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# barriersensbyls

Calculate price and sensitivities for European or American barrier options using Monte Carlo simulations

## Syntax

[PriceSens,Paths,Times,Z] = barriersensbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates,BarrierSpec,Barrier)
[PriceSens,Paths,Times,Z] = barriersensbyls(___,Name,Value)

## Description

example

[PriceSens,Paths,Times,Z] = barriersensbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates,BarrierSpec,Barrier) calculates barrier option prices or sensitivities on a single underlying asset using the Longstaff-Schwartz model. barriersensbyls computes prices of European and American barrier options.

For American options, the Longstaff-Schwartz least squares method is used to calculate the early exercise premium.

example

[PriceSens,Paths,Times,Z] = barriersensbyls(___,Name,Value) adds optional name-value pair arguments.

## Examples

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Compute the price of an American down in put option using the following data:

Rates = 0.0325;
Settle = '01-Jan-2016';
Maturity = '01-Jan-2017';
Compounding = -1;
Basis = 1;

Define a RateSpec.

RateSpec = intenvset('ValuationDate',Settle,'StartDates',Settle,'EndDates',Maturity, ...
'Rates',Rates,'Compounding',Compounding,'Basis',Basis)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9680
Rates: 0.0325
EndTimes: 1
StartTimes: 0
EndDates: 736696
StartDates: 736330
ValuationDate: 736330
Basis: 1
EndMonthRule: 1

Define a StockSpec.

AssetPrice = 40;
Volatility = 0.20;
StockSpec = stockspec(Volatility,AssetPrice)
StockSpec = struct with fields:
FinObj: 'StockSpec'
Sigma: 0.2000
AssetPrice: 40
DividendType: []
DividendAmounts: 0
ExDividendDates: []

Calculate the delta and gamma of an American barrier down in put option.

Strike = 45;
OptSpec = 'put';
Barrier = 35;
BarrierSpec = 'DI';
AmericanOpt = 1;

OutSpec = {'delta','gamma'};

[Delta,Gamma] = barriersensbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,...
Maturity,BarrierSpec,Barrier,'NumTrials',2000,'AmericanOpt',AmericanOpt,'OutSpec',OutSpec)
Delta = -0.6346
Gamma = -0.3091

## Input Arguments

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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

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

Definition of the option as 'call' or 'put', specified as a character vector or string object with values 'call' or 'put'.

Data Types: char | double

Option strike price value, specified as an integer.

Data Types: double

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

Option exercise dates, specified as a nonnegative scalar integer, a date character vector, or a datetime object:

• For a European option, use a 1-by-1 vector of dates. 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, or if ExerciseDates is a 1-by-1 vector of serial date numbers or a cell array of date character vectors, the option can be exercised between Settle and the single listed date in ExerciseDates.

Data Types: double | char

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/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.

• 'UO' — Up Knock Out

This option gives the option holder the right, but not the obligation, to buy/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/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.

• 'DO' — Down Knock Up

This option gives the option holder the right, but not the obligation, to buy/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.

OptionBarrier TypePayoff if Barrier CrossedPayoff if Barrier not Crossed
Call/PutDown Knock-outWorthlessStandard Call/Put
Call/PutDown Knock-inCall/PutWorthless
Call/PutUp Knock-outWorthlessStandard Call/Put
Call/PutUp Knock-inStandard Call/PutWorthless

Data Types: char

Barrier value, specified as a scalar integer.

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 single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: Price = barriersensbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,Maturity,BarrierSpec,Barrier,Rebate,1000)

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Option type, specified as NINST-by-1 positive integer scalar flags with values:

• 0 — European

• 1 — American

### Note

For American options, the Longstaff-Schwartz least squares method is used to calculate the early exercise premium. For more information on the least squares method, see https://people.math.ethz.ch/%7Ehjfurrer/teaching/LongstaffSchwartzAmericanOptionsLeastSquareMonteCarlo.pdf.

Data Types: double

Rebate value, specified as a scalar integer. 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

Scalar number of independent sample paths (simulation trials), specified as a nonnegative integer.

Data Types: double

Scalar number of simulation periods per trial, specified as a nonnegative integer.

Data Types: double

Time series array of dependent random variates, specified as a NumPeriods-by-1-by-NumTrials 3-D time series array. The Z value generates the Brownian motion vector (that is, Wiener processes) that drives the simulation.

Data Types: double

Indicator for antithetic sampling, specified with a value of true or false.

Data Types: logical

Define outputs, specifying 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

## Output Arguments

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

Simulated paths of correlated state variables, returned as a NumPeriods + 1-by-1-by-NumTrials 3-D time series array of simulated paths of correlated state variables. Each row of Paths is the transpose of the state vector X(t) at time t for a given trial.

Observation times associated with simulated paths, returned as a NumPeriods + 1-by-1 column vector of observation times associated with the simulated paths. Each element of Times is associated with the corresponding row of Paths.

Time series array of dependent random variates, returned as a NumPeriods-by-1-by-NumTrials 3-D array when Z is specified as an input argument. If the Z input argument is not specified, then the Z output argument contains the random variates generated internally.

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### 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] Broadie, M., P. Glasserman and S. Kou. "A continuity correction for discrete barrier options." Mathematical Finance. Vol. 7.4 , 1997, pp. 3250–349.

[4] Moon, K.S. "Efficient Monte Carlo algorithm for pricing barrier options." Communications of the Korean Mathematical Society. Vol 23.2, 2008 pp. 85–294.

[5] Papatheodorou, B. “Enhanced Monte Carlo methods for pricing and hedging exotic options." University of Oxford thesis, 2005.

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