# Documentation

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# swaptionbybk

Price swaption from Black-Karasinski interest-rate tree

## Syntax

``````[Price,PriceTree] = swaptionbybk(BKTree,OptSpec,Strike,ExerciseDates,Spread,Settle,Maturity)``````
``````[Price,PriceTree] = swaptionbybk(___,Name,Value)``````

## Description

example

``````[Price,PriceTree] = swaptionbybk(BKTree,OptSpec,Strike,ExerciseDates,Spread,Settle,Maturity)``` prices swaption using a Black-Karasinski tree.```

example

``````[Price,PriceTree] = swaptionbybk(___,Name,Value)``` adds optional name-value pair arguments.```

## Examples

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This example shows how to price a 4-year call and put swaption using a BK interest-rate tree, assuming the interest rate is fixed at 7% annually.

```Rates =0.07 * ones (10,1); Compounding = 2; StartDates = ['jan-1-2007';'jul-1-2007';'jan-1-2008';'jul-1-2008';'jan-1-2009'; ... 'jul-1-2009'; 'jan-1-2010'; 'jul-1-2010';'jan-1-2011';'jul-1-2011']; EndDates =['jul-1-2007';'jan-1-2008';'jul-1-2008';'jan-1-2009';'jul-1-2009'; ... 'jan-1-2010'; 'jul-1-2010';'jan-1-2011';'jul-1-2011';'jan-1-2012']; ValuationDate = 'jan-1-2007'; % define the RateSpec RateSpec = intenvset('Rates', Rates, 'StartDates', StartDates, 'EndDates', EndDates,... 'Compounding', Compounding); % use BKVolSpec to compute the interest-rate volatility Volatility = 0.10*ones(10,1); AlphaCurve = 0.05*ones(10,1); AlphaDates = EndDates; BKVolSpec = bkvolspec(ValuationDate, EndDates, Volatility, AlphaDates, AlphaCurve); % use BKTimeSpec to specify the structure of the time layout for the BK interest-rate tree BKTimeSpec = bktimespec(ValuationDate, EndDates, Compounding); % build the BK tree BKTree = bktree(BKVolSpec, RateSpec, BKTimeSpec); % use the following arguments for a 1-year swap and 4-year swaption ExerciseDates = 'jan-1-2011'; SwapSettlement = ExerciseDates; SwapMaturity = 'jan-1-2012'; Spread = 0; SwapReset = 2 ; Principal = 100; OptSpec = {'call' ;'put'}; Strike= [ 0.07 ; 0.0725]; Basis=1; % price the swaption PriceSwaption = swaptionbybk(BKTree, OptSpec, Strike, ExerciseDates, ... Spread, SwapSettlement, SwapMaturity, 'SwapReset', SwapReset, 'Basis', Basis, ... 'Principal', Principal)```
```PriceSwaption = 0.3634 0.4798 ```

This example shows how to price a 4-year call and put swaption with receing and paying legs using a BK interest-rate tree, assuming the interest rate is fixed at 7% annually. Build a tree with the following data.

```Rates =0.07 * ones (10,1); Compounding = 2; StartDates = ['jan-1-2007';'jul-1-2007';'jan-1-2008';'jul-1-2008';'jan-1-2009'; ... 'jul-1-2009'; 'jan-1-2010'; 'jul-1-2010';'jan-1-2011';'jul-1-2011']; EndDates =['jul-1-2007';'jan-1-2008';'jul-1-2008';'jan-1-2009';'jul-1-2009'; ... 'jan-1-2010'; 'jul-1-2010';'jan-1-2011';'jul-1-2011';'jan-1-2012']; ValuationDate = 'jan-1-2007'; ```

Define the `RateSpec`.

```RateSpec = intenvset('Rates', Rates, 'StartDates', StartDates, 'EndDates', EndDates,... 'Compounding', Compounding) ```
```RateSpec = struct with fields: FinObj: 'RateSpec' Compounding: 2 Disc: [10x1 double] Rates: [10x1 double] EndTimes: [10x1 double] StartTimes: [10x1 double] EndDates: [10x1 double] StartDates: [10x1 double] ValuationDate: 733043 Basis: 0 EndMonthRule: 1 ```

Use `BKVolSpec` to compute interest-rate volatility.

```Volatility = 0.10*ones(10,1); AlphaCurve = 0.05*ones(10,1); AlphaDates = EndDates; BKVolSpec = bkvolspec(ValuationDate, EndDates, Volatility, AlphaDates, AlphaCurve); ```

Use `BKTimeSpec` to specify the structure of the time layout for a BK tree.

```BKTimeSpec = bktimespec(ValuationDate, EndDates, Compounding); ```

Build the BK tree.

