lookbacksensbyls

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

Syntax

[PriceSens,Paths,Times,Z] = lookbacksensbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates)

[PriceSens,Paths,Times,Z] = lookbacksensbyls(___,Name,Value)

Description

[PriceSens,Paths,Times,Z] = lookbacksensbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates) returns prices or sensitivities of lookback options using the Longstaff-Schwartz model for Monte Carlo simulations. lookbacksensbyls computes prices of European and American lookback options.

For American options, the Longstaff-Schwartz least squares method calculates the early exercise premium.

lookbacksensbyls calculates values of fixed- and floating-strike lookback options. To compute the value of a floating-strike lookback option, Strike must be specified as NaN.

Note

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

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

Examples

Compute the Price and Delta of a European Floating Lookback Option Using Monte Carlo Simulation

Define the RateSpec.

StartDates = datetime(2013,1,1);
EndDates = datetime(2014,1,1);
Rates = 0.41;
Compounding = -1;
RateSpec = intenvset('ValuationDate', StartDates, 'StartDates', StartDates,...
'EndDates', EndDates, 'Rates', Rates,'Compounding', Compounding)

RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: -1
             Disc: 0.6637
            Rates: 0.4100
         EndTimes: 1
       StartTimes: 0
         EndDates: 735600
       StartDates: 735235
    ValuationDate: 735235
            Basis: 0
     EndMonthRule: 1

Define the StockSpec with continuous dividend yield.

AssetPrice = 120;
Sigma = 0.3;
Yield = 0.045;
StockSpec = stockspec(Sigma, AssetPrice, 'Continuous', Yield)

StockSpec = struct with fields:
             FinObj: 'StockSpec'
              Sigma: 0.3000
         AssetPrice: 120
       DividendType: {'continuous'}
    DividendAmounts: 0.0450
    ExDividendDates: []

Define the floating lookback option.

Settle   = datetime(2013,1,1);
Maturity = datetime(2013,7,1); 
OptSpec = 'call';
Strike = NaN;

Compute the price and delta of the European floating lookback option.

OutSpec = {'price', 'delta'};
[Price, Delta] = lookbacksensbyls(RateSpec, StockSpec, OptSpec, Strike, Settle, Maturity,...
'OutSpec', OutSpec)

Price = 
27.0768

Delta = 
0.2256

Compute the Price and Delta of a European Fixed Lookback Option Using Monte Carlo Simulation

Define the RateSpec.

StartDates = datetime(2013,1,1);
EndDates = datetime(2015,1,1);
Rates = 0.1;
Compounding = -1;
RateSpec = intenvset('ValuationDate', StartDates, 'StartDates', StartDates,...
'EndDates', EndDates, 'Rates', Rates,'Compounding', Compounding)

RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: -1
             Disc: 0.8187
            Rates: 0.1000
         EndTimes: 2
       StartTimes: 0
         EndDates: 735965
       StartDates: 735235
    ValuationDate: 735235
            Basis: 0
     EndMonthRule: 1

Define the StockSpec.

AssetPrice = 103;
Sigma = 0.30;
StockSpec = stockspec(Sigma, AssetPrice)

StockSpec = struct with fields:
             FinObj: 'StockSpec'
              Sigma: 0.3000
         AssetPrice: 103
       DividendType: []
    DividendAmounts: 0
    ExDividendDates: []

Define the fixed lookback option.

Settle   = datetime(2013,1,1);
Maturity = datetime(2013,7,1); 
OptSpec = 'call';
Strike = 99;

Compute the price and delta of the European fixed lookback option.

OutSpec = {'price', 'delta'};                                 
[Price, Delta] = lookbacksensbyls(RateSpec, StockSpec, OptSpec, Strike, Settle, Maturity,...
'OutSpec', OutSpec)

Price = 
22.7227

Delta = 
1.1349

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 can handle several types of underlying assets. For example, for physical commodities the price is represented by StockSpec.Asset, the volatility is represented by StockSpec.Sigma, and the convenience yield is represented by StockSpec.DividendAmounts.

Data Types: struct

OptSpec — Definition of option

character vector with values 'call' or 'put' | cell array of character vectors

Definition of option as 'call' or 'put', specified as a NINST-by-1 cell array of character vectors.

Data Types: char | cell

Strike — Option strike price values

integer | vector of integers

Option strike price values, specified as an integer using a NINST-by-1 vector of strike price values.

Data Types: single | double

Settle — Settlement or trade date

datetime array | string array | date character vector

Settlement or trade date for the lookback option, specified as a NINST-by-1 vector using a datetime array, string array, or date character vectors.

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

ExerciseDates — Matrix of exercise callable or puttable dates for European or American options

datetime array | string array | date character vector

Matrix of exercise callable or puttable dates for European or American options, specified as a datetime array, string array, or date character vectors as follows:

• European option — NINST-by-1 vector of exercise dates. For a European option, there is only one exercise date which is the option expiry date.

• American option — NINST-by-2 vector of exercise date boundaries. For each instrument, the option is exercised on any coupon date between or including the pair of dates on that row. If only one non-NaN date is listed, or if ExerciseDates is a NINST-by-1 vector of dates, the option is exercised between Settle and the single listed exercise date.

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

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: PriceSens = lookbacksensbyls(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr,'AmericanOpt',1,'OutSpec',{'All'})

AmericanOpt — Option type

0 European (default) | scalar with value [0,1]

Option type, specified as the comma-separated pair consisting of 'AmericanOpt' and an integer scalar flag with these 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

NumTrials — Scalar number of independent sample paths

1000 (default) | nonnegative scalar integer

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

Data Types: double

NumPeriods — Scalar number of simulation periods per trial

100 (default) | nonnegative scalar integer

Scalar number of simulation periods per trial, specified as the comma-separated pair consisting of 'NumPeriods' and a nonnegative integer. NumPeriods is considered only when pricing European lookback options. For American lookback options, NumPeriod is equal to the number of exercise days during the life of the option.

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 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) | scalar logical flag with value true or false

Indicator for antithetic sampling, specified as the comma-separated pair consisting of 'Antithetic' and a 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 should be 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 price or sensitivities of lookback option

scalar

Expected price or sensitivities (defined by OutSpec) of the lookback option, returned as a 1-by-1 array.

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. 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

Lookback Option

A lookback option is a path-dependent option based on the maximum or minimum value the underlying asset achieves during the entire life of the option.

Financial Instruments Toolbox™ software supports two types of lookback options: fixed and floating. Fixed lookback options have a specified strike price, while floating lookback options have a strike price determined by the asset path. For more information, see Lookback Option.

References

[1] Hull, J. C. Options, Futures, and Other Derivatives 5th Edition. Englewood Cliffs, NJ: Prentice Hall, 2002.

Version History

Introduced in R2014a

R2022b: Serial date numbers not recommended

Although lookbacksensbyls supports serial date numbers, datetime values are recommended instead. The datetime data type provides flexible date and time formats, storage out to nanosecond precision, and properties to account for time zones and daylight saving time.

To convert serial date numbers or text to datetime values, use the datetime function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum");
y = year(t)

y =

        2021

There are no plans to remove support for serial date number inputs.