NumericalIntegration

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

Description

Create and price a Vanilla instrument object with a Heston, Bates, or Merton model and a NumericalIntegration 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 object.

3. Use finpricer to specify a NumericalIntegration pricer object for the 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 instrument, see Choose Instruments, Models, and Pricers.

Creation

Syntax

NumericalIntegrationPricerObj = finpricer(PricerType,'Model',model,'DiscountCurve',ratecurve_obj,'SpotPrice',spotprice_value)

NumericalIntegrationPricerObj = finpricer(___,Name,Value)

Description

NumericalIntegrationPricerObj = finpricer(PricerType,'Model',model,'DiscountCurve',ratecurve_obj,'SpotPrice',spotprice_value) creates a NumericalIntegration pricer object by specifying PricerType and sets the properties for the required name-value pair arguments Model, DiscountCurve, and SpotPrice.

NumericalIntegrationPricerObj = finpricer(___,Name,Value) sets optional properties using additional name-value pairs in addition to the required arguments in the previous syntax. For example, NumericalIntegrationPricerObj = finpricer("NumericalIntegration",'Model',NIModel,'DiscountCurve',ratecurve_obj,'SpotPrice',1000,'DividendValue',100,'VolRiskPremium',0.9) creates a NumericalIntegration pricer object. You can specify multiple name-value pair arguments.

Input Arguments

PricerType — Pricer type

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

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

Data Types: char | string

Name-Value Arguments

Specify required and 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: NumericalIntegrationPricerObj = finpricer("NumericalIntegration",'Model',NIModel,'DiscountCurve',ratecurve_obj,'SpotPrice',1000,'DividendValue',100,'VolRiskPremium',0.9)

Required NumericalIntegration Name-Value Pair Arguments

Model — Model

model object

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

Data Types: object

DiscountCurve — ratecurve object for discounting cash flows

Read-only: 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 a 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 NumericalIntegration Name-Value Pair Arguments

DividendValue — Dividend yield

0 (default) | scalar numeric

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

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.

  For more information on the LittleTrap, see [1] and also the Little Trap formulation is defined by C_j and D_j, see 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

AbsTol — Absolute error tolerance for numerical integration

1e-10 (default) | numeric

Absolute error tolerance for numerical integration, specified as the comma-separated pair consisting of 'AbsTol' and a scalar numeric value.

Data Types: double

RelTol — Relative error tolerance for numerical integration

1e-6 (default) | numeric

Relative error tolerance for numerical integration, specified as the comma-separated pair consisting of 'RelTol' and a scalar numeric value.

Data Types: double

IntegrationRange — Numerical integration range used to approximate continuous integral over [0 Inf]

[1e-9 Inf] (default) | vector

Numerical integration range used to approximate the continuous integral over [0 Inf], specified as the comma-separated pair consisting of 'IntegrationRange' and a 1-by-2 vector representing [LowerLimit UpperLimit].

Data Types: double

Framework — Framework for computing option prices and sensitivities using numerical integration of models

"heston1993" (default) | string with values "heston1993" or "lewis2001" | character vector with values 'heston1993' or 'lewis2001'

Framework for computing option prices and sensitivities using the numerical integration of models, specified as the comma-separated pair consisting of 'Framework' and a scalar string or character vector with the following values:

• "heston1993" or 'heston1993' — Method used in Heston (1993)

• "lewis2001" or 'lewis2001' — Method used in Lewis (2001)

Data Types: char | string

Output Arguments

NumericalIntegrationPricerObj — NumericalIntegration pricer

NumericalIntegration object

NumericalIntegration pricer, returned as a NumericalIntegration object.

Properties

Model — Model

model object

Model, returned as a model object.

Data Types: object

DiscountCurve — ratecurve object for discounting cash flows

object

ratecurve object for discounting cash flows, returned as a ratecurve 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 numeric

Dividend yield, returned as a scalar numeric.

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

AbsTol — Absolute error tolerance for numerical integration

1e-10 (default) | numeric

Absolute error tolerance for numerical integration, returned as a scalar numeric value.

Data Types: double

RelTol — Relative error tolerance for numerical integration

1e-6 (default) | numeric

Relative error tolerance for numerical integration, returned as a scalar numeric value.

Data Types: double

IntegrationRange — Numerical integration range used to approximate continuous integral over [0 Inf]

[1e-9 Inf] (default) | vector

Numerical integration range used to approximate the continuous integral over [0 Inf], returned as a 1-by-2 vector representing [LowerLimit UpperLimit].

Data Types: double

Framework — Framework for computing option prices and sensitivities using numerical integration of models

"heston1993" (default) | string with value "heston1993" or "lewis2001"

Framework for computing option prices and sensitivities using the numerical integration of models, returned as a scalar string.

Data Types: string

Object Functions

price

Compute price for equity instrument with NumericalIntegration pricer

Examples

Use NumericalIntegration Pricer and Merton Model to Price Vanilla Instrument

This example shows the workflow to price a Vanilla instrument when you use a Merton model and a NumericalIntegration pricing method.