```BKTree = bktree(BKVolSpec, RateSpec, BKTimeSpec); ```

Define the arguments for a 1-year swap and 4-year swaption.

```ExerciseDates = 'jan-1-2011'; SwapSettlement = ExerciseDates; SwapMaturity = 'jan-1-2012'; Spread = 0; SwapReset = [2 2]; % 1st column represents swaption receiving leg, 2nd column represents swaption paying leg Principal = 100; OptSpec = {'call' ;'put'}; Strike= [ 0.07 ; 0.0725]; Basis= [1 3]; % 1st column represents swaption receiving leg, 2nd column represents swaption paying leg ```

Price the swaption.

```PriceSwaption = swaptionbybk(BKTree, OptSpec, Strike, ExerciseDates, ... Spread, SwapSettlement, SwapMaturity, 'SwapReset', SwapReset, 'Basis', Basis, ... 'Principal', Principal) ```
```PriceSwaption = 0.3634 0.4798 ```

## Input Arguments

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Interest-rate tree structure, specified by using `bKtree`.

Data Types: `struct`

Definition of the option as `'call'` or `'put'`, specified as a `NINST`-by-`1` cell array of character vectors. For more information, see Definitions.

Data Types: `char` | `cell`

Strike swap rate values, specified as a `NINST`-by-`1` vector.

Data Types: `double`

Exercise dates for the swaption, specified as a `NINST`-by-`1` vector or `NINST`-by-`2` using serial date numbers or date character vectors, depending on the option type.

• For a European option, `ExerciseDates` are a `NINST`-by-`1` vector of exercise dates. Each row is the schedule for one option. When using a European option, there is only one `ExerciseDate` on the option expiry date.

• For an American option, `ExerciseDates` are a `NINST`-by-`2` vector of exercise date boundaries. For each instrument, the option can be exercised on any coupon date between or including the pair of dates on that row. If only one non-`NaN` date is listed, or if `ExerciseDates` is `NINST`-by-`1`, the option can be exercised between the `ValuationDate` of the tree and the single listed `ExerciseDate`.

Data Types: `double` | `char` | `cell`

Number of basis points over the reference rate, specified as a `NINST`-by-`1` vector.

Data Types: `double`

Settlement date (representing the settle date for each swap), specified as a `NINST`-by-`1` vector of serial date numbers or date character vectors. The `Settle` date for every swaption is set to the `ValuationDate` of the BK Tree. The swap argument `Settle` is ignored. The underlying swap starts at the maturity of the swaption.

Data Types: `double` | `char`

Maturity date for each swap, specified as a `NINST`-by-`1` vector of dates using serial date numbers or date character vectors.

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 single quotes (`' '`). You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Example: ```[Price,PriceTree] = swaptionbybk(BKTree,OptSpec, ExerciseDates,Spread,Settle,Maturity,'SwapReset',4,'Basis',5,'Principal',10000)```

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(Optional) Option type, specified as `NINST`-by-`1` positive integer flags with values:

• `0` — European

• `1` — American

Data Types: `double`

Reset frequency per year for the underlying swap, specified as a `NINST`-by-`1` vector or `NINST`-by-`2` matrix representing the reset frequency per year for each leg. If `SwapReset` is `NINST`-by-`2`, the first column represents the receiving leg, while the second column represents the paying leg.

Data Types: `double`

Day-count basis representing the basis used when annualizing the input forward rate tree for each instrument, specified as a `NINST`-by-`1` vector or `NINST`-by-`2` matrix representing the basis for each leg. If `Basis` is `NINST`-by-`2`, the first column represents the receiving leg, while the second column represents the paying leg.

• 0 = actual/actual

• 1 = 30/360 (SIA)

• 2 = actual/360

• 3 = actual/365

• 4 = 30/360 (PSA)

• 5 = 30/360 (ISDA)

• 6 = 30/360 (European)

• 7 = actual/365 (Japanese)

• 8 = actual/actual (ICMA)

• 9 = actual/360 (ICMA)

• 10 = actual/365 (ICMA)

• 11 = 30/360E (ICMA)

• 12 = actual/365 (ISDA)

• 13 = BUS/252

Data Types: `double`

Notional principal amount, specified as a `NINST`-by-`1` vector.

Data Types: `double`

Derivatives pricing options structure, specified using `derivset`.

Data Types: `struct`

## Output Arguments

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Expected prices of the swaptions at time 0, returned as a `NINST`-by-`1` vector.

Tree structure of instrument prices, returned as a MATLAB® structure of trees containing vectors of swaption instrument prices and a vector of observation times for each node. Within `PriceTree`:

• `PriceTree.PTree` contains the clean prices.

• `PriceTree.tObs` contains the observation times.

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### Call Swaption

A Call swaption or Payer swaption allows the option buyer to enter into an interest-rate swap in which the buyer of the option pays the fixed rate and receives the floating rate.

### Put Swaption

A Put swaption or Receiver swaption allows the option buyer to enter into an interest-rate swap in which the buyer of the option receives the fixed rate and pays the floating rate.