# 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

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.```
````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 = '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

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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 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`

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 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 serial date number.

Data Types: `double`

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