lookbackbyls
Price European or American lookback options using Monte Carlo simulations
Syntax
Description
[
returns prices of lookback options using the Longstaff-Schwartz model for Monte Carlo
simulations. Price
,Paths
,Times
,Z
]
= lookbackbyls(RateSpec
,StockSpec
,OptSpec
,Strike
,Settle
,ExerciseDates
)lookbackbyls
computes prices of European and American
lookback options.
For American options, the Longstaff-Schwartz least squares method calculates the early exercise premium.
lookbackbyls
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 price
lookback options. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.
Examples
Compute the Price for a Floating Lookback Option Using Monte Carlo Simulation
Define the RateSpec
.
StartDates = datetime(2013,1,1); EndDates = datetime(2014,1,1); Rates = 0.042; Compounding = -1; RateSpec = intenvset('ValuationDate', StartDates, 'StartDates', StartDates,... 'EndDates', EndDates, 'Rates', Rates, 'Compounding', Compounding)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9589
Rates: 0.0420
EndTimes: 1
StartTimes: 0
EndDates: 735600
StartDates: 735235
ValuationDate: 735235
Basis: 0
EndMonthRule: 1
Define the StockSpec
.
AssetPrice = 50; Sigma = 0.36; StockSpec = stockspec(Sigma, AssetPrice)
StockSpec = struct with fields:
FinObj: 'StockSpec'
Sigma: 0.3600
AssetPrice: 50
DividendType: []
DividendAmounts: 0
ExDividendDates: []
Define the floating lookback option.
Settle = datetime(2013,1,1);
Maturity = datetime(2013,4,1);
OptSpec = 'put';
Strike = NaN;
Compute the price of the European floating lookback option.
Price = lookbackbyls(RateSpec, StockSpec, OptSpec, Strike, Settle, Maturity)
Price = 6.6471
Compute the Price of a Fixed Lookback Option Using Monte Carlo Simulation
Define the RateSpec
.
StartDates = 'Jan-1-2013'; EndDates = 'Jan-1-2014'; Rates = 0.045; Compounding = -1; RateSpec = intenvset('ValuationDate', StartDates, 'StartDates', StartDates,... 'EndDates', EndDates, 'Rates', Rates,'Compounding', Compounding)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9560
Rates: 0.0450
EndTimes: 1
StartTimes: 0
EndDates: 735600
StartDates: 735235
ValuationDate: 735235
Basis: 0
EndMonthRule: 1
Define the StockSpec
.
AssetPrice = 102; Sigma = 0.45; StockSpec = stockspec(Sigma, AssetPrice)
StockSpec = struct with fields:
FinObj: 'StockSpec'
Sigma: 0.4500
AssetPrice: 102
DividendType: []
DividendAmounts: 0
ExDividendDates: []
Define the fixed lookback option.
Settle = 'Jan-1-2013'; Maturity = 'July-1-2013'; OptSpec = 'call'; Strike = 98;
Compute the price of the European fixed lookback option.
Price = lookbackbyls(RateSpec, StockSpec, OptSpec, Strike, Settle, Maturity)
Price = 30.2368
Input Arguments
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 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, lookbackbyls
also
accepts serial date numbers as inputs, but they are not recommended.
ExerciseDates
— Exercise callable or puttable dates for European or American options
datetime array | string array | date character vector
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 ifExerciseDates
is aNINST
-by-1
vector of dates, the option is exercised betweenSettle
and the single listed exercise date.
To support existing code, lookbackbyls
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: Price =
lookbackbyls(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr,'AmericanOpt',1)
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
— European1
— 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, NumPeriods
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 of 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
Output Arguments
Price
— Expected price of lookback option
scalar
Expected price of the lookback option, returned as a
1
-by-1
scalar.
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 R2014aR2022b: Serial date numbers not recommended
Although lookbackbyls
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.
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