asianbyls

Price European or American Asian options using Monte Carlo simulations

Syntax

Price = asianbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates)

Price = asianbyls(___,Name,Value)

[Price,Paths,Times,Z] = asianbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates)

[Price,Paths,Times,Z] = asianbyls(___,Name,Value)

Description

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.

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

[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

Compute the Price of an Asian Option Using the Longstaff-Schwartz Model

Define the RateSpec.

Rates = 0.05;
StartDate = 'Jan-1-2013';
EndDate = 'Jan-1-2014';
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 = 'Jan-1-2013';
ExerciseDates = 'Jan-1-2014';
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

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 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

OptSpec — Definition of option

character vector with a value of 'call' or 'put'

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

Data Types: char

Strike — Option strike price value

nonnegative scalar integer

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

Settle — Settlement or trade date

date character vector | nonnegative scalar integer

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

Data Types: double | char

ExerciseDates — Option exercise dates

date character vector | nonnegative scalar integer

Option exercise dates, specified as a nonnegative scalar integer or date character vector:

• 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 serial date numbers or cell array of date character vectors, the option can be exercised between Settle and the single listed ExerciseDates.

Data Types: double | char

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

'AmericanOpt' — Option type

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

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

'AvgType' — Average types

arithmetic (default) | character vector with values of arithmetic or geometric

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

Data Types: char

'AvgPrice' — Average price of underlying asset at Settle

numeric

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

'AvgDate' — Date averaging period begins

serial date number

Date averaging period begins, specified as the comma-separated pair consisting of 'AvgDate' and a scalar serial date number.

Data Types: double

'NumTrials' — Simulation trials

1000 (default) | numeric

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

Data Types: double

'NumPeriods' — Simulation periods per trial

100 (default) | numeric

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

'Z' — Dependent random variates

nonnegative integer

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

'Antithetic' — Indicator for antithetic sampling

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

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

Price — Expected price of Asian option

scalar

Expected price of the Asian 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 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.

Z — Dependent random variates

vector

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.

More About

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.