Bachelier

Create Bachelier model object for Vanilla, Spread, or Binary instrument

Since R2021a

Description

Create and price a Vanilla, Spread, or Binary instrument object with a Bachelier model using this workflow:

1. Use fininstrument to create a Vanilla, Spread, or Binary instrument object.

2. Use finmodel to specify a Bachelier model object for the Vanilla, Spread, or Binary instrument object.

3. Use finpricer to specify an AssetMonteCarlo pricing method for the Vanilla, Spread, or Binary 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, Spread, or Binary instrument, see Choose Instruments, Models, and Pricers.

Creation

Syntax

BachelierModelObj = finmodel(ModelType,'Volatility',volatility_value)

BachelierModelObj = finmodel(ModelType,Name,Value)

Description

BachelierModelObj = finmodel(ModelType,'Volatility',volatility_value) creates a Bachelier model object by specifying ModelType and the required name-value pair argument Volatility to set properties using name-value pair arguments. For example, BachelierModelObj = finmodel("Bachelier",'Volatility',0.063) creates a Bachelier model object.

BachelierModelObj = finmodel(ModelType,Name,Value) sets optional properties using an additional name-value pair argument in addition to the required arguments in the previous syntax. For example, BachelierModelObj = finmodel("Bachelier",'Volatility',0.063,'Correlation',Corr) creates a Bachelier model object with a specified Correlation value.

Input Arguments

ModelType — Model type

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

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

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: BachelierModelObj = finmodel("Bachelier",'Volatility',0.063,'Correlation',Corr)

Volatility — Volatility value

nonnegative numeric

Volatility value, specified as the comma-separated pair consisting of 'Volatility' and a scalar nonnegative numeric.

Data Types: double

Correlation — Correlation for underlying assets

1 (default) | positive semidefinite matrix

Correlation for the underlying assets, specified as the comma-separated pair consisting of 'Correlation' and a positive semidefinite matrix. For more information on creating a positive semidefinite matrix, see nearcorr.

Data Types: double

Properties

Volatility — Volatility value

nonnegative numeric

Volatility value, returned as a scalar nonnegative numeric.

Data Types: double

Correlation — Correlation for underlying assets

1 (default) | positive semidefinite matrix

Correlation for the underlying assets, returned as a matrix.

Data Types: double

Examples

Use Bachelier Model and AssetMonteCarlo Pricer to Price American Option for Vanilla Instrument

This example shows the workflow to price an American option for a Vanilla instrument when you use a Bachelier model and an AssetMonteCarlo pricing method.

Create Vanilla Instrument Object

Use fininstrument to create a Vanilla instrument object.

VanillaOpt = fininstrument("Vanilla",'Strike',105,'ExerciseDate',datetime(2022,9,15),'OptionType',"call",'ExerciseStyle',"american",'Name',"vanilla_option")

VanillaOpt = 
  Vanilla with properties:

       OptionType: "call"
    ExerciseStyle: "american"
     ExerciseDate: 15-Sep-2022
           Strike: 105
             Name: "vanilla_option"

Create Bachelier Model Object

Use finmodel to create a Bachelier model object.

BachelierModel = finmodel("Bachelier","Volatility",0.2)

BachelierModel = 
  Bachelier with properties:

     Volatility: 0.2000
    Correlation: 1

Create ratecurve Object

Create a flat ratecurve object using ratecurve.

Settle = datetime(2018,9,15);
Maturity = datetime(2023,9,15);
Rate = 0.035;
myRC = ratecurve('zero',Settle,Maturity,Rate,'Basis',12)

myRC = 
  ratecurve with properties:

                 Type: "zero"
          Compounding: -1
                Basis: 12
                Dates: 15-Sep-2023
                Rates: 0.0350
               Settle: 15-Sep-2018
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create AssetMonteCarlo Pricer Object

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

outPricer = finpricer("AssetMonteCarlo",'DiscountCurve',myRC,"Model",BachelierModel,'SpotPrice',150,'simulationDates',datetime(2022,9,15))

outPricer = 
  BachelierMonteCarlo with properties:

           DiscountCurve: [1x1 ratecurve]
               SpotPrice: 150
         SimulationDates: 15-Sep-2022
               NumTrials: 1000
           RandomNumbers: []
                   Model: [1x1 finmodel.Bachelier]
            DividendType: "continuous"
           DividendValue: 0
        MonteCarloMethod: "standard"
    BrownianMotionMethod: "standard"

Price Vanilla Instrument

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

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

Price = 57.3776

outPR = 
  priceresult with properties:

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

outPR.Results

ans=1×7 table
    Price      Delta       Gamma       Lambda     Rho       Theta        Vega    
    ______    _______    __________    ______    ______    _______    ___________

    57.378    0.99107    -1.579e-14    2.5909    291.94    -2.5576    -2.1316e-10

More About

Bachelier Model

The Bachelier model assumes that the underlying asset follows a random walk with constant drift and volatility, and that the distribution of asset returns is a normal distribution.

The Bachelier model assumes that the underlying asset follows a normal distribution of returns, which is different from the Black-Scholes model that assumes log-normal distribution. The Bachelier model is commonly used for options on assets with low volatility or short-term options where the assumption of log-normality may not hold.

$$d{S}_t=\mu S_{t}dt+{\sigma }_{B}d{W}_t$$

where

S_t is the underlying asset.

μ is the drift.

W_t is the Brownian motion.

σ_B is the Bachelier volatility constant where σ_B > 0.

Version History

Introduced in R2021a