# optByBatesFFT

Option price by Bates model using FFT and FRFT

## Description

example

[Price,StrikeOut] = optByBatesFFT(Rate,AssetPrice,Settle,Maturity,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV,MeanJ,JumpVol,JumpFreq) computes vanilla European option price by Bates model, using Carr-Madan FFT and Chourdakis FRFT methods.

example

[Price,StrikeOut] = optByBatesFFT(___,Name,Value) adds optional name-value pair arguments.

## Examples

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Use optByBatesFFT to calibrate the FFT strike grid, compute option prices, and plot an option price surface.

Define Option Variables and Bates Model Parameters

AssetPrice = 80;
Rate = 0.03;
DividendYield = 0.02;
OptSpec = 'call';

V0 = 0.04;
ThetaV = 0.05;
Kappa = 1.0;
SigmaV = 0.2;
RhoSV = -0.7;
MeanJ = 0.02;
JumpVol = 0.08;
JumpFreq = 2;

Compute Option Prices for the Entire FFT (or FRFT) Strike Grid, Without Specifying Strike

Compute option prices and also output the corresponding strikes. If the Strike input is empty ([]), option prices will be computed on the entire FFT (or FRFT) strike grid. The FFT (or FRFT) strike grid is determined as exp(log-strike grid), where each column of the log-strike grid has NumFFT points with LogStrikeStep spacing that are roughly centered around each element of log(AssetPrice). The default value for NumFFT is 2^12. In addition to the prices in the first output, the optional last output contains the corresponding strikes.

Settle = datenum('29-Jun-2017');
Maturity = datemnth(Settle, 6);
Strike = []; % Strike is not specified (will use the entire FFT strike grid)

% Compute option prices for the entire FFT strike grid
[Call, Kout] = optByBatesFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
V0, ThetaV, Kappa, SigmaV, RhoSV, MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield);

% Show the lowest and highest strike values on the FFT strike grid
format
MinStrike = Kout(1) % Lowest possible strike in the current FFT strike grid
MinStrike = 2.9205e-135
MaxStrike = Kout(end) % Highest possible strike in the current FFT strike grid
MaxStrike = 1.8798e+138
% Show a subset of the strikes and corresponding option prices
Range = (2046:2052);
[Kout(Range) Call(Range)]
ans = 7×2

50.4929   29.4990
58.8640   21.4545
68.6231   12.8544
80.0000    5.3484
93.2631    1.2404
108.7251    0.1648
126.7505    0.0152

Change the Number of FFT (or FRFT) Points and Compare with optByBatesNI

Try a different number of FFT (or FRFT) points, and compare the results with direct numerical integration. Unlike optByBatesFFT, which uses FFT (or FRFT) techniques for fast computation across the whole range of strikes, the optByBatesNI function uses direct numerical integration and it is typically slower, especially for multiple strikes. However, the values computed by optByBatesNI can serve as a benchmark for adjusting the settings for optByBatesFFT.

% Try a smaller number of FFT (or FRFT) points
% (e.g. for faster performance or smaller memory footprint)
NumFFT = 2^10; % Smaller than the default value of 2^12
Strike = []; % Strike is not specified (will use the entire FFT strike grid)
[Call, Kout] = optByBatesFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
V0, ThetaV, Kappa, SigmaV, RhoSV, MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield, 'NumFFT', NumFFT);

% Compare with numerical integration method
Range = (510:516);
Strike = Kout(Range);
CallFFT = Call(Range);
CallNI = optByBatesNI(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
V0, ThetaV, Kappa, SigmaV, RhoSV, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield);
Error = abs(CallFFT-CallNI);
table(Strike, CallFFT, CallNI, Error)
ans=7×4 table
Strike     CallFFT       CallNI         Error
______    _________    ___________    _________

12.696       66.237         66.696      0.45912
23.449        55.86         56.103      0.24239
43.312       36.418         36.541      0.12246
80       5.4029         5.3484     0.054469
147.76     0.044921      0.0010864     0.043835
272.93    0.0094655    -7.8249e-08    0.0094656
504.11    0.0024986    -3.3873e-07    0.0024989

Make Further Adjustments to FFT (or FRFT)

If the values in the output CallFFT are significantly different from those in CallNI, try making adjustments to optByBatesFFT settings, such as CharacteristicFcnStep, LogStrikeStep, NumFFT, DampingFactor, and so on. Note that if (LogStrikeStep * CharacteristicFcnStep) is 2*pi / NumFFT, FFT is used. Otherwise, FRFT is used.