Create Vanilla Instrument Object

Use fininstrument to create a Vanilla instrument object.

VanillaOpt = fininstrument("Vanilla",'ExerciseDate',datetime(2020,3,15),'ExerciseStyle',"european",'Strike',105,'Name',"vanilla_option")

VanillaOpt = 
  Vanilla with properties:

       OptionType: "call"
    ExerciseStyle: "european"
     ExerciseDate: 15-Mar-2020
           Strike: 105
             Name: "vanilla_option"

Create Merton Model Object

Use finmodel to create a Merton model object.

MertonModel = finmodel("Merton",'Volatility',0.45,'MeanJ',0.02,'JumpVol',0.07,'JumpFreq',0.09)

MertonModel = 
  Merton with properties:

    Volatility: 0.4500
         MeanJ: 0.0200
       JumpVol: 0.0700
      JumpFreq: 0.0900

Create ratecurve Object

Create a flat ratecurve object using ratecurve.

myRC = ratecurve('zero',datetime(2019,9,15),datetime(2020,3,15),0.02)

myRC = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 0
                Dates: 15-Mar-2020
                Rates: 0.0200
               Settle: 15-Sep-2019
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create NumericalIntegration Pricer Object

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

outPricer = finpricer("numericalintegration",'Model',MertonModel,'DiscountCurve',myRC,'SpotPrice',100,'DividendValue',.01,'VolRiskPremium',0.9,'LittleTrap',false,'AbsTol',0.5,'RelTol',0.4,'Framework',"lewis2001")

outPricer = 
  NumericalIntegration with properties:

                Model: [1×1 finmodel.Merton]
        DiscountCurve: [1×1 ratecurve]
            SpotPrice: 100
         DividendType: "continuous"
        DividendValue: 0.0100
               AbsTol: 0.5000
               RelTol: 0.4000
     IntegrationRange: [1.0000e-09 Inf]
    CharacteristicFcn: @characteristicFcnMerton76
            Framework: "lewis2001"
       VolRiskPremium: 0.9000
           LittleTrap: 0

Price Vanilla Instrument

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

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

Price = 
10.7325

outPR = 
  priceresult with properties:

       Results: [1×6 table]
    PricerData: []

outPR.Results

ans=1×6 table
    Price     Delta      Gamma       Theta      Rho       Vega 
    ______    ______    ________    _______    ______    ______

    10.732    0.5058    0.012492    -12.969    19.815    27.954

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 (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 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 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

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

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 for 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:

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

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 Wiener 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 for 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 Wiener 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

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

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 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 prices 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).

Numerical Integration Method Under Heston (1993) Framework

Numerical integration is used to evaluate the continuous integral for the inverse Fourier transform.

The numerical integration method under the Heston (1993) framework is based on the following expressions

$$\begin{array}{l}Call(K)=S_t{e}^{-q\tau }{P}_1-K{e}^{-r\tau }{P}_2\\ Put(K)=Call(K)+K{e}^{-r\tau }-S_t{e}^{-q\tau }\\ P_j=\frac{1}{2}+\frac{1}{\pi }{\displaystyle \underset{0}{\overset{\infty }{\int }}\mathrm{Re}\left[\frac{e^{-i\varphi \ln (K)}{f}_j(\varphi )}{i\varphi }\right]d\varphi }\end{array}$$

Here:

r is the continuous risk-free rate.

q is the continuous dividend yield.

S_t is the asset price at time t.

K is the strike.

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.

f_j(ϕ) is the characteristic function for P_j(j = 1,2).

P₁ is the probability of S_t > K under the asset price measure for the model.

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

Where j = 1,2 so that f₁(ϕ) and f₂(ϕ) are the characteristic functions for probabilities P₁ and P₂, respectively.

Choose this framework by specifying the default value "Heston1993" for the Framework name-value pair argument.

Numerical Integration Method Under Lewis Framework

Numerical integration is used to evaluate the continuous integral for the inverse Fourier transform.

The numerical integration method under the Lewis (2001) framework is based on the following expressions:

$$\begin{array}{l}Call(k)=S_t{e}^{-q\tau }-\frac{\sqrt{K}{e}^{-\tau t}}{\pi }{\displaystyle \underset{0}{\overset{\infty }{\int }}\mathrm{Re}\left[K^{-iu}{f}_2\left(\varphi =\left(u-\frac{i}{2}\right)\right)\frac{1}{u^2+\frac{1}{4}}\right]}du\\ Put(K)=Call(K)=K{e}^{-\tau t}-S_t{e}^{-q\tau }\end{array}$$

Here

r is the continuous risk-free rate.

q is the continuous dividend yield.

S_t is the asset price at time t.

K is the strike.

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.

u is the characteristic function variable for integration, where $$\varphi =\left(u-\frac{i}{2}\right)$$.

f₂(ϕ) is the characteristic function for P₂.

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

Choose this framework by specifying the value "Lewis2001" for the Framework name-value pair argument.

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.

Version History

Introduced in R2020a