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optByHestonFD

Option price by Heston model using finite differences

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Syntax

[Price,PriceGrid,AssetPrices,Variances,Times] = optByHestonFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV)
[Price,PriceGrid,AssetPrices,Variances,Times] = optByHestonFD(___,Name,Value)

Description

[Price,PriceGrid,AssetPrices,Variances,Times] = optByHestonFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV) computes a vanilla European or American option price by the Heston model, using the alternating direction implicit (ADI) method.

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.

example

[Price,PriceGrid,AssetPrices,Variances,Times] = optByHestonFD(___,Name,Value) specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax.

example

Examples

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Open Live Script

Define the option variables and Heston model parameters. 

AssetPrice = 10;
Strike = 10;
Rate = 0.1;
Settle = datetime(2017,1,1);
ExerciseDates = datetime(2017,4,2);

V0 = 0.0625;
ThetaV = 0.16;
Kappa = 5.0;
SigmaV = 0.9;
RhoSV = 0.1;

Compute the American put option price.

OptSpec = 'Put';
Price = optByHestonFD(Rate, AssetPrice, Settle, ...
ExerciseDates, OptSpec, Strike, V0, ThetaV, Kappa, SigmaV, RhoSV, 'AmericanOpt', 1)
Price = 
0.5188

Input Arguments

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

Data Types: double

Current underlying asset price, specified as numeric value using a scalar numeric.

Data Types: double

Option settlement date, specified as a scalar datetime, string, or date character vector.

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

Option exercise dates, specified as a datetime array, string array, or date character vectors:

  • For a European option, there is only one ExerciseDates value and this is the option expiry date.

  • For an American option, use a 1-by-2 vector of exercise date boundaries. The option can be exercised on any tree date between or including the pair of dates on that row. If only one non-NaN date is listed, the option can be exercised between the Settle date and the single listed ExerciseDate.

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

Definition of the option, specified as a scalar using a cell array of character vectors or string arrays with values 'call' or 'put'.

Data Types: cell | string

Option strike price value, specified as a scalar numeric.

Data Types: double

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

Data Types: double

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

Data Types: double

Mean revision speed for the variance of the underlying asset, specified as a scalar numeric.

Data Types: double

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

Data Types: double

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

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,PriceGrid,AssetPrices,Variances,Times] = optByHestonD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV,'Basis',7)

Day-count basis 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

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

Note

If you enter a value for DividendYield, then set DividendAmounts and ExDividendDates = [ ] or do not enter them. If you enter values for DividendAmounts and ExDividendDates, then set DividendYield = 0.

Data Types: double

Cash dividend amounts, specified as the comma-separated pair consisting of 'DividendAmounts' and a NDIV-by-1 vector.

Note

Each dividend amount must have a corresponding ex-dividend date. If you enter values for DividendAmounts and ExDividendDates, then set DividendYield = 0.

Data Types: double

Ex-dividend dates, specified as the comma-separated pair consisting of 'ExDividendDates' and an NDIV-by-1 vector using a datetime array, string array, or date character vectors.

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

Maximum price for the price grid boundary, specified as the comma-separated pair consisting of 'AssetPriceMax' and a positive scalar.

Data Types: single | double

Maximum variance to use for the variance grid boundary, specified as the comma-separated pair consisting of 'VarianceMax' as a scalar numeric.

Data Types: double

Size of the asset grid for finite difference grid, specified as the comma-separated pair consisting of 'AssetGridSize' and a scalar numeric.

Data Types: double

Number of nodes for the variance grid for finite difference grid, specified as the comma-separated pair consisting of 'VarianceGridSize' and a scalar numeric.

Data Types: double

Number of nodes of the time grid for finite difference grid, specified as the comma-separated pair consisting of 'TimeGridSize' and a positive numeric scalar.

Data Types: double

Option type, specified as the comma-separated pair consisting of 'AmericanOpt' and a scalar flag with one of these values:

  • 0 — European

  • 1 — American

Data Types: double

Output Arguments

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Option price, returned as a scalar numeric.

Grid containing prices calculated by the finite difference method, returned as a three-dimensional grid with size AssetGridSize ⨉ VarianceGridSize ⨉ TimeGridSize. The depth is not necessarily equal to the TimeGridSize, because exercise and ex-dividend dates are added to the time grid. PriceGrid(:, :, end) contains the price for t = 0.

Prices of the asset corresponding to the first dimension of PriceGrid, returned as a vector.

Variances corresponding to the second dimension of PriceGrid, returned as a vector.

Times corresponding to the third dimension of PriceGrid, returned as a vector.

More About

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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: 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 allows modeling the implied volatility smiles observed in the market.

The stochastic differential equation is:

dSt=(r−q)Stdt+vtStdWtdvt=κ(θ−vt)dt+σvvtdWtvE[dWtdWtv]=pdt

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

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).

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

p is the correlation between the Wiener processes Wt and Wvt 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 R2018b

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See Also

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