Strike = []; % Strike is not specified (will use the entire FFT or FRFT strike grid)
[Call, Kout] = optByBatesFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
V0, ThetaV, Kappa, SigmaV, RhoSV, MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield, 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, ...
'LogStrikeStep', 0.001);

% Compare with numerical integration method
Strike = Kout(Range);
CallFFT = Call(Range);
CallNI = optByBatesNI(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
V0, ThetaV, Kappa, SigmaV, RhoSV, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield);
Error = abs(CallFFT-CallNI);
table(Strike, CallFFT, CallNI, Error)
ans=7×4 table
Strike    CallFFT    CallNI      Error
______    _______    ______    __________

79.76     5.4682     5.4682    1.5355e-08
79.84     5.4281     5.4281    1.4833e-08
79.92     5.3882     5.3882    1.4244e-08
80     5.3484     5.3484     1.359e-08
80.08     5.3088     5.3088    1.2875e-08
80.16     5.2693     5.2693    1.2101e-08
80.24       5.23       5.23    1.1272e-08

% Save the final FFT (or FRFT) strike grid for future reference. For
% example, it provides information about the range of |Strike| inputs for
% which the FFT (or FRFT) operation is valid.
FFTStrikeGrid = Kout;
MinStrike = FFTStrikeGrid(1) % Strike cannot be less than MinStrike
MinStrike = 47.9437
MaxStrike = FFTStrikeGrid(end) % Strike cannot be greater than MaxStrike
MaxStrike = 133.3566

Compute the Option Price for a Single Strike

Once the desired FFT (or FRFT) settings are determined, use the Strike input to specify the strikes (rather than providing an empty array). If the specified strikes do not match a value on the FFT (or FRFT) strike grid, the outputs are interpolated on the specified strikes.

Settle = datenum('29-Jun-2017');
Maturity = datemnth(Settle, 6);
Strike = 80;

Call = optByBatesFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
V0, ThetaV, Kappa, SigmaV, RhoSV,  MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield, 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, ...
'LogStrikeStep', 0.001)
Call = 5.3484

Compute the Option Prices for a Vector of Strikes

Use the Strike input to specify the strikes.

Settle = datenum('29-Jun-2017');
Maturity = datemnth(Settle, 6);
Strike = (76:2:84)';

Call = optByBatesFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
V0, ThetaV, Kappa, SigmaV, RhoSV,  MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield, 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, ...
'LogStrikeStep', 0.001)
Call = 5×1

7.5765
6.4020
5.3484
4.4173
3.6073

Compute the Option Prices for a Vector of Strikes and a Vector of Dates of the Same Lengths

Use the Strike input to specify the strikes. Also, the Maturity input can be a vector, but it must match the length of the Strike vector if the ExpandOutput name-value pair argument is not set to "true".

Settle = datenum('29-Jun-2017');
Maturity = datemnth(Settle, [12 18 24 30 36]); % Five maturities
Strike = [76 78 80 82 84]'; % Five strikes

Call = optByBatesFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
V0, ThetaV, Kappa, SigmaV, RhoSV,  MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield, 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, ...
'LogStrikeStep', 0.001) % Five values in vector output
Call = 5×1

9.7516
10.3931
10.8865
11.2990
11.6491

Expand the Outputs for a Surface

Set the ExpandOutput name-value pair argument to "true" to expand the outputs into NStrikes-by-NMaturities matrices. In this case, they are square matrices.

