Heston

Create Heston model object for Vanilla, Asian, Barrier, DoubleBarrier, Lookback, PartialLookback, VarianceSwap, Touch, DoubleTouch, Cliquet, or Binary instrument

Since R2020a

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Description

Create and price a Vanilla, Asian, Barrier, DoubleBarrier, Lookback, PartialLookback, VarianceSwap, Touch, DoubleTouch, Cliquet, or Binary instrument object with a Heston model using this workflow:

Use fininstrument to create a Vanilla, Barrier, Lookback, PartialLookback, Asian, DoubleBarrier, VarianceSwap, Binary, Touch, Cliquet, or DoubleTouch instrument object.
Use finmodel to specify a Heston model object for the Vanilla, Asian, Barrier, DoubleBarrier, Lookback, PartialLookback, VarianceSwap, Touch, DoubleTouch, Cliquet, or Binary instrument object.
Use finpricer to specify a FiniteDifference, NumericalIntegration, or FFT pricing method for the Vanilla instrument object.
Use finpricer to specify an AssetMonteCarlo pricing method for the Vanilla, Asian, Barrier, DoubleBarrier, Lookback, PartialLookback, Touch, DoubleTouch, Cliquet, or Binary instrument object.

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, AsianBarrier, DoubleBarrier, Lookback, PartialLookback, VarianceSwap, Touch, DoubleTouch, Cliquet, or Binary instrument, see Choose Instruments, Models, and Pricers.

Creation

Syntax

HestonModelObj = finmodel(ModelType,'V0'v0_value,'ThetaV',thetav_value,'Kappa',kappa_value,'SigmaV',sigmav_value,'RhoSV',rhosv_value)

Description

HestonModelObj = finmodel(ModelType,'V0'v0_value,'ThetaV',thetav_value,'Kappa',kappa_value,'SigmaV',sigmav_value,'RhoSV',rhosv_value) creates a Black model object by specifying ModelType and the required name-value pair arguments V0, ThetaV, Kappa, SigmaV, and RhoSV to set properties using required name-value pair arguments. For example, HestonModelObj = finmodel("Heston",'V0',0.032,'ThetaV',0.1,'Kappa',0.003,'SigmaV',0.2,'RhoSV',0.9) creates a Heston model object.

example

Input Arguments

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`ModelType` — Model type
string with value `"Heston"` | character vector with value `'Heston'`

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

Data Types: char | string

Name-Value Arguments

Specify required 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: HestonModelObj = finmodel("Heston",'V0',0.032,'ThetaV',0.1,'Kappa',0.003,'SigmaV',0.2,'RhoSV',0.9)

`V0` — Initial variance of underlying asset
numeric

Initial variance of the underlying asset, specified as the comma-separated pair consisting of 'V0' and a scalar numeric value.

Data Types: double

`ThetaV` — Long-term variance of underlying asset
numeric

Long-term variance of the underlying asset, specified as the comma-separated pair consisting of 'ThetaV' and a scalar numeric value.

Data Types: double

`Kappa` — Mean revision speed for the variance of underlying asset
numeric

Mean revision speed for the underlying asset, specified as the comma-separated pair consisting of 'Kappa' and a scalar numeric value.

Data Types: double

`SigmaV` — Volatility of the variance of underlying asset
numeric

Volatility of the variance of the underlying asset, specified as the comma-separated pair consisting of 'SigmaV' and 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 the comma-separated pair consisting of 'RhoSV' and a scalar numeric value.

Data Types: double

Properties

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`V0` — Initial variance of underlying asset
numeric

Initial variance of the underlying asset, returned as a scalar numeric value.

Data Types: double

`ThetaV` — Long-term variance of underlying asset
numeric

Long-term variance of the underlying asset, returned as a scalar numeric value.

Data Types: double

`Kappa` — Mean revision speed for the variance of underlying asset
numeric

Mean revision speed for the underlying asset, returned as a scalar numeric value.

Data Types: double

`SigmaV` — Volatility of the variance of underlying asset
numeric

Volatility of the variance of the underlying asset, returned 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, returned as a scalar numeric value.

Data Types: double

Examples

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Use `Heston` Model and `FFT` Pricer to Price `Vanilla` Instrument

Open Live Script

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

Use `Heston` Model and `AssetMonteCarlo` Pricer to Price `Lookback` Instrument

Open Live Script

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

Create Lookback Instrument Object

Use fininstrument to create a Lookback instrument object.

