# swaptionbycir

Price swaption from Cox-Ingersoll-Ross interest-rate tree

## Syntax

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

## Description

example

``````[Price,PriceTree] = swaptionbycir(CIRTree,OptSpec,Strike,ExerciseDates,Spread,Settle,Maturity)``` prices swaption with a Cox-Ingersoll-Ross (CIR) tree using a CIR++ model with the Nawalka-Beliaeva (NB) approach.```

example

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

## Examples

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Define a 3-year put swaption.

```Rates =0.075 * ones (10,1); Compounding = 2; StartDates = [datetime(2017,1,1);datetime(2017,7,1);datetime(2018,1,1);datetime(2018,7,1);datetime(2019,1,1);datetime(2019,7,1);datetime(2020,1,1);datetime(2020,7,1);datetime(2021,1,1);datetime(2021,7,1)]; EndDates = [datetime(2017,7,1);datetime(2018,1,1);datetime(2018,7,1);datetime(2019,1,1);datetime(2019,7,1);datetime(2020,1,1);datetime(2020,7,1);datetime(2021,1,1);datetime(2021,7,1);datetime(2022,1,1)]; ValuationDate = datetime(2017,1,1); ```

Create a `RateSpec` using the `intenvset` function.

`RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates', ValuationDate, 'EndDates',EndDates,'Rates', Rates, 'Compounding', Compounding); `

Create a `CIR` tree.

```NumPeriods = length(EndDates); Alpha = 0.03; Theta = 0.02; Sigma = 0.1; Maturity = datetime(2023,1,1); CIRTimeSpec = cirtimespec(ValuationDate, Maturity, NumPeriods); CIRVolSpec = cirvolspec(Sigma, Alpha, Theta); CIRT = cirtree(CIRVolSpec, RateSpec, CIRTimeSpec)```
```CIRT = struct with fields: FinObj: 'CIRFwdTree' VolSpec: [1x1 struct] TimeSpec: [1x1 struct] RateSpec: [1x1 struct] tObs: [0 0.6000 1.2000 1.8000 2.4000 3 3.6000 4.2000 4.8000 5.4000] dObs: [736696 736915 737134 737353 737572 737791 738010 738229 738448 738667] FwdTree: {1x10 cell} Connect: {[3x1 double] [3x3 double] [3x5 double] [3x7 double] [3x9 double] [3x11 double] [3x13 double] [3x14 double] [3x15 double]} Probs: {[3x1 double] [3x3 double] [3x5 double] [3x7 double] [3x9 double] [3x11 double] [3x13 double] [3x14 double] [3x15 double]} ```

Use the following arguments for a 1-year swap and a 3-year swaption.

```ExerciseDates = datetime(2020,1,1); SwapSettlement = ExerciseDates; SwapMaturity = datetime(2022,1,1); Spread = 0; SwapReset = 2 ; Principal = 100; OptSpec = 'put'; Strike= 0.04; Basis=1;```

Price the swaption.

```[Price,PriceTree] = swaptionbycir(CIRT,OptSpec,Strike,ExerciseDates,Spread,SwapSettlement,SwapMaturity,'SwapReset',SwapReset, ... 'Basis',Basis,'Principal',Principal)```
```Price = 3.1537 ```
```PriceTree = struct with fields: FinObj: 'CIRPriceTree' PTree: {1x11 cell} tObs: [0 0.6000 1.2000 1.8000 2.4000 3 3.6000 4.2000 4.8000 5.4000 6] Connect: {[3x1 double] [3x3 double] [3x5 double] [3x7 double] [3x9 double] [3x11 double] [3x13 double] [3x14 double] [3x15 double]} Probs: {[3x1 double] [3x3 double] [3x5 double] [3x7 double] [3x9 double] [3x11 double] [3x13 double] [3x14 double] [3x15 double]} ```

## Input Arguments

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

Data Types: `struct`

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

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

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 a `NINST`-by-`2` vector using a datetime array, string array, 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`.

To support existing code, `swaptionbycir` also accepts serial date numbers as inputs, but they are not recommended.

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 using a datetime array, string array, or date character vectors. The `Settle` date for every swaption is set to the `ValuationDate` of the CIR tree. The swap argument `Settle` is ignored. The underlying swap starts at the maturity of the swaption.

To support existing code, `swaptionbycir` also accepts serial date numbers as inputs, but they are not recommended.

Maturity date for each swap, specified as a `NINST`-by-`1` vector using a datetime array, string array, or date character vectors.

To support existing code, `swaptionbycir` also accepts serial date numbers as inputs, but they are not recommended.

### Name-Value Arguments

Specify optional pairs of arguments as `Name1=Value1,...,NameN=ValueN`, where `Name` is the argument name and `Value` is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose `Name` in quotes.

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

Option type, specified as the comma-separated pair consisting of `'AmericanOpt'`and `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 the comma-separated pair consisting of `'SwapReset'` and 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, and 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 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.

• 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 the comma-separated pair consisting of `'Principal'` and a `NINST`-by-`1` vector.

Data Types: `double`

## 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.

• `PriceTree.Connect` contains the connectivity vectors. Each element in the cell array describes how nodes in that level connect to the next. For a given tree level, there are `NumNodes` elements in the vector, and they contain the index of the node at the next level that the middle branch connects to. Subtracting 1 from that value indicates where the up-branch connects to, and adding 1 indicated where the down branch connects to.

• `PriceTree.Probs` contains the probability arrays. Each element of the cell array contains the up, middle, and down transition probabilities for each node of the level.

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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.

 Cox, J., Ingersoll, J., and S. Ross. "A Theory of the Term Structure of Interest Rates." Econometrica. Vol. 53, 1985.

 Brigo, D. and F. Mercurio. Interest Rate Models - Theory and Practice. Springer Finance, 2006.

 Hirsa, A. Computational Methods in Finance. CRC Press, 2012.

 Nawalka, S., Soto, G., and N. Beliaeva. Dynamic Term Structure Modeling. Wiley, 2007.

 Nelson, D. and K. Ramaswamy. "Simple Binomial Processes as Diffusion Approximations in Financial Models." The Review of Financial Studies. Vol 3. 1990, pp. 393–430.