spreadsensbyls

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

Syntax

PriceSens = spreadsensbyls(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr)

PriceSens = spreadsensbyls(___,Name,Value)

[PriceSens,Paths,Times,Z] = spreadsensbyls(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr)

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

Description

PriceSens = spreadsensbyls(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.

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

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

[PriceSens,Paths,Times,Z] = spreadsensbyls(___,Name,Value) returns the PriceSens, Paths, Times, and Z and adds optional name-value pair arguments.

Examples

Compute the Price and Sensitivities of a Spread Option Using Monte Carlo Simulation

Define the spread option dates.

Settle = '01-Jun-2012';
Maturity = '01-Sep-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;

Define the spread option.

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: 735113
       StartDates: 735021
    ValuationDate: 735021
            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 and sensitivities using Monte Carlo simulation based on the Longstaff-Schwartz model.

OutSpec = {'Price', 'Delta', 'Gamma'};
[Price, Delta, Gamma] = spreadsensbyls(RateSpec, StockSpec1, StockSpec2, ...
Settle, Maturity, OptSpec, Strike, Corr, 'OutSpec', OutSpec)

Price = 11.0799

Delta = 1×2

    0.6626   -0.5972

Gamma = 1×2

    0.0209    0.0240

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

StockSpec1 — Stock specification for underlying asset 1

structure

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

StockSpec2 — Stock specification for underlying asset 2

structure

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

Settle — Settlement date for spread option

date character vector | nonnegative scalar integer

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

Data Types: char | double

Maturity — Maturity date for spread option

date character vector | nonnegative scalar integer

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

Data Types: char | double

OptSpec — Definition of option

character vector with values 'call' or 'put'

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

Data Types: char

Strike — Option strike price value

nonnegative scalar integer

Option strike price value, specified as a scalar integer.

Data Types: single | double

Corr — Correlation between underlying asset prices

scalar integer

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.

Example: PriceSens = spreadbyls(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr,'AmericanOpt',1)

'AmericanOpt' — Option type

0 European (default) | scalar with values [0,1]

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

• 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

'NumTrials' — Number of independent sample paths

1000 (default) | nonnegative scalar integer

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

Data Types: single | double

'NumPeriods' — Number of simulation periods per trial

100 (default) | nonnegative scalar integer

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

Data Types: single | 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-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

'Antithetic' — Indicator for antithetic sampling

false (default) | 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 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

PriceSens — Expected price or sensitivities of spread option

scalar

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

Paths — Simulated paths of correlated state variables

vector

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.

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-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.

More About

Spread Option

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:

$$\max (X1-X2-K,0)$$

where:

K is the strike price.

For more information, see Spread Option.

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.