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.

[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

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

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

Data Types: double | char | datetime | string

ExerciseDates — Option exercise dates

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

Option exercise dates, specified as a serial date number, a date character vector, a datetime array, or a string 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 serial date numbers, date character vectors, datetime arrays, or string arrays. 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.

Data Types: double | char | cell | datetime | string

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

Ex-dividend dates, specified as the comma-separated pair consisting of 'ExDividendDates' and a NDIV-by-1 vector.

Data Types: double | char | string | datetime

'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.