# optSensByLocalVolFD

Option price and sensitivities by local volatility model, using finite differences

## Description

example

[PriceSens,PriceGrid,AssetPrices,Times] = optSensByLocalVolFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,ImpliedVolData) compute option price and sensitivities by the local volatility model, using the Crank-Nicolson method.

example

[PriceSens,PriceGrid,AssetPrices,Times] = optSensByLocalVolFD(___,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.

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 and sensitivities.

OptSpec = 'Call';
[Delta,Gamma,Lambda,Theta,Price]  = optSensByLocalVolFD(Rate, AssetPrice, ...
Settle, ExerciseDates, OptSpec, Strike, ImpliedVolData, 'DividendYield',DividendYield, ...
'OutSpec',["Delta" "Gamma" "Lambda" "Theta" "Price"])
Delta = 0.5519
Gamma = 0.0091
Lambda = 4.9994
Theta = -20.9529
Price = 65.1319

## Input Arguments

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

Data Types: double

Current underlying asset price, specified as a scalar numeric.

Data Types: double

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

Data Types: double | char | datetime | string

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, datetimes, or strings. 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

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

Data Types: char | string

Option strike price value, specified as a nonnegative scalar.

Data Types: double

A 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: PriceSens = Price = optByLocalVolFD(Rate,AssetPrice,Settle, ExerciseDates,OptSpec,Strike,ImpliedVolData,'AssetGridSize',1000,'OutSpec',{'delta','gamma','vega','lambda','rho','theta','price'})

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

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.

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

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

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: double

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

Data Types: double

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

Data Types: double

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

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 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 (MATLAB) and Interpolating Gridded Data (MATLAB).

Data Types: char | string

Define outputs, specified as the comma-separated pair consisting of 'OutSpec' and an NOUT- by-1 or a 1-by-NOUT cell array of character vectors with possible values of 'price', 'delta', 'gamma', 'vega', 'lambda', 'rho', and 'theta'.

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

Data Types: cell | string

## Output Arguments

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

Grid containing prices calculated by the finite difference method, returned as a grid that is two-dimensional with size AssetGridSizeTimeGridSize. 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.

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

Times corresponding to second dimension of the 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.

### 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]:

$\begin{array}{l}{\sigma }_{loc}^{2}\left(K,\tau \right)=\frac{{\sigma }_{imp}^{2}+2\tau {\sigma }_{imp}\frac{\partial {\sigma }_{imp}}{\partial \tau }+2\left(\tau -d\right)K\tau {\sigma }_{imp}\frac{\partial {\sigma }_{imp}}{\partial K}}{{\left(1+K{d}_{1}\sqrt{\tau }\frac{\partial {\sigma }_{imp}}{\partial K}\right)}^{2}+{K}^{2}\tau {\sigma }_{imp}\left(\frac{{\partial }^{2}{\sigma }_{imp}}{\partial {K}^{2}}-{d}_{1}\sqrt{\tau }{\left(\frac{\partial {\sigma }_{imp}}{\partial K}\right)}^{2}\right)}\\ {d}_{1}=\frac{\mathrm{ln}\left({S}_{0}/K\right)+\left(\left(\tau -d\right)+{\sigma }_{imp}^{2}/2\right)\tau }{{\sigma }_{imp}\sqrt{\tau }}\end{array}$

## 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.