VannaVolga

Create VannaVolga pricer object for Vanilla, Barrier, DoubleBarrier, Touch, or DoubleTouch instrument using BlackScholes model

Since R2020b

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Description

Create and price a Vanilla, Barrier, DoubleBarrier, Touch, or DoubleTouch instrument object with a BlackScholes model and a VannaVolga pricing method using this workflow:

Use fininstrument to create a Vanilla, Barrier, DoubleBarrier, Touch, or DoubleTouch instrument object.
Use finmodel to specify the BlackScholes model for the Vanilla, Barrier, DoubleBarrier, Touch, or DoubleTouch instrument object.
Use finpricer to specify the VannaVolga pricer object for the Vanilla, Barrier, DoubleBarrier, Touch, or DoubleTouch 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 instruments, models, and pricing methods for a Vanilla, Barrier, DoubleBarrier, Touch, or DoubleTouch instrument, see Choose Instruments, Models, and Pricers.

Creation

Syntax

VannaVolgaPricerObj = finpricer(PricerType,'DiscountCurve',ratecurve_obj,'Model',model,'SpotPrice',spot_price,'VolatilityRR',volatilityrr_value,'VolatilityBF',volatilitybf_value)

Description

VannaVolgaPricerObj = finpricer(PricerType,'DiscountCurve',ratecurve_obj,'Model',model,'SpotPrice',spot_price,'VolatilityRR',volatilityrr_value,'VolatilityBF',volatilitybf_value) creates a VannaVolga pricer object by specifying PricerType and sets properties using the required name-value pair arguments DiscountCurve, Model, SpotPrice, VolatilityRR, and VolatilityBF. For example, VannaVolgaPricerObj = finpricer("VannaVolga",'DiscountCurve',ratecurve_obj,'Model',BSModel,'SpotPrice',Spot,'VolatilityRR',VolRR,'VolatilityBF',VolBF) creates a VannaVolga pricer object.

example

VannaVolgaPricerObj = finpricer(___,Name,Value) sets optional properties using additional name-value pairs in addition to the required arguments in the previous syntax. For example, VannaVolgaPricerObj = finpricer("VannaVolga",'DiscountCurve',ratecurve_obj,'Model',BSModel,'SpotPrice',Spot,'VolatilityRR',VolRR,'VolatilityBF',VolBF,'DividendValue',0.0210) creates a VannaVolga pricer object. You can specify multiple name-value pair arguments.

example

Input Arguments

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`PricerType` — Pricer type
string with value `"VannaVolga"` | character vector with value `'VannaVolga'`

Pricer type, specified as a string with the value "VannaVolga" or a character vector with the value 'VannaVolga'.

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: VannaVolgaPricerObj = finpricer("VannaVolga",'DiscountCurve',ratecurve_obj,'Model',BSModel,'SpotPrice',Spot,'VolatilityRR',VolRR,'VolatilityBF',VolBF,'DividendValue',0.0210)

`DiscountCurve` — `ratecurve` object for discounting cash flows
`ratecurve` object

ratecurve object for discounting cash flows, specified as the comma-separated pair consisting of 'DiscountCurve' and the name of a previously created ratecurve object.

Data Types: object

`Model` — Model object
`BlackScholes` model object

Model object, specified as the comma-separated pair consisting of 'Model' and the name of a previously created BlackScholes model object using finmodel.

Data Types: object

`SpotPrice` — Current price of underlying asset
numeric

Current price of the underlying asset, specified as the comma-separated pair consisting of 'SpotPrice' and a scalar numeric.

Data Types: double

`VolatilityRR` — 25-delta risk reversal (RR) volatility
numeric

25-delta risk reversal (RR) volatility, specified as the comma-separated pair consisting of 'VolatilityRR' and a scalar numeric.

Data Types: double

`VolatilityBF` — 25-delta butterfly (BF) volatility
numeric

25-delta butterfly (BF) volatility, specified as the comma-separated pair consisting of 'VolatilityBF' and a scalar numeric.

