optByLocalVolFD

Option price by local volatility model, using finite differences

Syntax

[Price,PriceGrid,AssetPrices,Times] = optByLocalVolFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,ImpliedVolData)

[Price,PriceGrid,AssetPrices,Times] = optByLocalVolFD(___,Name,Value)

Description

[Price,PriceGrid,AssetPrices,Times] = optByLocalVolFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,ImpliedVolData) compute a Vanilla European or American option price by the local volatility model, using the Crank-Nicolson 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.

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

Examples

Price a European Option Using the Local Volatility Model

Define the option variables.

AssetPrice = 590;
Strike = 590;
Rate = 0.06;
DividendYield = 0.0262;
Settle = '01-Jan-2018';
ExerciseDates = '01-Jan-2020';

Define the implied volatility surface data.

Maturity = ["06-Mar-2018" "05-Jun-2018" "12-Sep-2018" "10-Dec-2018" "01-Jan-2019" ...
"02-Jul-2019" "01-Jan-2020" "01-Jan-2021" "01-Jan-2022" "01-Jan-2023"];
Maturity = repmat(Maturity,10,1);
Maturity = Maturity(:);

ExercisePrice = AssetPrice.*[0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.30 1.40];
ExercisePrice = repmat(ExercisePrice,1,10)';

ImpliedVol = [...
    0.190; 0.168; 0.133; 0.113; 0.102; 0.097; 0.120; 0.142; 0.169; 0.200; ...
    0.177; 0.155; 0.138; 0.125; 0.109; 0.103; 0.100; 0.114; 0.130; 0.150; ...
    0.172; 0.157; 0.144; 0.133; 0.118; 0.104; 0.100; 0.101; 0.108; 0.124; ...
    0.171; 0.159; 0.149; 0.137; 0.127; 0.113; 0.106; 0.103; 0.100; 0.110; ...
    0.171; 0.159; 0.150; 0.138; 0.128; 0.115; 0.107; 0.103; 0.099; 0.108; ...
    0.169; 0.160; 0.151; 0.142; 0.133; 0.124; 0.119; 0.113; 0.107; 0.102; ...
    0.169; 0.161; 0.153; 0.145; 0.137; 0.130; 0.126; 0.119; 0.115; 0.111; ...
    0.168; 0.161; 0.155; 0.149; 0.143; 0.137; 0.133; 0.128; 0.124; 0.123; ...
    0.168; 0.162; 0.157; 0.152; 0.148; 0.143; 0.139; 0.135; 0.130; 0.128; ...
    0.168; 0.164; 0.159; 0.154; 0.151; 0.147; 0.144; 0.140; 0.136; 0.132];

ImpliedVolData = table(Maturity, ExercisePrice, ImpliedVol);

Compute the European call option price.

OptSpec = 'Call';
Price = optByLocalVolFD(Rate, AssetPrice, ...
Settle, ExerciseDates, OptSpec, Strike, ImpliedVolData, 'DividendYield',DividendYield)

Price = 65.1319

Input Arguments

Rate — Continuously compounded risk-free interest rate

scalar numeric

Continuously compounded risk-free interest rate, specified by a scalar numeric.

Data Types: double

AssetPrice — Current underlying asset price

scalar numeric

Current underlying asset price, specified as a scalar numeric.

Data Types: double

Settle — Settlement date

datetime scalar | string scalar | date character vector

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

To support existing code, optByLocalVolFD 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 dates. The American option can be exercised on any date between or including the pair of dates. If only one non-NaN date is listed, the option can be exercised between Settle and the single listed date in ExerciseDates.

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

OptSpec — Definition of option

character vector with values 'call' or 'put' | string array with values "call" or "put"

Definition of the option, specified as a character vector or string array with values 'call' or 'put'.

Data Types: char | string

Strike — Option strike price value

nonnegative scalar

Option strike price value, specified as a nonnegative scalar.

Data Types: double

ImpliedVolData — Table of maturity dates, strike or exercise prices, and corresponding implied volatilities

table

Table of maturity dates, strike or exercise prices, and their corresponding implied volatilities,specified as a NVOL-by-3 table.

