Price European or American spread options using Monte Carlo simulations

## Description

example

Price = spreadbyls(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr) returns the price of a European or American call or put spread option using Monte Carlo simulations.

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

Price = spreadbyls(___,Name,Value) returns the price of a European or American call or put spread option using Monte Carlo simulations using optional name-value pair arguments.

[Price,Paths,Times,Z] = spreadbyls(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr) returns the Price, Paths, Times, and Z of a European or American call or put spread option using Monte Carlo simulations.

[Price,Paths,Times,Z] = spreadbyls(___,Name,Value) returns the Price, Paths, Times, and Z of a European or American call or put spread option using Monte Carlo simulations using optional name-value pair arguments.

## Examples

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Settle = '01-Jan-2012';
Maturity = '01-April-2012';

Define asset 1. Price and volatility of RBOB gasoline

Price1gallon = 2.85;          % \$/gallon
Price1 = Price1gallon * 42;   % \$/barrel
Vol1 = 0.29;

Define asset 2. Price and volatility of WTI crude oil

Price2 = 93.20;         % \$/barrel
Vol2 = 0.36;

Define the correlation between the underlying asset prices of asset 1 and asset 2.

Corr = 0.42;

OptSpec = 'call';
Strike = 20;

Define the RateSpec.

rates = 0.05;
Compounding = -1;
Basis = 1;
RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle, ...
'EndDates', Maturity, 'Rates', rates, ...
'Compounding', Compounding, 'Basis', Basis)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9876
Rates: 0.0500
EndTimes: 0.2500
StartTimes: 0
EndDates: 734960
StartDates: 734869
ValuationDate: 734869
Basis: 1
EndMonthRule: 1

Define the StockSpec for the two assets.

StockSpec1 = stockspec(Vol1, Price1)
StockSpec1 = struct with fields:
FinObj: 'StockSpec'
Sigma: 0.2900
AssetPrice: 119.7000
DividendType: []
DividendAmounts: 0
ExDividendDates: []

StockSpec2 = stockspec(Vol2, Price2)
StockSpec2 = struct with fields:
FinObj: 'StockSpec'
Sigma: 0.3600
AssetPrice: 93.2000
DividendType: []
DividendAmounts: 0
ExDividendDates: []

Compute the spread option price using Monte Carlo simulation based on the Longstaff-Schwartz model.

Price = spreadbyls(RateSpec, StockSpec1, StockSpec2, Settle, ...
Maturity, OptSpec, Strike, Corr)
Price = 11.0799

## 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 1. For information on the stock specification, see stockspec.

stockspec can handle other types of underlying assets. For example, for physical commodities the price is represented by StockSpec.Asset, the volatility is represented by StockSpec.Sigma, and the convenience yield is represented by StockSpec.DividendAmounts.

Data Types: struct

Stock specification for underlying asset 2. For information on the stock specification, see stockspec.

stockspec can handle other types of underlying assets. For example, for physical commodities the price is represented by StockSpec.Asset, the volatility is represented by StockSpec.Sigma, and the convenience yield is represented by StockSpec.DividendAmounts.

Data Types: struct

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

Data Types: char | double

Maturity date for spread option, specified as a date character vector or a nonnegative scalar integer.

Data Types: char | double

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

Data Types: char

Option strike price value, specified, as a nonnegative scalar integer.

Data Types: single | double

Correlation between underlying asset prices, specified as a scalar integer.

Data Types: single | double

### 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.

Option type, specified as the comma-separated pair consisting of 'AmericanOpt' and an integer scalar flag with value:

• 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

Scalar number of independent sample paths (simulation trials), specified as the comma-separated pair consisting of 'NumTrials' and a nonnegative integer.

Data Types: single | double

Scalar number of simulation periods per trial, specified as the comma-separated pair consisting of 'NumPeriods' and a nonnegative integer. NumPeriods is considered only when pricing European basket options. For American spread options, NumPeriod is equal to the number of exercise days during the life of the option.

Data Types: single | double

Time series array of dependent random variates, specified as the comma-separated pair consisting of 'Z' and a NumPeriods-by-2-by-NumTrials 3-D array. The Z value generates the Brownian motion vector (that is, Wiener processes) that drives the simulation.

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

## Output Arguments

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

Simulated paths of correlated state variables, returned as a NumPeriods + 1-by-2-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.

Time series array of dependent random variates, returned as a NumPeriods-by-2-by-NumTrials 3-D array when Z is specified as an input argument. If the Z input argument is not specified, then the Z output argument contains the random variates generated internally.

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A spread option is an option written on the difference of two underlying assets.

For example, a European call on the difference of two assets X1 and X2 would have the following pay off at maturity:

$\mathrm{max}\left(X1-X2-K,0\right)$

where:

K is the strike price.

## References

[1] Carmona, R., Durrleman, V. “Pricing and Hedging Spread Options.” SIAM Review. Vol. 45, No. 4, pp. 627–685, Society for Industrial and Applied Mathematics, 2003.