Calculate price and sensitivities for European or American basket options using Monte Carlo simulations

## Description

example

[PriceSens,Paths,Times,Z] = basketsensbyls(RateSpec,BasketStockSpec,OptSpec,Strike,Settle,ExerciseDates) calculates price and sensitivities for European or American basket options using the Longstaff-Schwartz model.

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

example

[PriceSens,Paths,Times,Z] = basketsensbyls(___,Name,Value) specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax.

## Examples

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Find a European put basket option of two stocks. The basket contains 50% of each stock. The stocks are currently trading at \$90 and \$75, with annual volatilities of 15%. Assume that the correlation between the assets is zero. On May 1, 2009, an investor wants to buy a one-year put option with a strike price of \$80. The current annualized, continuously compounded interest is 5%. Use this data to compute price and delta of the put basket option with the Longstaff-Schwartz approximation model.

Settle = datetime(2009,5,1);
Maturity  = datetime(2010,5,1);

% Define RateSpec
Rate = 0.05;
Compounding = -1;
RateSpec = intenvset('ValuationDate', Settle, 'StartDates',...
Settle, 'EndDates', Maturity, 'Rates', Rate, 'Compounding', Compounding);

% Define the Correlation matrix. Correlation matrices are symmetric,
% and have ones along the main diagonal.
NumInst  = 2;
InstIdx = ones(NumInst,1);
Corr = diag(ones(NumInst,1), 0);

AssetPrice =  [90; 75];
Volatility = 0.15;
Quantity = [0.50; 0.50];

% Compute the price of the put basket option. Calculate also the delta
% of the first stock.
OptSpec = {'put'};
Strike = 80;
OutSpec = {'Price','Delta'};
UndIdx = 1; % First element in the basket

Strike, Settle, Maturity,'OutSpec', OutSpec,'UndIdx', UndIdx)
PriceSens = 0.9822
Delta = -0.0995

Compute the Price and Delta of the basket with a correlation of -20%:

NewCorr = [1 -0.20; -0.20 1];

% Compute the price and delta of the put basket option.
Strike, Settle, Maturity,'OutSpec', OutSpec,'UndIdx', UndIdx)
PriceSens = 0.7814
Delta = -0.0961

## 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

Data Types: struct

Definition of the option as 'call' or 'put', specified as a character vector or a 2-by-1 cell array of character vectors.

Data Types: char | cell

Option strike price value, specified as one of the following:

• For a European or Bermuda option, Strike is a scalar (European) or 1-by-NSTRIKES (Bermuda) vector of strike prices.

• For an American option, Strike is a scalar vector of the strike price.

Data Types: double

Settlement or trade date for the basket option, specified as a scalar datetime, string, or date character vector.

To support existing code, basketsensbyls also accepts serial date numbers as inputs, but they are not recommended.

Option exercise dates, specified as a datetime array, string array, or date character vectors:

• For a European or Bermuda option, ExerciseDates is a 1-by-1 (European) or 1-by-NSTRIKES (Bermuda) vector of exercise dates. For a European option, there is only one ExerciseDate on the option expiry date.

• For an American option, ExerciseDates is a 1-by-2 vector of exercise date boundaries. The option exercises on any date between, or including, the pair of dates on that row. If there is only one non-NaN date, or if ExerciseDates is 1-by-1, the option exercises between the Settle date and the single listed ExerciseDate.

To support existing code, basketsensbyls also accepts serial date numbers as inputs, but they are not recommended.

### 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.

Option type, specified as the comma-separated pair consisting of 'AnericanOpt' and a NINST-by-1 positive integer scalar flags with values:

• 0 — European/Bermuda

• 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: double

Number of simulation periods per trial, specified as the comma-separated pair consisting of 'NumPeriods' and a scalar nonnegative integer.

Note

NumPeriods is considered only when pricing European basket options. For American and Bermuda basket options, NumPeriod equals the number of exercise days during the life of the option.

Data Types: double

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

Data Types: double

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

Data Types: 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 a 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 is 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

Index of the underlying instrument to compute the sensitivity, specified as the comma-separated pair consisting of 'UndIdx' and a scalar numeric.

Data Types: double

## Output Arguments

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Expected prices or sensitivities (defined using OutSpec) for basket option, returned as a NINST-by-1 matrix.

Simulated paths of correlated state variables, returned as a NumPeriods + 1-by-1-by-NumTrials 3-D time series array of simulated paths of correlated state variables. 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-1-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 basket option is an option on a portfolio of several underlying equity assets.

Payout for a basket option depends on the cumulative performance of the collection of the individual assets. A basket option tends to be cheaper than the corresponding portfolio of plain vanilla options for these reasons:

• If the basket components correlate negatively, movements in the value of one component neutralize opposite movements of another component. Unless all the components correlate perfectly, the basket option is cheaper than a series of individual options on each of the assets in the basket.

• A basket option minimizes transaction costs because an investor has to purchase only one option instead of several individual options.

## References

[1] Longstaff, F.A., and E.S. Schwartz. “Valuing American Options by Simulation: A Simple Least-Squares Approach.” The Review of Financial Studies. Vol. 14, No. 1, Spring 2001, pp. 113–147.

## Version History

Introduced in R2009b

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