Data Types: table

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 = optByLocalVolFD(Rate,AssetPrice,Settle, ExerciseDates,OptSpec,Strike,ImpliedVolData,'AssetGridSize',1000)

Basis — Day-count basis

0 (default) | numeric values: 0,1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13

Day-count basis, specified as the comma-separated pair consisting of 'Basis' and a scalar using one of the supported values:

• 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) | 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.

For each dividend amount, there must be a corresponding ExDividendDates date. If you enter values for DividendAmounts and ExDividendDates, then set DividendYield = 0.

Note

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

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

AssetPriceMax — Maximum price for price grid boundary

if unspecified, AssetPriceMax values are calculated using asset distributions 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: double

AssetGridSize — Size of asset grid for finite difference grid

400 (default) | positive scalar

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

Data Types: double

TimeGridSize — Size of time grid for finite difference grid

100 (default) | positive scalar

Size of the time grid for a finite difference grid, specified as the comma-separated pair consisting of 'TimeGridSize' and a positive 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 positive integer scalar flag with one of these values:

• 0 — European

• 1 — American

Data Types: double

InterpMethod — Method of interpolation for estimating the implied volatility surface from ImpliedVolData

'linear' (default) | character vector with values 'linear', 'makima', 'spline', or 'tpaps' | string with values "linear", "makima", "spline", or "tpaps"

Method of interpolation for estimating the implied volatility surface from ImpliedVolData, specified as the comma-separated pair consisting of 'InterpMethod' and a character vector or string array with one of the following values:

• 'linear' — Linear interpolation

• 'makima' — Modified Akima cubic Hermite interpolation

• 'spline' — Cubic spline interpolation

• 'tpaps' — Thin-plate smoothing spline interpolation

Note

The 'tpaps' method uses the thin-plate smoothing spline functionality from Curve Fitting Toolbox™.

The 'makima' and 'spline' methods work only for gridded data. For scattered data, use the 'linear' or 'tpaps' methods.

For more information on gridded or scattered data and details on interpolation methods, see Gridded and Scattered Sample Data and Interpolating Gridded Data.

Data Types: char | string

Output Arguments

Price — Option price

scalar numeric

Option price, returned as a scalar numeric.

PriceGrid — Grid containing prices calculated by finite difference method

grid

Grid containing prices calculated by the finite difference method, returned as a grid that is two-dimensional with size AssetGridSize ⨉ TimeGridSize. The number of columns does not have to be equal to the TimeGridSize, because ExerciseDates and ExDividendDates are added to the time grid. PriceGrid(:, :, end) contains the price for t = 0.

AssetPrices — Prices of asset

vector

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

Times — Times

vector

Times corresponding to second dimension of the 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.

Local Volatility Model

A local volatility model treats volatility as a function both of the current asset level and of time.

The local volatility can be estimated by using the Dupire formula [2]:

$$\begin{array}{l}{\sigma }_{loc}^2(K,\tau )=\frac{{\sigma }_{imp}^2+2\tau {\sigma }_{imp}\frac{\partial {\sigma }_{imp}}{\partial \tau }+2(\tau -d)K\tau {\sigma }_{imp}\frac{\partial {\sigma }_{imp}}{\partial K}}{{\left(1+K{d}_1\sqrt{\tau }\frac{\partial {\sigma }_{imp}}{\partial K}\right)}^2+K^2\tau {\sigma }_{imp}\left(\frac{{\partial }^2{\sigma }_{imp}}{\partial K^2}-d_1\sqrt{\tau }{\left(\frac{\partial {\sigma }_{imp}}{\partial K}\right)}^2\right)}\\ d_1=\frac{\ln (S_0/K)+\left((\tau -d)+{\sigma }_{imp}^2/2\right)\tau }{{\sigma }_{imp}\sqrt{\tau }}\end{array}$$

References

[1] Andersen, L. B., and R. Brotherton-Ratcliffe. "The Equity Option Volatility Smile: An Implicit Finite-Difference Approach." Journal of Computational Finance. Vol. 1, Number 2, 1997, pp. 5–37.

[2] Dupire, B. "Pricing with a Smile." Risk. Vol. 7, Number 1, 1994, pp. 18–20.

Version History

Introduced in R2018b

R2022b: Serial date numbers not recommended

Although optByLocalVolFD 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.