# optstocksensbyls

Calculate price and sensitivities for European, Bermudan, or American vanilla options using Monte Carlo simulations

## Description

example

PriceSens = optstocksensbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates) returns vanilla option prices or sensitivities using the Longstaff-Schwartz model. optstocksensbyls computes prices or sensitivities 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

PriceSens = optstocksensbyls(___,Name,Value)adds optional name-value pair arguments.

example

[PriceSens,Path,Times,Z] = optstocksensbyls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates) returns vanilla option prices or sensitivities using the Longstaff-Schwartz model.

example

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

## Examples

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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;
DivType = 'continuous';
DivAmounts = 0.04;
StockSpec = stockspec(Sigma, AssetPrice, DivType, DivAmounts)
StockSpec = struct with fields:
FinObj: 'StockSpec'
Sigma: 0.1000
AssetPrice: 100
DividendType: {'continuous'}
DividendAmounts: 0.0400
ExDividendDates: []

Define the vanilla option.

OptSpec = 'call';
Settle = 'jan-1-2013';
ExerciseDates = 'jan-1-2015';
Strike = 105;

Compute the Delta sensitivity for the vanilla option using the Longstaff-Schwartz model.

Antithetic = true;
OutSpec = {'Delta'};
PriceSens = optstocksensbyls(RateSpec, StockSpec, OptSpec, Strike, ...
Settle, ExerciseDates,'Antithetic', Antithetic, 'OutSpec', OutSpec)
PriceSens = 0.3945

To display the output for Price, Delta, Path, and Times, use the following:

OutSpec = {'Price','Delta'};
[Price, Delta, Path, Times] = optstocksensbyls(RateSpec, StockSpec, OptSpec, Strike, ...
Settle, ExerciseDates,'Antithetic', Antithetic, 'OutSpec', OutSpec);

## 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.

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:

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

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

• 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 a serial date number.

Data Types: double | char

Option exercise date, 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 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 = optstocksensbyls(RateSpec,StockSpec, OptSpec,Strike,Settle,ExerciseDates,'AmericanOpt','1','NumTrials','2000','OutSpec',{'Price','Delta','Gamma'})

Option type, specified as the comma-separated pair consisting of 'AmericanOpt' and a positive integer scalar flag 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 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' and 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 the comma-separated pair consisting of 'Z' and a NumPeriods-by-1-by-NumTrials 3-D time series array.

Data Types: single | double

Indicator for antithetic sampling, specified as the comma-separated pair consisting of 'Antithetic' and a value of true or false.

Data Types: logical

Define outputs, specified as the comma-separated pair consisting of 'OutSpec' and a NOUT- by-1 or 1-by-NOUT cell array of character vectors with possible values of 'Price', 'Delta', 'Gamma', 'Vega', 'Lambda', 'Rho', 'Theta', and 'All'.

OutSpec = {'All'} specifies that the output should be Delta, Gamma, Vega, Lambda, Rho, Theta, and Price, in that order. This is the same as specifying OutSpec to include each sensitivity:

Example: OutSpec = {'delta','gamma','vega','lambda','rho','theta','price'}

Data Types: char | cell

## Output Arguments

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Expected price or sensitivities (defined by OutSpec) of the vanilla option, returned as a 1-by-1 array.

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.

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### 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.