Documentation

Calculate European spread option prices or sensitivities using Bjerksund-Stensland pricing model

## Syntax

``PriceSens = spreadbybjs(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr)``
``PriceSens = spreadsensbybjs(___,Name,Value)``

## Description

example

````PriceSens = spreadbybjs(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr)` returns the European spread option prices or sensitivities using the Bjerksund-Stensland pricing model.```
````PriceSens = spreadsensbybjs(___,Name,Value)` 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 Kirk model.

```OutSpec = {'Price', 'Delta', 'Gamma'}; [Price, Delta, Gamma] = spreadsensbybjs(RateSpec, StockSpec1, StockSpec2, Settle, ... Maturity, OptSpec, Strike, Corr, 'OutSpec', OutSpec)```
```Price = 11.2000 ```
```Delta = 1×2 0.6737 -0.6082 ```
```Gamma = 1×2 0.0190 0.0216 ```

## 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.

If `Strike` is equal to zero the function computes the price and sensitivities of an exchange option.

Data Types: `single` | `double`

Correlation between underlying asset prices, specified as an integer using a `NINST`-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`.

Example: ```PriceSens = spreadsensbykirk(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr,OutSpec,{'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

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Expected prices or sensitivities values (defined by `OutSpec`) of the spread option, returned as a `NINST`-by-`1` or `NINST`-by-`2` 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] Bjerksund, Petter, Stensland, Gunnar. “Closed form spread option valuation.” Department of Finance, NHH, 2006.