛ_p is the annual frequency (intensity) of the Poisson process P_tfor (ƛ_p ≥ 0).

σ is the volatility of the asset price for (σ > 0).

The characteristic function $$f_{Merton{76}_j}(\varphi )$$ for j = 1 (asset price measure) and j = 2 (risk-neutral measure) is

$$\begin{array}{l}{f}_{Merton{76}_j}=f_{B{S}_j}\exp \left({\lambda }_p\tau {\left(1+{\mu }_j\right)}^{m_j+\frac{1}{2}}\left[{\left(1+{\mu }_j\right)}^{i\varphi }{e}^{{\delta }^2\left(m_{j}i\varphi +\frac{{(i\varphi )}^2}{2}\right)}-1\right]-{\lambda }_p\tau {\mu }_{j}i\varphi \right)\\ \text{where for }j=1,2:\\ f_{B{S}_1}(\varphi )=\frac{f_{B{S}_2}\left(\varphi -i\right)}{f_{B{S}_2}\left(-i\right)}\\ f_{BS2}(\varphi )=\exp \left(i\varphi \left[\ln S_t+\left(r-q-\frac{{\sigma }^2}{2}\right)\tau \right]-\frac{{\varphi }^2{\sigma }^2}{2}\tau \right)\\ m_1=\frac{1}{2},m_2=-\frac{1}{2}\end{array}$$

Here:

ϕ is the characteristic function variable.

τ is the time to maturity (τ = T- t).

i is the unit imaginary number ( i² = -1).

Carr-Madan Formulation

The Carr-Madan (1999) formulation is a popular modified implementation of Heston (1993) framework.

Rather than computing the probabilities P₁ and P₂ as intermediate steps, Carr and Madan developed an alternative expression so that taking its inverse Fourier transform gives the option price itself directly.

$$\begin{array}{l}Call(k)=\frac{e^{-\alpha k}}{\pi }{\displaystyle {\int }_0^{\infty }\mathrm{Re}\left[e^{-iuk}\psi (u)\right]}du\\ \psi (u)=\frac{e^{-r\tau }{f}_2(\varphi =(u-(\alpha +1)i))}{{\alpha }^2+\alpha -u^2+iu(2\alpha +1)}\\ Put(K)=Call(K)+K{e}^{-r\tau }-S_t{e}^{-q\tau }\end{array}$$

Here:

r is the continuous risk-free rate.

q is the continuous dividend yield.

S_t is the asset price at time t.

τ is time to maturity (τ = T-t).

Call(K) is the call price at strike K.

Put(K) is the put price at strike K

i is a unit imaginary number (i²= -1)

ϕ is the characteristic function variable.

α is the damping factor.

u is the characteristic function variable for integration, where ϕ = (u - (α+1)i).

f₂(ϕ) is the characteristic function for P₂.

P₂ is the probability of S_t > K under the risk-neutral measure for the model.

To apply FFT or FRFT to this formulation, the characteristic function variable for integration u is discretized into NumFFT(N) points with the step size CharacteristicFcnStep (Δu), and the log-strike k is discretized into N points with the step size LogStrikeStep(Δk).

The discretized characteristic function variable for integration u_j(for j = 1,2,3,…,N) has a minimum value of 0 and a maximum value of (N-1) (Δu), and it approximates the continuous integration range from 0 to infinity.

The discretized log-strike grid k_n(for n = 1, 2, 3, N) is approximately centered around ln(S_t), with a minimum value of

$$\ln (S_t)-\frac{N}{2}\Delta k$$

and a maximum value of

$$\ln (S_t)+\left(\frac{N}{2}-1\right)\Delta k$$

Where the minimum allowable strike is

$$S_t\exp \left(-\frac{N}{2}\Delta k\right)$$

and the maximum allowable strike is

$$S_t\exp \left[\left(\frac{N}{2}-1\right)\Delta k\right]$$

As a result of the discretization, the expression for the call option becomes

$$Call(k_n)=\Delta u\frac{e^{-\alpha k_n}}{\pi }{\displaystyle \sum _{j=1}^N\mathrm{Re}\left[e^{-i\Delta k\Delta u(j-1)(n-1)e^{i{u}_j}\left[\frac{N\Delta k}{2}-\ln (S_t)\right]}\psi (u_j)\right]}{w}_j$$

Here:

Δu is the step size of the discretized characteristic function variable for integration.

Δk is the step size of the discretized log strike.

N is the number of FFT or FRFT points.

w_j is the weights for quadrature used for approximating the integral.

FFT is used to evaluate the above expression if Δk and Δu are subject to the following constraint:

$$\Delta k\Delta u=\left(\frac{2\pi }{N}\right)$$

Otherwise, the functions use the FRFT method described in Chourdakis (2005).

References

[1] Albrecher, H., P. Mayer, W. Schoutens, and J. Tistaert. “The Little Heston Trap.” Working Paper, Linz and Graz University of Technology, K.U. Leuven, ING Financial Markets, 2006.