[Call, Kout] = optByBatesFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
V0, ThetaV, Kappa, SigmaV, RhoSV,  MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield, 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, ...
'LogStrikeStep', 0.001, 'ExpandOutput', true) % (5 x 5) matrix output
Call = 5×5

9.7516   11.4387   12.8395   14.0588   15.1361
8.6554   10.3931   11.8344   13.0890   14.1980
7.6432    9.4149   10.8865   12.1693   13.3046
6.7153    8.5035    9.9952   11.2990   12.4553
5.8705    7.6581    9.1594   10.4771   11.6491

Kout = 5×5

76    76    76    76    76
78    78    78    78    78
80    80    80    80    80
82    82    82    82    82
84    84    84    84    84

Compute Option Prices for a Vector of Strikes and a Vector of Dates of Different Lengths

When ExpandOutput is "true", NStrikes do not have to match NMaturities (that is, the output NStrikes-by-NMaturities matrix can be rectangular).

Settle = datenum('29-Jun-2017');
Maturity = datemnth(Settle, 12*(0.5:0.5:3)'); % Six maturities
Strike = (76:2:84)'; % Five strikes

Call = optByBatesFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
V0, ThetaV, Kappa, SigmaV, RhoSV,  MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield, 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, ...
'LogStrikeStep', 0.001, 'ExpandOutput', true) % (5 x 6) matrix output
Call = 5×6

7.5765    9.7516   11.4387   12.8395   14.0588   15.1361
6.4020    8.6554   10.3931   11.8344   13.0890   14.1980
5.3484    7.6432    9.4149   10.8865   12.1693   13.3046
4.4173    6.7153    8.5035    9.9952   11.2990   12.4553
3.6073    5.8705    7.6581    9.1594   10.4771   11.6491

Compute the Option Prices for a Vector of Strikes and a Vector of Asset Prices

When ExpandOutput is "true", the output can also be a NStrikes-by-NAssetPrices rectangular matrix by accepting a vector of asset prices.

Settle = datenum('29-Jun-2017');
Maturity = datemnth(Settle, 12); % Single maturity
ManyAssetPrices = [70 75 80 85]; % Four asset prices
Strike = (76:2:84)'; % Five strikes

Call = optByBatesFFT(Rate, ManyAssetPrices, Settle, Maturity, OptSpec, Strike, ...
V0, ThetaV, Kappa, SigmaV, RhoSV,  MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield, 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, ...
'LogStrikeStep', 0.001, 'ExpandOutput', true) % (5 x 4) matrix output
Call = 5×4

4.2033    6.6918    9.7516   13.2808
3.5558    5.8112    8.6554   11.9993
2.9906    5.0181    7.6432   10.7934
2.5018    4.3096    6.7153    9.6652
2.0825    3.6818    5.8705    8.6158

Plot an Option Price Surface

Use the Strike input to specify the strikes. Increase the value for NumFFT to support a wider range of strikes. Also, the Maturity input can be a vector. Set ExpandOutput to "true" to output the surface as a NStrikes-by-NMaturities matrix.

Settle = datenum('29-Jun-2017');
Maturity = datemnth(Settle, 12*[1/12 0.25 (0.5:0.5:3)]');
Times = yearfrac(Settle, Maturity);
Strike = (2:2:200)';

% Increase |NumFFT| to support a wider range of strikes
NumFFT = 2^13;

Call = optByBatesFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
V0, ThetaV, Kappa, SigmaV, RhoSV,  MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield, 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, ...
'LogStrikeStep', 0.001, 'ExpandOutput', true);

[X,Y] = meshgrid(Times,Strike);

figure;
surf(X,Y,Call);
title('Price');
xlabel('Years to Option Expiry');
ylabel('Strike');
view(-112,34);
xlim([0 Times(end)]);
zlim([0 80]);

## Input Arguments

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Continuously compounded risk-free interest rate, specified as a scalar decimal value.

Data Types: double

Current underlying asset price, specified as numeric value using a scalar or a NINST-by-1 or NColumns-by-1 vector.

For more information on the proper dimensions for AssetPrice, see the name-value pair argument ExpandOutput.

