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Calculate price and sensitivities of European or American spread options using finite difference method

## Description

example

PriceSens = spreadsensbyfd(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr) returns the price and sensitivities of European or American call or put spread options using the Alternate Direction Implicit (ADI) finite difference method. The spread is between the asset defined in StockSpec1 minus the asset defined in StockSpec2.

[PriceSens,PriceGrid,AssetPrice1,AssetPrice2,Times] = spreadsensbyfd(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr) returns the PriceSens, PriceGrid, AssetPrice1, AssetPrice2, and Times for European or American call or put spread options using the Alternate Direction Implicit (ADI) finite difference method. The spread is between the asset defined in StockSpec1 minus the asset defined in StockSpec2.

[PriceSens,PriceGrid,AssetPrice1,AssetPrice2,Times] = spreadsensbyfd(___,Name,Value) returns the PriceSens, PriceGrid, AssetPrice1, AssetPrice2, and Times and adds optional name-value pair arguments.

## Examples

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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;

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 based on the Alternate Direction Implicit (ADI) finite difference method.

OutSpec = {'Price', 'Delta', 'Gamma'};
[Price, Delta, Gamma, PriceGrid, AssetPrice1, AssetPrice2, Times] = ...
Maturity, OptSpec, Strike, Corr, 'OutSpec', OutSpec);

Display the price and sensitivities.

Price
Price = 11.1998
Delta
Delta = 1×2

0.6736   -0.6082

Gamma
Gamma = 1×2

0.0190    0.0214

Plot the finite difference grid.

mesh(AssetPrice1, AssetPrice2, PriceGrid(:, :, 1)');
title('Spread Option Prices for Range of Underlying Prices');
xlabel('Price of underlying asset 1');
ylabel('Price of underlying asset 2');

## 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 dates for the spread option, specified as date character vectors or as serial date numbers using a NINST-by-1 vector or cell array of character vector dates.

Data Types: char | cell | double

Maturity date for spread option, specified as date character vectors or as serial date numbers using a NINST-by-1 vector or cell array of character vector dates.

Data Types: char | cell | double

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

Data Types: char | cell

Option strike price values, specified as an integer using a NINST-by-1 vector of strike price values.

Data Types: single | double

Correlation between underlying asset prices, specified as an integer using aNINST-by-1 vector.

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.

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

Minimum price for price grid boundary, specified as the comma-separated pair consisting of 'AssetPriceMin' and a 1-by-2 array. The first entry in the array corresponds to the first asset defined by StockSpec1 and the second entry corresponds to the second asset defined by StockSpec2.

For the finite difference method, the composition of the grid affects the quality of the output and the execution time. It is highly recommended to use the optional arguments AssetPriceMin, AssetPriceMax, PriceGridSize, and TimeGridSize to control the composition of the grid to ensure the quality of the output and a reasonable execution time.

Data Types: single | double

Maximum price for price grid boundary, specified as the comma-separated pair consisting of 'AssetPriceMax' and a 1-by-2 array. The first entry in the array corresponds to the first asset defined by StockSpec1 and the second entry corresponds to the second asset defined by StockSpec2.

For the finite difference method, the composition of the grid affects the quality of the output and the execution time. It is highly recommended to use the optional arguments AssetPriceMin, AssetPriceMax, PriceGridSize, and TimeGridSize to control the composition of the grid to ensure the quality of the output and a reasonable execution time.

Data Types: single | double

Size for finite difference grid, specified as the comma-separated pair consisting of 'PriceGridSize' and a 1-by-2 array. The first entry corresponds to the first asset defined by StockSpec1 and the second entry corresponds to the second asset defined by StockSpec2.

For the finite difference method, the composition of the grid affects the quality of the output and the execution time. It is highly recommended to use the optional argumentsAssetPriceMax, PriceGridSize, and TimeGridSize to control the composition of the grid to ensure the quality of the output and a reasonable execution time.

Data Types: single | double

Size of time grid for finite difference grid, specified as the comma-separated pair consisting of 'TimeGridSize' and a nonnegative integer.

For the finite difference method, the composition of the grid affects the quality of the output and the execution time. It is highly recommended to use the optional argumentsAssetPriceMax, PriceGridSize, and TimeGridSize to control the composition of the grid to ensure the quality of the output and a reasonable execution time.

Data Types: single | double

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

• 0 — European

• 1 — American

Data Types: single | double

## Output Arguments

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

Grid containing prices calculated by finite difference method, returned as a 3-D grid with a size of PriceGridSize(1) * PriceGridSize(2) * TimeGridSize. The price for t = 0 is contained in PriceGrid(:, :, 1).

Prices for first asset defined by StockSpec1, corresponding to the first dimension of PriceGrid, returned as a vector.

Prices for second asset defined by StockSpec2, corresponding to the second dimension of PriceGrid, returned as a vector.

Times corresponding to third dimension of PriceGrid, returned as a vector.

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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.

[2] Villeneuve, S., Zanette, A. “Parabolic ADI Methods for Pricing American Options on Two Stocks.” Mathematics of Operations Research. Vol. 27, No. 1, pp. 121–149, INFORMS, 2002.

[3] Ikonen, S., Toivanen, J. Efficient Numerical Methods for Pricing American Options Under Stochastic Volatility. Wiley InterScience, 2007.