Data Types: double

Optional VannaVolga Name-Value Pair Arguments

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`DividendType` — Dividend type
`"continuous"` (default) | string with value of `"continuous"` | character vector with value of `'continuous'`

Dividend type, specified as the comma-separated pair consisting of 'DividendType' and a string or character vector for a continuous dividend yield.

Data Types: char | string

`DividendValue` — Continuous dividend yield
`0` (default) | scalar numeric

Continuous dividend yield, specified as the comma-separated pair consisting of 'DividendValue' and a scalar numeric.

Note

When pricing currency (FX) options, specify the optional input argument 'DividendValue' as the continuously compounded risk-free interest rate in the foreign country.

Data Types: double

Properties

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`DiscountCurve` — `ratecurve` object for discounting cash flows
object

ratecurve object for discounting cash flows, returned as a ratecurve object.

Data Types: object

`Model` — Model
`BlackScholes` model object

Model, returned as a BlackScholes model object.

Data Types: object

`SpotPrice` — Current price of underlying asset
numeric

Current price of the underlying asset, returned as a scalar numeric.

Data Types: double

`VolatilityRR` — 25-delta risk reversal (RR) volatility
numeric

25-delta risk reversal (RR) volatility, returned as a scalar numeric.

Data Types: double

`VolatilityBF` — 25-delta butterfly (BF) volatility
numeric

25-delta butterfly (BF) volatility, returned as a scalar numeric.

Data Types: double

`DividendType` — Dividend type
`"continuous"` (default) | string with value of `"continuous"`

This property is read-only.

Dividend type, returned as a string.

Data Types: string

`DividendValue` — Continuous dividend yield
`0` (default) | scalar numeric

Continuous dividend yield, returned as a scalar numeric.

Data Types: double

Object Functions

price Compute price for equity instrument with VannaVolga pricer

Examples

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Use `VannaVolga` Pricer and `BlackScholes` Model to Price `DoubleBarrier` Instrument

Open Live Script

This example shows the workflow to price a DoubleBarrier instrument when you use a BlackScholes model and a VannaVolga pricing method.

Create DoubleBarrier Instrument Object

Use fininstrument to create a DoubleBarrier instrument object.

DoubleBarrierOpt = fininstrument("DoubleBarrier",'Strike',100,'ExerciseDate',datetime(2020,8,15),'OptionType',"call",'ExerciseStyle',"European",'BarrierType',"DKO",'BarrierValue',[110 80],'Name',"doublebarrier_option")

DoubleBarrierOpt = 
  DoubleBarrier with properties:

       OptionType: "call"
           Strike: 100
     BarrierValue: [110 80]
    ExerciseStyle: "european"
     ExerciseDate: 15-Aug-2020
      BarrierType: "dko"
           Rebate: [0 0]
             Name: "doublebarrier_option"

Create BlackScholes Model Object

Use finmodel to create a BlackScholes model object.

BlackScholesModel = finmodel("BlackScholes","Volatility",0.02)

BlackScholesModel = 
  BlackScholes with properties:

     Volatility: 0.0200
    Correlation: 1

Create ratecurve Object

Create a flat ratecurve object using ratecurve.

Settle = datetime(2019,9,15);
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-2019
         InterpMethod: "linear"
    ShortExtrapMethod: "next"
     LongExtrapMethod: "previous"

Create VannaVolga Pricer Object

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

VolRR = -0.0045;
VolBF = 0.0037;
RateF = 0.0210;
outPricer = finpricer("VannaVolga","DiscountCurve",myRC,"Model",BlackScholesModel,'SpotPrice',100,'DividendValue',RateF,'VolatilityRR',VolRR,'VolatilityBF',VolBF)

outPricer = 
  VannaVolga with properties:

    DiscountCurve: [1x1 ratecurve]
            Model: [1x1 finmodel.BlackScholes]
        SpotPrice: 100
     DividendType: "continuous"
    DividendValue: 0.0210
     VolatilityRR: -0.0045
     VolatilityBF: 0.0037

Price DoubleBarrier Instrument

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

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

Price = 
1.6450

outPR = 
  priceresult with properties:

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

outPR.Results

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

    1.645    0.82818    75.662    50.346    14.697    -1.3145    74.666

More About

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Vanna-Volga Method

The Vanna Volga method is an empirical procedure based on adding an analytically derived correction to the Black-Scholes price of the instrument.

