# optSensByHestonFD

Option price and sensitivities by Heston model using finite differences

## Syntax

## Description

`[`

computes a vanilla European or American option price and sensitivities by the Heston model,
using the alternating direction implicit (ADI) method.`PriceSens`

,`PriceGrid`

,`AssetPrices`

,`Variances`

,`Times`

] = optByHestonFD(`Rate`

,`AssetPrice`

,`Settle`

,`ExerciseDates`

,`OptSpec`

,`Strike`

,`V0`

,`ThetaV`

,`Kappa`

,`SigmaV`

,`RhoSV`

)

**Note**

Alternatively, you can use the `Vanilla`

object to calculate
price or sensitivities for vanilla options. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

`[`

specifies options using one or more name-value pair arguments in addition to the input
arguments in the previous syntax.`PriceSens`

,`PriceGrid`

,`AssetPrices`

,`Variances`

,`Times`

] = optByHestonFD(___,`Name,Value`

)

## Examples

### Compute an American Option Price and Sensitivities Using the Heston Model

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 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

`Rate`

— Continuously compounded risk-free interest rate

scalar decimal

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

**Data Types: **`double`

`AssetPrice`

— Current underlying asset price

scalar numeric

Current underlying asset price, specified as a scalar numeric.

**Data Types: **`double`

`Settle`

— Option settlement date

datetime scalar | string scalar | date character vector

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

To support existing code, `optSensByHestonFD`

also
accepts serial date numbers as inputs, but they are not recommended.

`ExerciseDates`

— Option exercise dates

datetime array | string array | date character vector

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, `optSensByHestonFD`

also
accepts serial date numbers as inputs, but they are not recommended.

`OptSpec`

— Definition of option

cell array of character vector with values `'call'`

or
`'put'`

| string array with values `"call"`

or
`"put"`

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`

`Strike`

— Option strike price value

scalar numeric

Option strike price value, specified as a scalar numeric.

**Data Types: **`double`

`V0`

— Initial variance of underlying asset

scalar numeric

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

**Data Types: **`double`

`ThetaV`

— Long-term variance of underlying asset

scalar numeric

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

**Data Types: **`double`

`Kappa`

— Mean revision speed for the variance of underlying asset

scalar numeric

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

**Data Types: **`double`

`SigmaV`

— Volatility of the variance of underlying asset

scalar numeric

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

**Data Types: **`double`

`RhoSV`

— Correlation between Weiner processes for underlying asset and its variance

scalar numeric

Correlation between the Weiner 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: **```
[PriceSens,PriceGrid,AssetPrices,Variances,Times] =
optSensByHestonFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV,'Basis',7)
```

`Basis`

— Day-count basis of instrument

`0`

(default) | numeric values: `0`

,`1`

, `2`

,
`3`

, `4`

, `6`

,
`7`

, `8`

, `9`

,
`10`

, `11`

, `12`

,
`13`

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`

`DividendYield`

— Continuously compounded underlying asset yield

`0`

(default) | scalar numeric

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`

`DividendAmounts`

— Cash dividend amounts

`[ ]`

(default) | vector

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`

`ExDividendDates`

— Ex-dividend dates

`[]`

(default) | datetime array | string array | date character vector

Ex-dividend dates, specified as the comma-separated pair consisting of
`'ExDividendDates'`

and a
`NDIV`

-by-`1`

vector using a datetime array,
string array, or date character vectors.

To support existing code, `optSensByHestonFD`

also
accepts serial date numbers as inputs, but they are not recommended.

`AssetPriceMax`

— Maximum price for price grid boundary

if unspecified, value is calculated based on asset price
distribution at maturity (default) | positive scalar

Maximum price for price grid boundary, specified as the comma-separated pair
consisting of `'AssetPriceMax'`

and a positive scalar.

**Data Types: **`single`

| `double`

`VarianceMax`

— Maximum variance to use for variance grid boundary

`1.0`

(default) | scalar numeric

Maximum variance to use for variance grid boundary, specified as the
comma-separated pair consisting of `'VarianceMax'`

as a scalar
numeric.

**Data Types: **`double`

`AssetGridSize`

— Size of asset grid for finite difference grid

`400`

(default) | scalar numeric

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`

`VarianceGridSize`

— Number of nodes for variance grid for finite difference grid

`200`

(default) | scalar numeric

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`

`TimeGridSize`

— Number of nodes of time grid for finite difference grid

`100`

(default) | positive numeric scalar

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`

`AmericanOpt`

— Option type

`0`

(European) (default) | scalar with values `[0,1]`

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`

`OutSpec`

— Define outputs

`["price"]`

(default) | cell array of character vectors with values `'price'`

,
`'delta'`

, `'gamma'`

, `'vega'`

,
`'rho'`

, `'theta'`

, and
`'vegalt'`

| string array with values `"price"`

, `"delta"`

,
`"gamma"`

, `"vega"`

, `"rho"`

,
`"theta"`

, and `"vegalt"`

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

`PriceSens`

— Option price and sensitivities

scalar numeric

Option price and sensitivities, returned as a scalar numeric.
`OutSpec`

determines the types and order of the outputs.

`PriceGrid`

— Grid containing prices calculated by the finite difference method

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`

.

`AssetPrices`

— Prices of the asset

vector

Prices of the asset corresponding to the first dimension of
`PriceGrid`

, returned as a vector.

`Variances`

— Variances

vector

Variances corresponding to the second dimension of `PriceGrid`

,
returned as a vector.

`Times`

— Times

vector

Times corresponding to the third dimension of `PriceGrid`

,
returned as a vector.

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

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

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:

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

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*

*v*_{0} is the initial variance
of the asset price at *t* = 0 for
(*v*_{0} > 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 Weiner 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 R2018b**

### R2022b: Serial date numbers not recommended

Although `optSensByHestonFD`

supports serial date numbers,
`datetime`

values are recommended instead. The
`datetime`

data type provides flexible date and time
formats, storage out to nanosecond precision, and properties to account for time
zones and daylight saving time.

To convert serial date numbers or text to `datetime`

values, use the `datetime`

function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum"); y = year(t)

y = 2021

There are no plans to remove support for serial date number inputs.

## See Also

`optstockbyfd`

| `optstocksensbyfd`

| `optByHestonFD`

| `optByLocalVolFD`

| `optSensByLocalVolFD`

| `optByBatesFD`

| `optSensByBatesFD`

| `optByMertonFD`

| `optSensByMertonFD`

| `Vanilla`

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