optSensByHestonFD

Option price and sensitivities by Heston model using finite differences

Syntax

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

[PriceSens,PriceGrid,AssetPrices,Variances,Times] = optByHestonFD(___,Name,Value)

Description

[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.

[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

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 = '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

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

serial date number | date character vector | datetime array | string array

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

ExerciseDates — Option exercise dates

serial date number | date character vector | string array | datetime array

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

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 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)

'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) | serial date number | date character vector | string array | datetime array

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

'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: $$\max (St-K,0)$$

• 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:

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

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 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.