optByHestonFFT

Option price by Heston model using FFT and FRFT

Syntax

[Price,StrikeOut] = optByHestonFFT(Rate,AssetPrice,Settle,Maturity,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV)

[Price,StrikeOut] = optByHestonFFT(___,Name,Value)

Description

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

Note

Alternatively, you can use the Vanilla object to price vanilla options. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

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

Examples

Workflow for Plotting an Option Price Surface Using the Heston Model

Use optByHestonFFT to calibrate a FFT strike grid and then compute option prices and plot an option price surface.

Define Option Variables and Heston 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;

Compute the 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 = datetime(2017,6,29);
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] =  optByHestonFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
    V0, ThetaV, Kappa, SigmaV, RhoSV, '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   29.4843
   58.8640   21.3767
   68.6231   12.5614
   80.0000    4.7008
   93.2631    0.6496
  108.7251    0.0144
  126.7505    0.0001

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

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

% 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] = optByHestonFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
    V0, ThetaV, Kappa, SigmaV, RhoSV, 'DividendYield', DividendYield, 'NumFFT', NumFFT);

% Compare with numerical integration method
Range = (510:516);
Strike = Kout(Range);
CallFFT = Call(Range);
CallNI = optByHestonNI(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, V0, ...
    ThetaV, Kappa, SigmaV, RhoSV, 'DividendYield', DividendYield);
Error = abs(CallFFT-CallNI);
table(Strike, CallFFT, CallNI, Error)

ans=7×4 table
    Strike     CallFFT       CallNI         Error  
    ______    _________    ___________    _________

    12.696       66.066         66.696      0.62964
    23.449       55.766         56.103      0.33672
    43.312       36.359         36.539      0.17974
        80       4.7727         4.7007     0.071928
    147.76     0.066156     2.3472e-08     0.066156
    272.93     0.013271    -2.5036e-09     0.013271
    504.11    0.0034504    -3.0876e-07    0.0034508

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 optByHestonFFT 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] = optByHestonFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
    V0, ThetaV, Kappa, SigmaV, RhoSV, 'DividendYield', DividendYield, ...
    'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, 'LogStrikeStep', 0.001);

% Compare with numerical integration method
Strike = Kout(Range);
CallFFT = Call(Range);
CallNI = optByHestonNI(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, V0, ...
    ThetaV, Kappa, SigmaV, RhoSV, 'DividendYield', DividendYield);
Error = abs(CallFFT-CallNI);
table(Strike, CallFFT, CallNI, Error)

ans=7×4 table
    Strike    CallFFT    CallNI      Error   
    ______    _______    ______    __________

    79.76      4.826      4.826    2.7708e-08
    79.84     4.7841     4.7841    3.0111e-08
    79.92     4.7423     4.7423    3.2376e-08
       80     4.7007     4.7007    3.4496e-08
    80.08     4.6593     4.6593    3.6457e-08
    80.16     4.6181     4.6181    3.8253e-08
    80.24      4.577      4.577    3.9872e-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 = datetime(2017,6,29);
Maturity = datemnth(Settle, 6);
Strike = 80; 

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

Call = 
4.7007

Compute the Option Prices for a Vector of Strikes

Use the Strike input to specify the strikes.

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

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

Call = 5×1

    7.0401
    5.8053
    4.7007
    3.7316
    2.8991

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 = datetime(2017,6,29);
Maturity = datemnth(Settle, [12 18 24 30 36]); % Five maturities
Strike = [76 78 80 82 84]'; % Five strikes

Call =  optByHestonFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
    V0, ThetaV, Kappa, SigmaV, RhoSV, 'DividendYield', DividendYield, 'NumFFT', NumFFT, ...
    'CharacteristicFcnStep', 0.065, 'LogStrikeStep', 0.001) % Five values in vector output

Call = 5×1

    8.9560
    9.3419
    9.6240
    9.8560
   10.0500

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] =  optByHestonFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
    V0, ThetaV, Kappa, SigmaV, RhoSV, 'DividendYield', DividendYield, 'NumFFT', NumFFT, ...
    'CharacteristicFcnStep', 0.065, 'LogStrikeStep', 0.001, ...
    'ExpandOutput', true) % (5 x 5) matrix outputs

