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 object.

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

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

Name-Value Arguments

Specify required 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: 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

Output Arguments

DupireObj — Dupire model

Dupire object

Dupire model, returned as a Dupire object.

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 FiniteDifference 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: [100×3 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: [1×1 ratecurve]
             Model: [1×1 finmodel.Dupire]
         SpotPrice: 100
    GridProperties: [1×1 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: [1×7 table]
    PricerData: [1×1 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

Dupire Model

The Dupire model assumes that the underlying asset follows a diffusion process, and that the volatility of the asset is a function of both the asset price and time.

The Dupire captures the local behavior of volatility by considering the implied volatility surface derived from option prices.

$$d{S}_t=\mu S_{t}dt+{\sigma }_{Dupire}(S_t,t)S_{t}d{W}_t$$

where

S_t is the underlying asset.

μ is the drift.

W_t is the Brownian motion.

σ_Dupire is the Dupire local volatility that varies with time and the underlying asset.

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 R2020a