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

[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.

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

Examples

Compute the Delta and Gamma of an American Barrier Down In Put Option

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

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 the 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

scalar 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 a 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 | cell

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.

• '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.

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

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

'AmericanOpt' — Option type

0 (European) (default) | values [0,1]

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

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' — 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

'NumTrials' — Number of independent sample paths

1000 (default) | nonnegative integer

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

Data Types: double

'NumPeriods' — Number of simulation periods per trial

100 (default) | nonnegative integer

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

Data Types: double

'Z' — Time series array of dependent random variates

vector

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

'Antithetic' — Indicator for antithetic sampling

false (default) | logical flag with value of true or false

Indicator for antithetic sampling, specified as the comma-separated pair consisting of 'Antithetic' and a scalar value of true or false.

Data Types: logical

'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

Output Arguments

PriceSens — Expected prices or sensitivities for barrier options

matrix

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

Paths — Simulated paths of correlated state variables

vector

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.

Times — Observation times associated with simulated paths

vector

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.

Z — Time series array of dependent random variates

vector

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

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.