Data Types: double

Option settlement date, specified as a NINST-by-1 or NColumns-by-1 vector using serial date numbers, date character vectors, datetime arrays, or string arrays. The Settle date must be before the Maturity date.

For more information on the proper dimensions for Settle, see the name-value pair argument ExpandOutput.

Data Types: double | char | datetime | string

Option maturity date, specified as a NINST-by-1 or NColumns-by-1 vector using serial date numbers, date character vectors, datetime arrays, or string arrays.

For more information on the proper dimensions for Maturity, see the name-value pair argument ExpandOutput.

Data Types: double | char | datetime | string

Definition of the option, specified as a NINST-by-1 or NColumns-by-1 vector using a cell array of character vectors or string arrays with values 'call' or 'put'.

For more information on the proper dimensions for OptSpec, see the name-value pair argument ExpandOutput.

Data Types: cell | string

Option strike price value, specified as a NINST-by-1, NRows-by-1, NRows-by-NColumns vector of strike prices.

If this input is an empty array ([]), option prices are computed on the entire FFT (or FRFT) strike grid, which is determined as exp(log-strike grid). Each column of the log-strike grid has 'NumFFT' points with 'LogStrikeStep' spacing that are roughly centered around each element of log(AssetPrice).

For more information on the proper dimensions for Strike, see the name-value pair argument ExpandOutput.

Data Types: double

Initial variance of the underling asset, specified as a scalar numeric value.

Data Types: double

Long-term variance of the underling asset, specified as a scalar numeric value.

Data Types: double

Mean revision speed for the underling asset, specified as a scalar numeric value.

Data Types: double

Volatility of the variance of the underling asset, specified as a scalar numeric value.

Data Types: double

Correlation between the Weiner processes for the underlying asset and its variance, specified as a scalar numeric value.

Data Types: double

Mean of the random percentage jump size (J), specified as a scalar decimal value where log(1+J) is normally distributed with mean (log(1+MeanJ)-0.5*JumpVol^2) and the standard deviation JumpVol.

Data Types: double

Standard deviation of log(1+J) where J is the random percentage jump size, specified as a scalar decimal value.

Data Types: double

Annual frequency of Poisson jump process, specified as a scalar numeric value.

Data Types: double

### Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: [Price, StrikeOut] = optByBatesFFT(Rate,AssetPrice,Settle,Maturity,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV,MeanJ,JumpVol,JumpFreq,'Basis',7)

Day-count of the instrument, specified as the comma-separated pair consisting of 'Basis' and a scalar using a supported value:

• 0 = actual/actual

• 1 = 30/360 (SIA)

• 2 = actual/360

• 3 = actual/365

• 4 = 30/360 (PSA)

• 5 = 30/360 (ISDA)

• 6 = 30/360 (European)

• 7 = actual/365 (Japanese)

• 8 = actual/actual (ICMA)

• 9 = actual/360 (ICMA)

• 10 = actual/365 (ICMA)

• 11 = 30/360E (ICMA)

• 12 = actual/365 (ISDA)

• 13 = BUS/252

Data Types: double

Continuously compounded underlying asset yield, specified as the comma-separated pair consisting of 'DividendYield' and a scalar numeric value.

Data Types: double

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

Data Types: double

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.

• false — Use the original Heston formation.

Data Types: logical

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

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

Data Types: double

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

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

Data Types: double

Type of quadrature, specified as the comma-separated pair consisting of 'Quadrature' and a single character vector or string array with a value of 'simpson' or 'trapezoidal'.

Data Types: char | string

Flag to expand the outputs, specified as the comma-separated pair consisting of 'ExpandOutput' and a logical:

• true — If true, the outputs are NRows-by- NColumns matrices. NRows is the number of strikes for each column and it is determined by the Strike input. For example, Strike can be a NRows-by-1 vector, or a NRows-by-NColumns matrix. If Strike is empty, NRows is equal to NumFFT. NColumns is determined by the sizes of AssetPrice, Settle, Maturity, and OptSpec, which must all be either scalar or NColumns-by-1 vectors.