The Vanna-Volga method consists of adjusting the Black-Scholes theoretical value (BSTV) by the cost of a portfolio which hedges three main risks associated to the volatility of the option: the Vega, the Vanna and the Volga.

The general formulation of the Vanna-Volga method suggests that the Vega, Vanna, and Volga values can be replicated by the weighted sum of at-the-money (ATM), risk-reversal (RR), and butterfly (BF) strategies.

$X_{i} = ω_{A T M} A T M_{i} + ω_{R R} R R_{i} + ω_{B F} B F_{i} (i = v e g a, v a n n a, v o l g a)$

Here, the weights are obtained by solving the system x = Aω

$\begin{array}{l} A T M_{v e g a} R R_{v e g a} B F_{v e g a} ω_{A T M} X_{v e g a} \\ A = A T M_{v a n n a} R R_{v a n n a} B F_{v a n n a,} ω = ω_{R R,} X = X_{v a n n a} \\ A T M_{v o l g a} R R_{v o l g a} B F_{v o l g a} ω_{B F} X_{v o l g a} \end{array}$

Given this replication, the Vanna-Volga method adjusts the Black-Scholes price of the option by the smile cost of the above weighted sum:

$\begin{array}{l} X^{V V} = X^{B S} + ω_{A T M} (A T M^{m k t} - A T M^{B S}) + ω_{R R} (R R^{m k t} - R R^{B S}) + ω_{B F} (B F^{m k t} - B F^{B S}) \\ = X^{B S} + x^{T} (^{A^{T}) - 1} I \\ = X^{B S} + X_{v e g a} Ω_{v e g a} + X_{v a n n a} Ω_{v a n n a} + X_{v o l g a} Ω_{v o l g a} \\ A T M^{m k t} - A T M^{B S} Ω_{v e g a} \\ where I = R R^{m k t} - R R^{B S}, Ω_{v a n n a} = (^{A^{T}) - 1} I \\ B F^{m k t} - B F^{m k t} Ω_{v o l g a} \end{array}$

The resulting correction or overhedge turns out to be too large. So, the option value is modified as follows:

$\begin{array}{l} X^{V V} = X^{B S} + X_{v e g a} Ω_{v e g a} + X_{v a n n a} Ω_{v a n n a} + X_{v o l g a} Ω_{v o l g a} \\ where, ρ_{v e g a} = \frac{1}{2} + \frac{1}{2} ρ, and ρ_{v o l g a} = \frac{1}{2} + \frac{1}{2} ρ . \end{array}$

ρ is the risk-neutral probability of not hitting the barrier.

References

[1] Bossens, Frédéric, Grégory Rayée, Nikos S. Skantzos, and Griselda Deelstra. "Vanna-Volga Methods Applied to FX Derivatives: From Theory to Market Practice." International Journal of Theoretical and Applied Finance. 13, no. 08 (December 2010): 1293–1324.

[2] Castagna, Antonio, and Fabio Mercurio. "The Vanna-Volga Method for Implied Volatilities." Risk. 20 (January 2007): 106–111.

[3] Castagna, Antonio, and Fabio Mercurio. "Consistent Pricing of FX Options." Working Papers Series, Banca IMI, 2006.

[4] Fisher, Travis. "Variations on the Vanna-Volga Adjustment." Bloomberg Research Paper, 2007.

[5] Wystup, Uwe. FX Options and Structured Products. Hoboken, NJ: Wiley Finance, 2006.

