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# swaptionbynormal

Price swaptions using Normal or Bachelier option pricing model

## Syntax

``Price = swaptionbynormal(RateSpec,OptSpec,Strike,Settle,ExerciseDates,Maturity,Volatility)``
``Price = swaptionbynormal(___,Name,Value)``

## Description

example

````Price = swaptionbynormal(RateSpec,OptSpec,Strike,Settle,ExerciseDates,Maturity,Volatility)` prices swaptions using the Normal or Bachelier option pricing model.```

example

````Price = swaptionbynormal(___,Name,Value)` adds optional name-value pair arguments. ```

## Examples

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Define the zero curve, and create a `RateSpec`.

```Settle = datenum('20-Jan-2016'); ZeroTimes = [.5 1 2 3 4 5 7 10 20 30]'; ZeroRates = [0.0052 0.0055 0.0061 0.0073 0.0094 0.0119 0.0168 0.0222 0.0293 0.0307]'; ZeroDates = datemnth(Settle,12*ZeroTimes); RateSpec = intenvset('StartDate',Settle,'EndDates',ZeroDates,'Rates',ZeroRates)```
```RateSpec = struct with fields: FinObj: 'RateSpec' Compounding: 2 Disc: [10x1 double] Rates: [10x1 double] EndTimes: [10x1 double] StartTimes: [10x1 double] EndDates: [10x1 double] StartDates: 736349 ValuationDate: 736349 Basis: 0 EndMonthRule: 1 ```

Define the swaption.

```ExerciseDate = datenum('20-Jan-2021'); Maturity = datenum('20-Jan-2026'); OptSpec = 'call'; LegReset = [1 1];```

Compute the par swap rate.

`[~,ParSwapRate] = swapbyzero(RateSpec,[NaN 0],Settle,Maturity,'LegReset',LegReset)`
```ParSwapRate = 0.0216 ```
```Strike = ParSwapRate; BlackVol = .3; NormalVol = BlackVol*ParSwapRate;```

Price with Black volatility.

`Price = swaptionbyblk(RateSpec,OptSpec,Strike,Settle,ExerciseDate,Maturity,BlackVol)`
```Price = 5.9756 ```

Price with Normal volatility.

`Price_Normal = swaptionbynormal(RateSpec,OptSpec,Strike,Settle,ExerciseDate,Maturity,NormalVol)`
```Price_Normal = 5.5537 ```

Create a `RateSpec`.

```Rate = 0.06; Compounding = -1; ValuationDate = 'Jan-1-2010'; EndDates = 'Jan-1-2020'; Basis = 1; RateSpec = intenvset('ValuationDate', ValuationDate,'StartDates', ValuationDate, ... 'EndDates', EndDates, 'Rates', Rate, 'Compounding', Compounding, 'Basis', Basis);```

Define the swaption.

```ExerciseDate = datenum('20-Jan-2021'); Maturity = datenum('20-Jan-2026'); Settle = 'Jan-1-2010'; OptSpec = 'call'; Strike = .09; NormalVol = .03; Reset = [1 4]; % 1st column represents receiving leg, 2nd column represents paying leg Basis = [1 7]; % 1st column represents receiving leg, 2nd column represents paying leg```

Price with Normal volatility.

`Price_Normal = swaptionbynormal(RateSpec,OptSpec,Strike,Settle,ExerciseDate,Maturity,NormalVol,'Reset',Reset,'Basis',Basis)`
```Price_Normal = 5.9084 ```

Define the `RateSpec`.

```Settle = datenum('20-Jan-2016'); ZeroTimes = [.5 1 2 3 4 5 7 10 20 30]'; ZeroRates = [0.0052 0.0055 0.0061 0.0073 0.0094 0.0119 0.0168 0.0222 0.0293 0.0307]'; ZeroDates = datemnth(Settle,12*ZeroTimes); RateSpec = intenvset('StartDate',Settle,'EndDates',ZeroDates,'Rates',ZeroRates)```
```RateSpec = struct with fields: FinObj: 'RateSpec' Compounding: 2 Disc: [10x1 double] Rates: [10x1 double] EndTimes: [10x1 double] StartTimes: [10x1 double] EndDates: [10x1 double] StartDates: 736349 ValuationDate: 736349 Basis: 0 EndMonthRule: 1 ```

Define the swaption instrument and price with `swaptionbyblk`.

```ExerciseDate = datenum('20-Jan-2021'); Maturity = datenum('20-Jan-2026'); OptSpec = 'call'; [~,ParSwapRate] = swapbyzero(RateSpec,[NaN 0],Settle,Maturity,'StartDate',ExerciseDate)```
```ParSwapRate = 0.0326 ```
```Strike = ParSwapRate; BlackVol = .3; NormalVol = BlackVol*ParSwapRate; Price = swaptionbyblk(RateSpec,OptSpec,Strike,Settle,ExerciseDate,Maturity,BlackVol)```
```Price = 3.6908 ```

Price the swaption instrument using `swaptionbynormal`.

`Price_Normal = swaptionbynormal(RateSpec,OptSpec,Strike,Settle,ExerciseDate,Maturity,NormalVol)`
```Price_Normal = 3.7602 ```

Price the swaption instrument using `swaptionbynormal` for a negative strike.

` Price_Normal = swaptionbynormal(RateSpec,OptSpec,-.005,Settle,ExerciseDate,Maturity,NormalVol)`
```Price_Normal = 16.3674 ```

## 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`.

If the discount curve for the paying leg is different than the receiving leg, `RateSpec` can be a `NINST`-by-`2` input variable of `RateSpec`s, with the second input being the discount curve for the paying leg. If only one curve is specified, then it is used to discount both legs.

Data Types: `struct`

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

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.

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.

Data Types: `char` | `cell`

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

Data Types: `double`

Settlement date (representing the settle date for each swaption), specified as a `NINST`-by-`1` vector of serial date numbers, or cell array of date character vectors, datetime objects, or string objects. `Settle` must not be later than `ExerciseDates`.

The `Settle` date input for `swaptionbynormal` is the valuation date on which the swaption (an option to enter into a swap) is priced. The swaption buyer pays this price on this date to hold the swaption.

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

Dates on which the swaption expires and the underlying swap starts, specified as a `NINST`-by-`1` vector of serial date numbers, or cell array of date character vectors, datetime objects, or string objects. There is only one `ExerciseDate` on the option expiry date. This is also the `StartDate` of the underlying forward swap.

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

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

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

Volatilities values (for normal volatility), specified as a `NINST`-by-`1` vector of numeric values.

For more information on the Normal model, see Work with Negative Interest Rates.

Data Types: `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: `Price = swaptionbynormal(OISCurve,OptSpec,Strike,Settle,ExerciseDate,Maturity,NormalVol,'Reset',4)`

Reset frequency per year for the underlying forward swap, specified as the comma-separated pair consisting of `'Reset'` and a `NINST`-by-`1` vector or `NINST`-by-`2` matrix representing the reset frequency per year for each leg. If `Reset` 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 of the instrument representing the basis used when annualizing the input term structure, specified as the comma-separated pair consisting of `'Basis'` and 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.

Values are:

• 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

For more information, see basis.

Data Types: `double`

Notional principal amount, specified as the comma-separated pair consisting of `'Principal'` and a `NINST`-by-`1` vector.

Data Types: `double`

The rate curve to be used in projecting the future cash flows, specified as the comma-separated pair consisting of `'ProjectionCurve'` and a rate curve structure. This structure must be created using `intenvset`. Use this optional input if the forward curve is different from the discount curve.

Data Types: `struct`

## Output Arguments

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

## More About

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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.

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