Documentation

# optstockbyls

Price European, Bermudan, or American vanilla options using Monte Carlo simulations

## Syntax

``Price = optstockbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates)``
``Price = optstockbyls(___,Name,Value)``
``````[Price,Path,Times,Z] = optstockbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates)``````
``````[Price,Path,Times,Z] = optstockbyls(___,Name,Value)``````

## Description

example

````Price = optstockbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates)` returns vanilla option prices using the Longstaff-Schwartz model. `optstockbyls` computes prices of European, Bermudan, and American vanilla options. For American and Bermudan options, the Longstaff-Schwartz least squares method is used to calculate the early exercise premium. ```

example

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

example

``````[Price,Path,Times,Z] = optstockbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates)``` returns vanilla option prices using the Longstaff-Schwartz model. ```

example

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

## Examples

collapse all

Define the `RateSpec`.

```StartDates = 'Jan-1-2013'; EndDates = 'Jan-1-2015'; Rates = 0.05; RateSpec = intenvset('ValuationDate', StartDates, 'StartDates', StartDates, ... 'EndDates', EndDates, 'Rates', Rates)```
```RateSpec = struct with fields: FinObj: 'RateSpec' Compounding: 2 Disc: 0.9060 Rates: 0.0500 EndTimes: 4 StartTimes: 0 EndDates: 735965 StartDates: 735235 ValuationDate: 735235 Basis: 0 EndMonthRule: 1 ```

Define the `StockSpec` for the asset.

```AssetPrice = 100; Sigma = 0.1; StockSpec = stockspec(Sigma, AssetPrice)```
```StockSpec = struct with fields: FinObj: 'StockSpec' Sigma: 0.1000 AssetPrice: 100 DividendType: [] DividendAmounts: 0 ExDividendDates: [] ```

Define the vanilla option.

```OptSpec = 'put'; Settle = 'Jan-1-2013'; ExerciseDates = 'Jan-1-2015'; Strike = 105;```

Compute the vanilla option price using the Longstaff-Schwartz model.

```Antithetic = true; Price = optstockbyls(RateSpec, StockSpec, OptSpec, Strike, Settle, ... ExerciseDates, 'Antithetic', Antithetic)```
```Price = 3.2292 ```

## Input Arguments

collapse all

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.

Data Types: `struct`

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

Data Types: `char`

Option strike price value, specified with nonnegative scalar integer:

• For a European option, use a scalar of strike price.

• For a Bermuda option, use a `1`-by-`NSTRIKES` vector of strike prices.

• For an American option, use a scalar of strike price.

Data Types: `single` | `double`

Settlement date or trade date for the vanilla option, specified as a date character vector or nonnegative scalar integer.

Data Types: `double` | `char`

Option exercise dates, specified as a date character vector or serial date number:

• 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 a Bermuda option, use a `1`-by-`NSTRIKES` vector of dates.

• 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 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 = optstockbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates,'AmericanOpt','1','NumTrials','2000')```

Option type, specified as the comma-separated pair consisting of `'AmericanOpt'` and positive integer scalar flags with values:

• `0` — European or Bermuda

• `1` — American

### Note

For American and Bermudan 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`

Simulation trials, specified as a scalar number of independent sample paths.

Data Types: `double`

Simulation periods per trial, specified as a scalar number. `NumPeriods` is considered only when pricing European vanilla options. For American and Bermuda vanilla options, `NumPeriod` 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 a be `NumPeriods`-by-`1`-by-`NumTrials` 3-D time series array.

Data Types: `single` | `double`

Indicator for antithetic sampling, specified with a value of `true` or `false`.

Data Types: `logical`

## Output Arguments

collapse all

Expected price of the vanilla 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 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, if `Z` is specified as an optional input argument, the same value is returned. Otherwise, `Z` contains the random variates generated internally.

collapse all

### Vanilla Option

A vanilla option is a category of options that includes only the most standard components.

A vanilla option has an expiration date and straightforward strike price. American-style options and European-style options are both categorized as vanilla options.

The payoff for a vanilla option is as follows:

• For a call: $\mathrm{max}\left(St-K,0\right)$

• For a put: $\mathrm{max}\left(K-St,0\right)$

where:

St is the price of the underlying asset at time t.

K is the strike price.