Heston

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

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:

1. Use fininstrument to create a Vanilla, Barrier, Lookback, PartialLookback, Asian, DoubleBarrier, VarianceSwap, Binary, Touch, Cliquet, or DoubleTouch instrument object.

2. Use finmodel to specify a Heston model object for the Vanilla, Asian, Barrier, DoubleBarrier, Lookback, PartialLookback, VarianceSwap, Touch, DoubleTouch, Cliquet, or Binary instrument object.

3. Choose a pricing method.

   • 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.

   • Use finpricer to specify an AssetReinforcementLearning pricing method for a Vanilla 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.

Input Arguments

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

Output Arguments

HestonModelObj — Heston model

Heston object

Heston model, returned as a Heston object.

Properties

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

Use Heston Model and FFT Pricer 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: [1×1 finmodel.Heston]
            DiscountCurve: [1×1 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: [1×7 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

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: [1×1 ratecurve]
               SpotPrice: 90
         SimulationDates: 15-Sep-2022
               NumTrials: 1000
           RandomNumbers: []
                   Model: [1×1 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: [1×8 table]
    PricerData: [1×1 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 

Use Heston Model and AssetReinforcementLearning Pricer to Price Vanilla Instrument

Since R2026a

This example shows the workflow to price a Vanilla instrument when using a Heston model and an AssetReinforcementLearning pricing method. Note, to use this functionality, you must have Reinforcement Learning Toolbox™ installed.

Create Vanilla Instrument Object

Use fininstrument to create a Vanilla instrument object.

VanillaOpt = fininstrument("Vanilla",ExerciseDate=datetime(2021,8,15),Strike=110,OptionType="put",ExerciseStyle="american",Name="vanilla_option")

VanillaOpt = 
  Vanilla with properties:

       OptionType: "put"
    ExerciseStyle: "american"
     ExerciseDate: 15-Aug-2021
           Strike: 110
             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=0.08,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 ratecurve object using ratecurve.

Settle = datetime(2019,1,1);
Type = 'zero';
ZeroTimes = [calmonths(6) calyears([1 2 3 4 5 7 10 20 30])]';
ZeroRates = [0.0052 0.0055 0.0061 0.0073 0.0094 0.0119 0.0168 0.0222 0.0293 0.0307]';
ZeroDates = Settle + ZeroTimes;

myRC = ratecurve('zero',Settle,ZeroDates,ZeroRates)

myRC = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 0
                Dates: [10×1 datetime]
                Rates: [10×1 double]
               Settle: 01-Jan-2019
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create AssetReinforcementLearning Pricer Object

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

SpotPrice = 100;
SimDates = [Settle+days(1):days(2):Settle+years(2)];

outPricer = finpricer("AssetReinforcementLearning",DiscountCurve=myRC,Model=HestonModel,SpotPrice=SpotPrice,SimulationDates=SimDates)

outPricer = 
  AssetReinforcementLearning with properties:

      DiscountCurve: [1×1 ratecurve]
          SpotPrice: 100
    SimulationDates: [02-Jan-2019    04-Jan-2019    06-Jan-2019    08-Jan-2019    10-Jan-2019    12-Jan-2019    14-Jan-2019    16-Jan-2019    18-Jan-2019    20-Jan-2019    22-Jan-2019    24-Jan-2019    26-Jan-2019    28-Jan-2019    …    ] (1×365 datetime)
          NumTrials: 1000
              Model: [1×1 finmodel.Heston]
       DividendType: "continuous"
      DividendValue: 0

Price Vanilla Instrument

Use price to compute the price for the Vanilla instrument.

[Price,priceResultData] = price(outPricer,VanillaOpt)

Price = 
15.5194

priceResultData = 
  priceresult with properties:

       Results: [1×1 table]
    PricerData: [1×1 struct]

priceResultData.PricerData

ans = struct with fields:
    SimulationTimes: [366×1 timetable]
              Paths: [366×2×1000 double]
      TrainingStats: [1×1 rl.train.result.rlTrainingResult]
              Agent: [1×1 rl.agent.rlLSPIAmericanOptionAgent]

More About

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:

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

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).

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

R2026a: Support for Vanilla Instrument using AssetReinforcementLearning pricing method

The Heston model supports the Vanilla instrument when using an AssetReinforcementLearning pricing method.