basketsensbyls
Calculate price and sensitivities for European or American basket options using Monte Carlo simulations
Syntax
Description
[
calculates price and sensitivities for European or American basket options using the
Longstaff-Schwartz model. PriceSens
,Paths
,Times
,Z
] = basketsensbyls(RateSpec
,BasketStockSpec
,OptSpec
,Strike
,Settle
,ExerciseDates
)
For American options, the Longstaff-Schwartz least squares method is used to calculate the early exercise premium.
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 = 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); % 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
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) or1
-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
datetime scalar | string scalar | date character vector
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.
ExerciseDates
— Option exercise dates
datetime array | string array | date character vector
Option exercise dates, specified as a datetime array, string array, or date character vectors:
For a European or Bermuda option,
ExerciseDates
is a1
-by-1
(European) or1
-by-NSTRIKES
(Bermuda) vector of exercise dates. For a European option, there is only oneExerciseDate
on the option expiry date.For an American option,
ExerciseDates
is a1
-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 ifExerciseDates
is1
-by-1
, the option exercises between theSettle
date and the single listedExerciseDate
.
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.
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/Bermuda1
— 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.
Version History
Introduced in R2009bR2022b: Serial date numbers not recommended
Although basketsensbyls
supports serial date numbers,
datetime
values are recommended instead. The
datetime
data type provides flexible date and time
formats, storage out to nanosecond precision, and properties to account for time
zones and daylight saving time.
To convert serial date numbers or text to datetime
values, use the datetime
function. For example:
t = datetime(738427.656845093,"ConvertFrom","datenum"); y = year(t)
y = 2021
There are no plans to remove support for serial date number inputs.
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