Call = 5×5

    8.9560   10.4543   11.7058   12.8009   13.7728
    7.7946    9.3419   10.6337   11.7644   12.7685
    6.7244    8.3028    9.6240   10.7828   11.8134
    5.7475    7.3379    8.6771    9.8560   10.9074
    4.8645    6.4474    7.7930    8.9840   10.0500

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 the 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 = datetime(2017,6,29);
Maturity = datemnth(Settle, 12*(0.5:0.5:3)'); % Six maturities
Strike = (76:2:84)'; % Five strikes

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

Call = 5×6

    7.0401    8.9560   10.4543   11.7058   12.8009   13.7728
    5.8053    7.7946    9.3419   10.6337   11.7644   12.7685
    4.7007    6.7244    8.3028    9.6240   10.7828   11.8134
    3.7316    5.7475    7.3379    8.6771    9.8560   10.9074
    2.8991    4.8645    6.4474    7.7930    8.9840   10.0500

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 = datetime(2017,6,29);
Maturity = datemnth(Settle, 12); % Single maturity
ManyAssetPrices = [70 75 80 85]; % Four asset prices
Strike = (76:2:84)'; % Five strikes

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

Call = 5×4

    3.2944    5.8047    8.9560   12.6052
    2.6413    4.8810    7.7946   11.2507
    2.0864    4.0575    6.7244    9.9738
    1.6230    3.3325    5.7475    8.7783
    1.2429    2.7028    4.8645    7.6676

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 = datetime(2017,6,29);
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 =  optByHestonFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ...
    V0, ThetaV, Kappa, SigmaV, RhoSV, '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

Rate — Continuously compounded risk-free interest rate

decimal

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

Data Types: double

AssetPrice — Current underlying asset price

numeric

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

Settle — Option settlement date

datetime array | string array | date character vector

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

To support existing code, optByHestonFFT also accepts serial date numbers as inputs, but they are not recommended.

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

Maturity — Option maturity date

datetime array | string array | date character vector

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

To support existing code, optByHestonFFT also accepts serial date numbers as inputs, but they are not recommended.

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

OptSpec — Definition of option

cell array of character vector with values 'call' or 'put' | string array with values "call" or "put"

Definition of the option as 'call' or 'put', 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

Strike — Option strike price value

numeric

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

V0 — Initial variance of underlying asset

numeric

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

Data Types: double

ThetaV — Long-term variance of underlying asset

numeric

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

Data Types: double

Kappa — Mean revision speed for the variance of underlying asset

numeric

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

Data Types: double

SigmaV — Volatility of the variance of underlying asset

numeric

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

Data Types: double

RhoSV — Correlation between Wiener processes for underlying asset and its variance

numeric

Correlation between the Wiener processes for the underlying asset and its variance, 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] = optByHestonFFT(Rate,AssetPrice,Settle,Maturity,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV,'Basis',7)

Basis — Day-count basis of instrument

0 (default) | numeric values: 0,1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13

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

For more information, see Basis.

Data Types: double

DividendYield — Continuously compounded underlying asset yield

0 (default) | numeric

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

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 values 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.

• false — Use the original Heston formation.

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, 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 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) | character vector with values:'simpson' or 'trapezoidal' | string array with values: "simpson" or "trapezoidal"

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

ExpandOutput — Flag to expand the outputs

false (outputs are NINST-by-1 vectors) (default) | logical with value of true or false

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

Price — Option prices

numeric

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

StrikeOut — Strikes corresponding to Price

numeric

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

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: $$\max (K-St,0)$$

where:

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 (square root of 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:

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

where

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 Wiener 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:

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

where

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

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

Carr-Madan Formulation

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.

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

where

τ 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

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

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

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

[4] 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

R2022b: Serial date numbers not recommended

Although optByHestonFFT supports serial date numbers, datetime values are recommended instead. The datetime data type provides flexible date and time formats, storage out to nanosecond precision, and properties to account for time zones and daylight saving time.

To convert serial date numbers or text to datetime values, use the datetime function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum");
y = year(t)

y =

        2021

There are no plans to remove support for serial date number inputs.