# asianbyls

Price European or American Asian options using Monte Carlo simulations

## Description

example

Price = asianbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates) returns fixed- and floating-strike Asian option prices using the Longstaff-Schwartz model. asianbyls computes prices of European and American Asian options.

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

To compute the value of a floating-strike Asian option, Strike should be specified as NaN. Fixed-strike Asian options are also known as average price options and floating-strike Asian options are also known as average strike options.

Note

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

Price = asianbyls(___,Name,Value) adds optional name-value pair arguments.

example

[Price,Paths,Times,Z] = asianbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates) returns fixed- and floating-strike Asian option Price, Paths, Times, and Z values using the Longstaff-Schwartz model. asianbyls computes prices of European and American Asian options.

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

## Examples

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Define the RateSpec.

Rates = 0.05;
StartDate = datetime(2013,1,1);
EndDate = datetime(2014,1,1);
RateSpec = intenvset('ValuationDate', StartDate, 'StartDates', StartDate, ...
'EndDates', EndDate,'Compounding', -1, 'Rates', Rates)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9512
Rates: 0.0500
EndTimes: 1
StartTimes: 0
EndDates: 735600
StartDates: 735235
ValuationDate: 735235
Basis: 0
EndMonthRule: 1

Define the StockSpec for the asset.

AssetPrice = 100;
Sigma = 0.2;
StockSpec = stockspec(Sigma, AssetPrice)
StockSpec = struct with fields:
FinObj: 'StockSpec'
Sigma: 0.2000
AssetPrice: 100
DividendType: []
DividendAmounts: 0
ExDividendDates: []

Define the Asian 'call' option.

Settle = datetime(2013,1,1);
ExerciseDates = datetime(2014,1,1);
Strike = 110;
OptSpec = 'call';

Compute the price for the European arithmetic average price for the Asian option using the Longstaff-Schwartz model.

NumTrials = 10000;
NumPeriods = 100;
AvgType = 'arithmetic';
Antithetic= true;
Price= asianbyls(RateSpec, StockSpec, OptSpec, Strike, Settle, ExerciseDates, ...
'NumTrials', NumTrials, 'NumPeriods', NumPeriods,'Antithetic', Antithetic, 'AvgType', AvgType)
Price = 1.9876

## 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 underlying asset, specified using StockSpec obtained from stockspec. For information on the stock specification, see stockspec.

stockspec can handle other types of underlying assets. For example, stocks, stock indices, and commodities. If dividends are not specified in StockSpec, dividends are assumed to be 0.

Data Types: struct

Definition of option, specified as 'call' or 'put' using a character vector

Data Types: char

Option strike price value, specified with a nonnegative scalar integer. To compute the value of a floating-strike Asian option, Strike should be specified as NaN. Floating-strike Asian options are also known as average strike options.

Data Types: double

Settlement or trade date for the Asian option, specified as a scalar datetime, string, or date character vector. By default, asianbyls calculates the price of Asian options based on averages that start on the settlement date.

To support existing code, asianbyls 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, use a 1-by-1 vector of dates. For a European option, there is only one ExerciseDates on the option expiry date.

• 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 dates, the option can be exercised between Settle and the single listed ExerciseDates.

To support existing code, asianbyls 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 = asianbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates,'NumTrials',NumTrials,'NumPeriods',NumPeriods,'Antithetic',Antithetic,'AvgType','arithmetic')

Option type, specified as the comma-separated pair consisting of 'AmericanOpt' and a 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: single | double

Average types, specified as the comma-separated pair consisting of 'AvgType' and arithmetic for arithmetic average, or geometric for geometric average.

Data Types: char

Average price of underlying asset at Settle, specified as the comma-separated pair consisting of 'AvgPrice' and a scalar numeric value. The AvgPrice is assumed to be calculated in the time window starting at AvgDate and ending on Settle. In other words, the average is backward looking.

Note

Use this argument when AvgDate < Settle.

Data Types: double

Date averaging period begins, specified as the comma-separated pair consisting of 'AvgDate' and a scalar datetime, string, or date character vector.

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

Simulation trials, specified as the comma-separated pair consisting of 'NumTrials' and a scalar number of independent sample paths.

Data Types: double

Simulation periods per trial, specified as the comma-separated pair consisting of 'NumPeriods' scalar numeric value. NumPeriods is considered only when pricing European Asian options. For American Asian options, NumPeriods is equal to the number of exercise days during the life of the option.

Data Types: double

Dependent random variates used to generate the Brownian motion vector (that is, Wiener processes) that drive the simulation, specified as the comma-separated pair consisting of 'Z' and a NumPeriods-by-2-by-NumTrials 3-D time series array.

Data Types: single | double

Logical flag to indicate antithetic sampling, specified as the comma-separated pair consisting of 'Antithetic' and a value of true or false.

Data Types: logical

## Output Arguments

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Expected price of the Asian option, returned as a 1-by-1 scalar.

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.

Observation times associated with the 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.

Dependent random variates, returned, if Z is specified as an optional input argument, the same value is returned. Otherwise, Z contains the random variates generated internally.

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### Asian Option

An Asian option is a path-dependent option with a payoff linked to the average value of the underlying asset during the life (or some part of the life) of the option.

Asian options are similar to lookback options in that there are two types of Asian options: fixed (average price option) and floating (average strike option). Fixed Asian options have a specified strike, while floating Asian options have a strike equal to the average value of the underlying asset over the life of the option. For more information, see Asian Option.

## Version History

Introduced in R2013b

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