basketsensbyls

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

Syntax

[PriceSens,Paths,Times,Z] = basketsensbyls(RateSpec,BasketStockSpec,OptSpec,Strike,Settle,ExerciseDates)

[PriceSens,Paths,Times,Z] = basketsensbyls(___,Name,Value)

Description

[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.

[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

Calculate Price and Sensitivities for Basket Options Using the Longstaff-Schwartz Model

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 = 'May-1-2009';
Maturity  = 'May-1-2010';

% 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);

% Define BasketStockSpec
AssetPrice =  [90; 75]; 
Volatility = 0.15;
Quantity = [0.50; 0.50];
BasketStockSpec = basketstockspec(Volatility, AssetPrice, Quantity, Corr);

% 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
                                     
[PriceSens, Delta] = basketsensbyls(RateSpec, BasketStockSpec, OptSpec,...
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];

% Define the new BasketStockSpec.
BasketStockSpec = basketstockspec(Volatility, AssetPrice, Quantity, NewCorr);

% Compute the price and delta of the put basket option. 
[PriceSens, Delta] = basketsensbyls(RateSpec, BasketStockSpec, OptSpec,...
Strike, Settle, Maturity,'OutSpec', OutSpec,'UndIdx', UndIdx)

PriceSens = 0.7814

Delta = -0.0961

Input Arguments

RateSpec — Interest-rate term structure

structure

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

BasketStockSpec — BasketStock specification

structure

BasketStock specification, specified using basketstockspec.

Data Types: struct

OptSpec — Definition of option

character vector with values 'call' or 'put' | cell array of character vectors with values 'call' or 'put'

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

Strike — Option strike price value

scalar numeric | vector

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

Settle — Settlement or trade date

serial date number | date character vector

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

Data Types: double | char

ExerciseDates — Option exercise dates

serial date number | date character vector

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

• 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.

Data Types: double | char | cell

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: PriceSens = basketsensbyls(RateSpec,BasketStockSpec,OptSpec, Strike,Settle,Maturity,'AmericanOpt',AmericanOpt,'NumTrials',NumTrial,'OutSpec','delta')

'AmericanOpt' — Option type

0 (European/Bermuda) (default) | values [0,1]

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

'NumPeriods' — Number of simulation periods per trial

100 (default) | nonnegative integer

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

'NumTrials' — Number of independent sample paths (simulation trials)

1000 (default) | nonnegative integer

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

Data Types: double

'Z' — Time series array of dependent random variates

vector

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

'Antithetic' — Indicator for antithetic sampling

false (default) | scalar logical flag with value of true or false

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

Data Types: logical

'OutSpec' — Define outputs

{'Price'} (default) | character vector with values 'Price', 'Delta', 'Gamma', 'Vega', 'Lambda', 'Rho', 'Theta', and 'All' | cell array of character vectors with values 'Price', 'Delta', 'Gamma', 'Vega', 'Lambda', 'Rho', 'Theta', and 'All'

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

'UndIdx' — Index of the underlying instrument to compute the sensitivity

[] (default) | scalar numeric

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

PriceSens — Expected prices or sensitivities for basket option

matrix

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

Paths — Simulated paths of correlated state variables

vector

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.

Times — Observation times associated with simulated paths

vector

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.

Z — Time series array of dependent random variates

vector

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.

More About

Basket Option

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.

For more information, see Basket Option.

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.