Version History

Introduced in R2020b

VannaVolga

Description

Creation

Syntax

Description

Input Arguments

`PricerType` — Pricer type
string with value `"VannaVolga"` | character vector with value `'VannaVolga'`

`DiscountCurve` — `ratecurve` object for discounting cash flows
`ratecurve` object

`Model` — Model object
`BlackScholes` model object

`SpotPrice` — Current price of underlying asset
numeric

`VolatilityRR` — 25-delta risk reversal (RR) volatility
numeric

`VolatilityBF` — 25-delta butterfly (BF) volatility
numeric

`DividendType` — Dividend type
`"continuous"` (default) | string with value of `"continuous"` | character vector with value of `'continuous'`

`DividendValue` — Continuous dividend yield
`0` (default) | scalar numeric

Properties

`DiscountCurve` — `ratecurve` object for discounting cash flows
object

`Model` — Model
`BlackScholes` model object

`SpotPrice` — Current price of underlying asset
numeric

`VolatilityRR` — 25-delta risk reversal (RR) volatility
numeric

`VolatilityBF` — 25-delta butterfly (BF) volatility
numeric

`DividendType` — Dividend type
`"continuous"` (default) | string with value of `"continuous"`

`DividendValue` — Continuous dividend yield
`0` (default) | scalar numeric

Object Functions

Examples

Use `VannaVolga` Pricer and `BlackScholes` Model to Price `DoubleBarrier` Instrument

More About

Vanna-Volga Method

References

Version History

See Also

Functions

Topics

VannaVolga

Description

Creation

Syntax

Description

Input Arguments

PricerType — Pricer type string with value "VannaVolga" | character vector with value 'VannaVolga'

DiscountCurve — ratecurve object for discounting cash flows ratecurve object

Model — Model object BlackScholes model object

SpotPrice — Current price of underlying asset numeric

VolatilityRR — 25-delta risk reversal (RR) volatility numeric

VolatilityBF — 25-delta butterfly (BF) volatility numeric

DividendType — Dividend type "continuous" (default) | string with value of "continuous" | character vector with value of 'continuous'

DividendValue — Continuous dividend yield 0 (default) | scalar numeric

Properties

DiscountCurve — ratecurve object for discounting cash flows object

Model — Model BlackScholes model object

SpotPrice — Current price of underlying asset numeric

VolatilityRR — 25-delta risk reversal (RR) volatility numeric

VolatilityBF — 25-delta butterfly (BF) volatility numeric

DividendType — Dividend type "continuous" (default) | string with value of "continuous"

DividendValue — Continuous dividend yield 0 (default) | scalar numeric

Object Functions

Examples

Use VannaVolga Pricer and BlackScholes Model to Price DoubleBarrier Instrument

More About

Vanna-Volga Method

References

Version History

See Also

Functions

Topics

`PricerType` — Pricer type
string with value `"VannaVolga"` | character vector with value `'VannaVolga'`

`DiscountCurve` — `ratecurve` object for discounting cash flows
`ratecurve` object

`Model` — Model object
`BlackScholes` model object

`SpotPrice` — Current price of underlying asset
numeric

`VolatilityRR` — 25-delta risk reversal (RR) volatility
numeric

`VolatilityBF` — 25-delta butterfly (BF) volatility
numeric

`DividendType` — Dividend type
`"continuous"` (default) | string with value of `"continuous"` | character vector with value of `'continuous'`

`DividendValue` — Continuous dividend yield
`0` (default) | scalar numeric

`DiscountCurve` — `ratecurve` object for discounting cash flows
object

`Model` — Model
`BlackScholes` model object

`SpotPrice` — Current price of underlying asset
numeric

`VolatilityRR` — 25-delta risk reversal (RR) volatility
numeric

`VolatilityBF` — 25-delta butterfly (BF) volatility
numeric

`DividendType` — Dividend type
`"continuous"` (default) | string with value of `"continuous"`

`DividendValue` — Continuous dividend yield
`0` (default) | scalar numeric

Use `VannaVolga` Pricer and `BlackScholes` Model to Price `DoubleBarrier` Instrument