barriersensbyfd

Calculate barrier option prices or sensitivities using finite difference method

Syntax

[PriceSens,PriceGrid,AssetPrices,Times] = barriersensbyfd(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates,BarrierSpec,Barrier)

[PriceSens,PriceGrid,AssetPrices,Times] = barriersensbyfd(___,Name,Value)

Description

[PriceSens,PriceGrid,AssetPrices,Times] = barriersensbyfd(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates,BarrierSpec,Barrier) calculates European and American barrier option prices or sensitivities of a single underlying asset using the finite difference method. barrierbyfd assumes that the barrier is continuously monitored.

[PriceSens,PriceGrid,AssetPrices,Times] = barriersensbyfd(___,Name,Value) adds optional name-value pair arguments. barriersesbyfd assumes that the barrier is continuously monitored.

Examples

Calculate Price and Sensitivities for a Barrier Down and Out Call Option Using Finite Difference Method

Create a RateSpec.

AssetPrice = 50;
Strike = 45;
Rate = 0.035;
Volatility = 0.30;
Settle = '01-Jan-2015';
Maturity = '01-Jan-2016';
Basis = 1;
 
RateSpec = intenvset('ValuationDate',Settle,'StartDates',Settle,'EndDates',...
Maturity,'Rates',Rate,'Compounding',-1,'Basis',Basis)

RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: -1
             Disc: 0.9656
            Rates: 0.0350
         EndTimes: 1
       StartTimes: 0
         EndDates: 736330
       StartDates: 735965
    ValuationDate: 735965
            Basis: 1
     EndMonthRule: 1

Create a StockSpec.

StockSpec = stockspec(Volatility,AssetPrice)

StockSpec = struct with fields:
             FinObj: 'StockSpec'
              Sigma: 0.3000
         AssetPrice: 50
       DividendType: []
    DividendAmounts: 0
    ExDividendDates: []

Calculate the Price, Delta, and Theta of a European Down and Out call option using the finite difference method.

Barrier = 40;
BarrierSpec = 'DO';
OptSpec = 'Call';
OutSpec = {'price';'delta';'theta'};
[Price, Delta, Theta] = barriersensbyfd(RateSpec,StockSpec,OptSpec,Strike,Settle,...
Maturity, BarrierSpec,Barrier,'Outspec',OutSpec)

Price = 8.5020

Delta = 0.8569

Theta = -1.8502

Input Arguments

RateSpec — Interest-rate term structure

structure

Interest-rate term structure (annualized and continuously compounded), specified by the RateSpec obtained from intenvset. For information on the interest-rate specification, see intenvset.

Data Types: struct

StockSpec — Stock specification for underlying asset

structure

Stock specification for the underlying asset. For information on the stock specification, see stockspec.

stockspec handles several types of underlying assets. For example, for physical commodities the price is StockSpec.Asset, the volatility is StockSpec.Sigma, and the convenience yield is StockSpec.DividendAmounts.

Data Types: struct

OptSpec — Definition of option

character vector with values 'call' or 'put' | string array with values 'call' or 'put'

Definition of an option as 'call' or 'put', specified as a character vector or string array with values "call" or "put".

Data Types: char | string

Strike — Option strike price value

numeric

Option strike price value, specified as a scalar numeric.

Data Types: double

Settle — Settlement or trade date

serial date number | date character vector | datetime object

Settlement or trade date for the barrier option, specified as a serial date number, a date character vector, or a datetime object.

Data Types: double | char | datetime

ExerciseDates — Option exercise dates

serial date number | date character vector | datetime object

Option exercise dates, specified as a serial date number, a date character vector, or datetime object:

• For a European option, there is only one ExerciseDates on the option expiry date which is the maturity of the instrument.

• For an American option, use a 1-by-2 vector of exercise date boundaries. The option can be exercised on any date between or including the pair of dates on that row. 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 | datetime

BarrierSpec — Barrier option type

character vector with values: 'UI', 'UO', 'DI', 'DO'

Barrier option type, specified as a character vector with the following values:

• 'UI' — Up Knock-in

  This option becomes effective when the price of the underlying asset passes above the barrier level. It gives the option holder the right, but not the obligation, to buy or sell (call/put) the underlying security at the strike price if the underlying asset goes above the barrier level during the life of the option. Note, barrierbyfd does not support American knock-in barrier options.

• 'UO' — Up Knock-out

  This option gives the option holder the right, but not the obligation, to buy or sell (call/put) the underlying security at the strike price as long as the underlying asset does not go above the barrier level during the life of the option. This option terminates when the price of the underlying asset passes above the barrier level. Usually, with an up-and-out option, the rebate is paid if the spot price of the underlying reaches or exceeds the barrier level.

