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barriersensbyls

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

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Syntax

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

Description

[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.

Note

Alternatively, you can use the Barrier object to calculate price or sensitivities for Barrier options. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

example

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

example

Examples

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Open Live Script

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

Rates = 0.0325;
Settle = datetime(2016,1,1);
Maturity = datetime(2017,1,1);
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 array with values "call" or "put".

Data Types: char | string

Option strike price value, specified as a scalar numeric.

Data Types: double

Settlement or trade date for the barrier option, specified as a scalar datetime, string, or date character vector.

To support existing code, barriersensbyls also accepts serial date numbers as inputs, but they are not recommended.

Option exercise dates, specified as a datetime array, string array, or date character vectors:

  • 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.

To support existing code, barriersensbyls also accepts serial date numbers as inputs, but they are not recommended.

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.

  • '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.

  • '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.

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 level, specified as a scalar numeric.

Data Types: double

Name-Value Arguments

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

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

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 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

Number of independent sample paths (simulation trials), specified as the comma-separated pair consisting of 'NumTrials' and a scalar nonnegative integer.

Data Types: double

Number of simulation periods per trial, specified as the comma-separated pair consisting of 'NumPeriods' and a scalar nonnegative integer.

Data Types: double

Time series array of dependent random variates, specified as the comma-separated pair consisting of 'Z' and 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 as the comma-separated pair consisting of 'Antithetic' and a scalar value of true or false.

Data Types: logical

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

Number of days between monitoring barriers, specified as a scalar integer. The default is 0 which indicates that the barrier is continuously monitored.

Data Types: double

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.

More About

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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. For more information, see Barrier 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.

Version History

Introduced in R2016b

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See Also

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