Dupire

Create Dupire model object for local volatility for Vanilla instrument

Description

Create and price a Vanilla instrument object with a Dupire model using this workflow:

1. Use fininstrument to create a Vanilla instrument object.

2. Use finmodel to specify a Dupire model object for the Vanilla instrument.

3. Use finpricer to specify a FiniteDifference pricing method for the Vanilla instrument.

For more information on this workflow, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

For more information on the available pricing methods for a Vanilla instrument, see Choose Instruments, Models, and Pricers.

Creation

Syntax

DupireObj = finmodel(ModelType,'ImpliedVolData',impliedvoldata_value)

Description

DupireObj = finmodel(ModelType,'ImpliedVolData',impliedvoldata_value) creates a Dupire model object by specifying ModelType and the required name-value pair argument ImpliedVolData to set properties using name-value pair arguments. For example, DupireObj = finmodel("Dupire",'ImpliedVolData',voldata_table) creates a Dupire model object.

Input Arguments

ModelType — Model type

string with value "Dupire" | character vector with value 'Dupire'

Model type, specified as the string with the value "Dupire" or the character vector with the value 'Dupire'.

Data Types: char | string

Dupire Name-Value Pair Arguments

Specify required 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: DupireObj = finmodel("Dupire",'ImpliedVolData',voldata_table)

'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 the comma-separated pair consisting of 'ImpliedVolData' and an NVOL-by-3 table.

Data Types: table

Properties

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

table

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

Data Types: table

Examples

Use Dupire Model and Finite Difference Pricer to Price Vanilla Instrument

This example shows the workflow to price a Vanilla instrument when you use a Dupire model and a FiniteDifference pricing method.

Create Vanilla Instrument Object

Use fininstrument to create a Vanilla instrument object.

VanillaOpt = fininstrument("Vanilla",'ExerciseDate',datetime(2020,1,1),'Strike',105,'ExerciseStyle',"american",'Name',"vanilla_option")

VanillaOpt = 
  Vanilla with properties:

       OptionType: "call"
    ExerciseStyle: "american"
     ExerciseDate: 01-Jan-2020
           Strike: 105
             Name: "vanilla_option"

Create Dupire Model Object

Define the implied volatility surface data.

AssetPrice = 590;
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);

Use finmodel to create a Dupire model object.

DupireModel = finmodel("Dupire",'ImpliedVolData',ImpliedVolData)

DupireModel = 
  Dupire with properties:

    ImpliedVolData: [100x3 table]

Create ratecurve Object

Create a flat ratecurve object using ratecurve.

Settle = datetime(2018,1,1);
Maturity = datetime(2020,9,1);
Rate = 0.06;
myRC = ratecurve('zero',Settle,Maturity,Rate)

myRC = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 0
                Dates: 01-Sep-2020
                Rates: 0.0600
               Settle: 01-Jan-2018
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create FiniteDifference Pricer Object

Use finpricer to create a FiniteDifference pricer object and use the ratecurve object for the 'DiscountCurve' name-value pair argument.

outPricer = finpricer("FiniteDifference",'Model',DupireModel,'DiscountCurve',myRC,'SpotPrice',100,'DividendValue',0.0262,'DividendType',"continuous")

outPricer = 
  FiniteDifference with properties:

     DiscountCurve: [1x1 ratecurve]
             Model: [1x1 finmodel.Dupire]
         SpotPrice: 100
    GridProperties: [1x1 struct]
      DividendType: "continuous"
     DividendValue: 0.0262

Price Vanilla Instrument

Use price to compute the price and sensitivities for the Vanilla instrument.

[Price, outPR] = price(outPricer,VanillaOpt,["all"])

Price = 15.5930

outPR = 
  priceresult with properties:

       Results: [1x7 table]
    PricerData: [1x1 struct]

outPR.Results

ans=1×7 table
    Price      Delta       Gamma      Lambda     Theta      Rho      Vega 
    ______    _______    _________    ______    _______    ______    _____

    15.593    0.55004    0.0091484    3.5275    -3.3431    78.792    49.33

More About

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.