• false — If false, the outputs are NINST-by-1 vectors. Also, the inputs Strike, AssetPrice, Settle, Maturity, and OptSpec must all be either scalar or NINST-by-1 vectors.

Data Types: logical

## Output Arguments

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Option prices, returned as a NINST-by-1, or NRows-by-NColumns, depending on ExpandOutput.

Strikes corresponding to Price, returned as a NINST-by-1, or NRows-by-NColumns, depending on ExpandOutput.

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### 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: $\mathrm{max}\left(St-K,0\right)$

• For a put: $\mathrm{max}\left(K-St,0\right)$

where:

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

K is the strike price.

### 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) were also added to model sudden asset price movements.

The stochastic differential equation is:

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

where

r is the continuous risk-free rate.

q is the continuous dividend yield.

St is the asset price at time t.

vt 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 $\mathrm{ln}\left(1+{\mu }_{J}\right)-\frac{{\delta }^{2}}{2}$ and the standard deviation δ, and (1+J) has a lognormal distribution:

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

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).

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

p is the correlation between the Weiner processes Wt and ${W}_{t}^{v}$ for (-1 ≤ p ≤ 1).

μJ is the mean of J for (μJ > -1).

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

${\lambda }_{p}$ is the annual frequency (intensity) of Poisson process Pt for (${\lambda }_{p}$ ≥ 0).

The characteristic function ${f}_{Bate{s}_{j}\left(\varphi \right)}$ for j = 1 (asset price mean measure) and j =2 (risk-neutral measure) is:

where

ϕ is the characteristic function variable.

ƛVolRisk 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 Cj and Dj under “The Little Heston Trap” by Albrecher et al. (2007) are:

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

The Carr and 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.

$\begin{array}{l}Call\left(k\right)=\frac{{e}^{-\alpha k}}{\pi }{\int }_{0}^{\infty }\mathrm{Re}\left[{e}^{-iuk}\psi \left(u\right)\right]du\\ \psi \left(u\right)=\frac{{e}^{-r\tau }{f}_{2}\left(\varphi =\left(u-\left(\alpha +1\right)i\right)\right)}{{\alpha }^{2}+\alpha -{u}^{2}+iu\left(2\alpha +1\right)}\\ Put\left(K\right)=Call\left(K\right)+K{e}^{-r\tau }-{S}_{t}{e}^{-q\tau }\end{array}$

where

r is the continuous risk-free rate.

q is the continuous dividend yield.

St 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 St > 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 CharacteristicFcnStepu), and the log-strike k is discretized into N points with the step size LogStrikeStepk).

The discretized characteristic function variable for integration, uj(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, kn(for n = 1, 2, 3, N) is approximately centered around ln(St), with a minimum value of

$\mathrm{ln}\left({S}_{t}\right)-\frac{N}{2}\Delta k$

and a maximum value of

$\mathrm{ln}\left({S}_{t}\right)+\left(\frac{N}{2}-1\right)\Delta k$

Where the minimum allowable strike is

${S}_{t}\mathrm{exp}\left(-\frac{N}{2}\Delta k\right)$

and the maximum allowable strike is

${S}_{t}\mathrm{exp}\left[\left(\frac{N}{2}-1\right)\Delta k\right]$

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

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

where

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

Δk is the step size of discretized log-strike.

N is the number of FFT/FRFT points

wj 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., Mayer, P., Schoutens, W., and Tistaert, J. "The Little Heston Trap." Working Paper, Linz and Graz University of Technology, K.U. Leuven, ING Financial Markets, 2006.

[2] Bates, D. S. “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options.” The Review of Financial Studies. Vol 9. No. 1. 1996.

[3] Carr, P. and D.B. Madan. “Option Valuation Using the Fast Fourier Transform.” Journal of Computational Finance. Vol 2. No. 4. 1999.

[4] Chourdakis, K. “Option Pricing Using Fractional FFT.” Journal of Computational Finance. 2005.

[5] Heston, S. L. “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.” The Review of Financial Studies. Vol 6. No. 2. 1993.

## Version History

Introduced in R2018a