• 'DI' — Down Knock-in

  This option becomes effective when the price of the underlying stock passes below the barrier level. It gives the option holder the right, but not the obligation, to buy or sell (call/put) the underlying security at the strike price if the underlying security goes below the barrier level during the life of the option. With a down-and-in option, the rebate is paid if the spot price of the underlying does not reach the barrier level during the life of the option. Note, barrierbyfd does not support American knock-in barrier options.

• 'DO' — Down Knock-up

  This option gives the option holder the right, but not the obligation, to buy or sell (call/put) the underlying asset at the strike price, as long as the underlying asset does not go below the barrier level during the life of the option. This option terminates when the price of the underlying security passes below the barrier level. Usually the option holder receives a rebate amount if the option expires worthless.

Option	Barrier Type	Payoff if Barrier Crossed	Payoff if Barrier not Crossed
Call/Put	Down Knock-out	Worthless	Standard Call/Put
Call/Put	Down Knock-in	Call/Put	Worthless
Call/Put	Up Knock-out	Worthless	Standard Call/Put
Call/Put	Up Knock-in	Standard Call/Put	Worthless

Data Types: char

Barrier — Barrier level

numeric

Barrier level, specified as a scalar numeric.

Data Types: double

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 = barriersensbyfd(RateSpec,StockSpec,OptSpec,Strike,Settle,Maturity,BarrierSpec,Barrier,Rebate,1000,AmericanOpt,1)

'Rebate' — Rebate value

0 (default) | numeric

Rebate value, specified as the comma-separated pair consisting of 'Rebate' and a scalar numeric. For Knock-in options, the Rebate is paid at expiry. For Knock-out options, the Rebate is paid when the Barrier is reached.

Data Types: double

'OutSpec' — Define outputs

{'Price'} (default) | character vector with values 'Price', 'Delta', 'Gamma', 'Vega', 'Lambda', 'Rho', 'Theta', and 'All' | cell array of character vectors with values 'Price', 'Delta', 'Gamma', 'Vega', 'Lambda', 'Rho', 'Theta', and 'All'

Define outputs, specified as the comma-separated pair consisting of 'OutSpec' and a NOUT- by-1 or a 1-by-NOUT cell array of character vectors with possible values of 'Price', 'Delta', 'Gamma', 'Vega', 'Lambda', 'Rho', 'Theta', and 'All'.

OutSpec = {'All'} specifies that the output is Delta, Gamma, Vega, Lambda, Rho, Theta, and Price, in that order. This is the same as specifying OutSpec to include each sensitivity.

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

Data Types: char | cell

'AssetGridSize' — Size of asset grid used for finite difference grid

400 (default) | positive numeric

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

Data Types: double

'TimeGridSize' — Size of time grid used for finite difference grid

100 (default) | positive numeric

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

Data Types: double

'AmericanOpt' — Option type

0 European (default) | integer with values 0 or 1

Option type, specified as the comma-separated pair consisting of 'AmericanOpt' and a scalar flag with one of the following values:

• 0 — European

• 1 — American

Data Types: logical

Output Arguments

PriceSens — Expected prices or sensitivities values for barrier options

matrix

Expected prices or sensitivities (defined using OutSpec) for barrier options, returned as a NINST-by-1 matrix.

PriceGrid — Grid containing prices calculated by finite difference method

grid

Grid containing prices calculated by the finite difference method, returned as a two-dimensional grid with size PriceGridSize*length(Times). The number of columns does not have to be equal to the TimeGridSize, because ex-dividend dates in the StockSpec are added to the time grid. The price for t = 0 is contained in PriceGrid(:, end).

AssetPrices — Prices of the asset defined by StockSpec

vector

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

Times — Times corresponding to second dimension of PriceGrid

vector

Times corresponding to the second dimension of the PriceGrid, returned as a vector.

More About

Barrier Option

A Barrier option has not only a strike price but also a barrier level and sometimes a rebate.

A rebate is a fixed amount that is paid if the option cannot be exercised because the barrier level has been reached or not reached. The payoff for this type of option depends on whether the underlying asset crosses the predetermined trigger value (barrier level), indicated by Barrier, during the life of the option.

References

[1] Hull, J. Options, Futures and Other Derivatives. Fourth Edition. Prentice Hall, 2000, pp. 646–649.

[2] Aitsahlia, F., L. Imhof, and T.L. Lai. “Pricing and hedging of American knock-in options.” The Journal of Derivatives. Vol. 11.3 , 2004, pp. 44–50.

[3] Rubinstein M. and E. Reiner. “Breaking down the barriers.” Risk. Vol. 4(8), 1991, pp. 28–35.