LookbackOpt = fininstrument("Lookback",'Strike',105,'ExerciseDate',datetime(2022,9,15),'OptionType',"put",'ExerciseStyle',"american",'Name',"lookback_option")

LookbackOpt = 
  Lookback with properties:

       OptionType: "put"
           Strike: 105
      AssetMinMax: NaN
    ExerciseStyle: "american"
     ExerciseDate: 15-Sep-2022
             Name: "lookback_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',0.08,'RhoSV',0.9)

HestonModel = 
  Heston with properties:

        V0: 0.0320
    ThetaV: 0.1000
     Kappa: 0.0030
    SigmaV: 0.0800
     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 AssetMonteCarlo Pricer Object

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

outPricer = finpricer("AssetMonteCarlo",'DiscountCurve',myRC,"Model",HestonModel,'SpotPrice',90,'simulationDates',datetime(2022,9,15))

outPricer = 
  HestonMonteCarlo with properties:

           DiscountCurve: [1x1 ratecurve]
               SpotPrice: 90
         SimulationDates: 15-Sep-2022
               NumTrials: 1000
           RandomNumbers: []
                   Model: [1x1 finmodel.Heston]
            DividendType: "continuous"
           DividendValue: 0
        MonteCarloMethod: "standard"
    BrownianMotionMethod: "standard"

Price Lookback Instrument

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

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

Price = 
21.9733

outPR = 
  priceresult with properties:

       Results: [1x8 table]
    PricerData: [1x1 struct]

outPR.Results

ans=1×8 table
    Price      Delta     Gamma    Lambda       Rho       Theta      Vega     VegaLT
    ______    _______    _____    _______    _______    _______    ______    ______

    21.973    -0.7701      0      -3.1542    -215.94    0.28812    99.825    1.447

More About

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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 allows modeling the implied volatility smiles observed in the market.

The Heston model assumes that the volatility of the underlying asset follows a stochastic process. This stochastic process is typically modeled as a mean-reverting process, allowing volatility to fluctuate around a long-term average.

The stochastic differential equation is:

$\begin{array}{l} d S_{t} = (r - q) S_{t} d t + \sqrt{v_{t}} S_{t} d W_{t} \\ d v_{t} = κ (θ - v_{t}) d t + σ_{v} \sqrt{v_{t}} d W_{t}^{v} \\ E [d W_{t} d W_{t}^{v}] = p d t \end{array}$

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).

References

[1] 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, Number 2, 1993.

Version History

Introduced in R2020a

Heston

Description

Creation

Syntax

Description

Input Arguments

`ModelType` — Model type
string with value `"Heston"` | character vector with value `'Heston'`

`V0` — Initial variance of underlying asset
numeric

`ThetaV` — Long-term variance of underlying asset
numeric

`Kappa` — Mean revision speed for the variance of underlying asset
numeric

`SigmaV` — Volatility of the variance of underlying asset
numeric

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

Properties

`V0` — Initial variance of underlying asset
numeric

`ThetaV` — Long-term variance of underlying asset
numeric

`Kappa` — Mean revision speed for the variance of underlying asset
numeric

`SigmaV` — Volatility of the variance of underlying asset
numeric

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

Examples

Use `Heston` Model and `FFT` Pricer to Price `Vanilla` Instrument

Use `Heston` Model and `AssetMonteCarlo` Pricer to Price `Lookback` Instrument

More About

Heston Stochastic Volatility Model

References

Version History

See Also

Functions

Topics

Heston

Description

Creation

Syntax

Description

Input Arguments

ModelType — Model type string with value "Heston" | character vector with value 'Heston'

V0 — Initial variance of underlying asset numeric

ThetaV — Long-term variance of underlying asset numeric

Kappa — Mean revision speed for the variance of underlying asset numeric

SigmaV — Volatility of the variance of underlying asset numeric

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

Properties

V0 — Initial variance of underlying asset numeric

ThetaV — Long-term variance of underlying asset numeric

Kappa — Mean revision speed for the variance of underlying asset numeric

SigmaV — Volatility of the variance of underlying asset numeric

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

Examples

Use Heston Model and FFT Pricer to Price Vanilla Instrument

Use Heston Model and AssetMonteCarlo Pricer to Price Lookback Instrument

More About

Heston Stochastic Volatility Model

References

Version History

See Also

Functions

Topics

`ModelType` — Model type
string with value `"Heston"` | character vector with value `'Heston'`

`V0` — Initial variance of underlying asset
numeric

`ThetaV` — Long-term variance of underlying asset
numeric

`Kappa` — Mean revision speed for the variance of underlying asset
numeric

`SigmaV` — Volatility of the variance of underlying asset
numeric

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

`V0` — Initial variance of underlying asset
numeric

`ThetaV` — Long-term variance of underlying asset
numeric

`Kappa` — Mean revision speed for the variance of underlying asset
numeric

`SigmaV` — Volatility of the variance of underlying asset
numeric

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

Use `Heston` Model and `FFT` Pricer to Price `Vanilla` Instrument

Use `Heston` Model and `AssetMonteCarlo` Pricer to Price `Lookback` Instrument