# optSensByHestonFD

Option price and sensitivities by Heston model using finite differences

## Description

example

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

example

[PriceSens,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.

## Examples

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Define the option variables and Heston model parameters.

AssetPrice = 10;
Strike = 10;
Rate = 0.1;
Settle = '01-Jan-2017';
ExerciseDates = '02-Apr-2017';

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

Compute the American put option price and sensitivities.

OptSpec = 'Put';
[Price,Delta,Gamma,Rho,Theta,Vega,VegaLT] = optSensByHestonFD(Rate, AssetPrice, Settle, ExerciseDates, ...
OptSpec, Strike, V0, ThetaV, Kappa, SigmaV, RhoSV, 'AmericanOpt', 1, ...
'OutSpec', ["Price" "Delta" "Gamma" "Rho" "Theta" "Vega" "VegaLT"])
Price = 0.5188
Delta = -0.4472
Gamma = 0.2822
Rho = -0.9234
Theta = -1.1614
Vega = 0.8998
VegaLT = 1.0921

## 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 a scalar numeric.

Data Types: double

Option settlement date, specified as a scalar using serial date numbers, date character vectors, datetime arrays, or string arrays.

Data Types: double | char | datetime | string

Option exercise dates, specified as a serial date number, date character vector, string array, or datetime array:

• 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.

Data Types: double | char | string | datetime

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 Weiner processes for the underlying asset and its variance, specified as a scalar numeric.

Data Types: double

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: [PriceSens,PriceGrid,AssetPrices,Variances,Times] = optSensByHestonFD(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

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 a NDIV-by-1 vector of serial date numbers, character vectors, string arrays, or datetime arrays.

Data Types: double | char | string | datetime

Maximum price for 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 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

Define outputs, specified as the comma-separated pair consisting of 'OutSpec' and an NOUT- by-1 or a 1-by-NOUT string array or cell array of character vectors with the supported values.

### Note

'vega' is the sensitivity with respect to the initial volatility sqrt(V0). In contrast, 'vegalt' is the sensitivity with respect to the long-term volatility sqrt(ThetaV).

Example: OutSpec = {'price','delta','gamma','vega','rho','theta','vegalt'}

Data Types: string | cell

## Output Arguments

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Option price and sensitivities, returned as a scalar numeric. OutSpec determines the types and order of the outputs.

Grid containing prices calculated by the finite difference method, returned as a three-dimensional grid with size AssetGridSizeVarianceGridSizeTimeGridSize. 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.

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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: $\mathrm{max}\left(St-K,0\right)$

• For a put: $\mathrm{max}\left(K-St,0\right)$

where:

St is the price of the underlying asset at time t.

K is the strike price.

### 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:

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

where

r is the continuous risk-free rate.

q is the continuous dividend yield.

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 Weiner 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.