Price options on floating-rate notes for Black-Karasinski interest-rate tree

`[`

prices options on floating-rate notes from a Black-Karasinski interest rate tree.
`Price`

,`PriceTree`

]
= optfloatbybdt(`BKTree`

,`OptSpec`

,`Strike`

,`ExerciseDates`

,`AmericanOpt`

,`Spread`

,`Settle`

,`Maturity`

)`optfloatbybk`

computes prices of options on vanilla floating-rate
notes.

`[`

adds optional name-value pair arguments. `Price`

,`PriceTree`

]
= optfloatbybdt(___,`Name,Value`

)

Define the interest-rate term structure.

Rates = [0.03;0.034;0.038;0.04]; ValuationDate = 'Jan-1-2012'; StartDates = ValuationDate; EndDates = {'Jan-1-2013'; 'Jan-1-2014'; 'Jan-1-2015'; 'Jan-1-2016'}; Compounding = 1;

Create the `RateSpec`

.

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

`RateSpec = `*struct with fields:*
FinObj: 'RateSpec'
Compounding: 1
Disc: [4x1 double]
Rates: [4x1 double]
EndTimes: [4x1 double]
StartTimes: [4x1 double]
EndDates: [4x1 double]
StartDates: 734869
ValuationDate: 734869
Basis: 0
EndMonthRule: 1

Build the BK tree.

VolDates = ['1-Jan-2013'; '1-Jan-2014'; '1-Jan-2015';'1-Jan-2016']; VolCurve = 0.01; AlphaDates = '01-01-2016'; AlphaCurve = 0.1; BKVolSpec = bkvolspec(RateSpec.ValuationDate,VolDates,VolCurve,... AlphaDates,AlphaCurve); BKTimeSpec = bktimespec(RateSpec.ValuationDate,VolDates,Compounding); BKT = bktree(BKVolSpec,RateSpec,BKTimeSpec)

`BKT = `*struct with fields:*
FinObj: 'BKFwdTree'
VolSpec: [1x1 struct]
TimeSpec: [1x1 struct]
RateSpec: [1x1 struct]
tObs: [0 1 2 3]
dObs: [734869 735235 735600 735965]
CFlowT: {[4x1 double] [3x1 double] [2x1 double] [4]}
Probs: {[3x1 double] [3x3 double] [3x5 double]}
Connect: {[2] [2 3 4] [2 3 4 5 6]}
FwdTree: {[1.0300] [1.0387 1.0380 1.0373] [1x5 double] [1x7 double]}

The floater instrument has a spread of 10, a period of one year, and matures on Jan-1-2016.

Spread = 10; Settle = 'Jan-1-2012'; Maturity = 'Jan-1-2016'; Period = 1;

Define the option for the floating-rate note.

OptSpec = {'call'}; Strike = 95; ExerciseDates = 'Jan-1-2016'; AmericanOpt = [0;1];

Compute the price of the call options.

```
Price = optfloatbybk(BKT,OptSpec,Strike,ExerciseDates,AmericanOpt,...
Spread,Settle,Maturity)
```

`Price = `*2×1*
4.2740
5.3655

`BKTree`

— Interest-rate tree structurebinomial tree structure

Interest-rate tree specified as a structure by using `bktree`

.

**Data Types: **`struct`

`OptSpec`

— Definition of option character vector | cell array of character vectors

Definition of option as `'call'`

or `'put'`

specified
as a `NINST`

-by-`1`

cell array of
character vectors for `'call'`

or `'put'`

.

**Data Types: **`cell`

| `char`

`Strike`

— Option strike price valuesnonnegative integer | vector of nonnegative integers

Option strike price values specified nonnegative integers using
as `NINST`

-by-`NSTRIKES`

vector
of strike price values.

**Data Types: **`single`

| `double`

`ExerciseDates`

— Exercise date for option (European, Bermuda, or American) serial date number | vector of serial date numbers | date character vector | cell array of date character vectors

Exercise date for option (European, Bermuda, or American) specified
as serial date numbers or date character vectors using a `NINST`

-by-`NSTRIKES`

or `NINST`

-by-`2`

vector
of for the option exercise dates.

If a European or Bermuda option, the

`ExerciseDates`

is a`1`

-by-`1`

(European) or`1`

-by-`NSTRIKES`

(Bermuda) vector of exercise dates. For a European option, there is only one`ExerciseDate`

on the option expiry date.If an American option, then

`ExerciseDates`

is a`1`

-by-`2`

vector of exercise date boundaries. The option exercises on any date between or including the pair of dates on that row. If there is only one non-`NaN`

date, or if`ExerciseDates`

is`1`

-by-`1`

, the option exercises between the`Settle`

date and the single listed`ExerciseDate`

.

**Data Types: **`double`

| `char`

| `cell`

`AmericanOpt`

— Option typescalar | vector of positive integers

`[0,1]`

Option type specified as `NINST`

-by-`1`

positive
integer scalar flags with values:

`0`

— European/Bermuda`1`

— American

**Data Types: **`single`

| `double`

`Spread`

— Number of basis points over the reference ratenonnegative integer | vector of nonnegative integers

Number of basis points over the reference rate specified as
a vector of nonnegative integers for the number of instruments (`NINST`

)-by-`1`

).

**Data Types: **`single`

| `double`

`Settle`

— Settlement dates of floating-rate note`ValuationDate`

of BK tree (default) | serial date number | vector of serial date numbers | date character vector | cell array of date character vectorsSettlement dates of floating-rate note specified as serial date numbers or date character
vectors using a `NINST`

-by-`1`

vector of dates.

The `Settle`

date for every floating-rate note is set to the
`ValuationDate`

of the BK tree. The floating-rate note argument
`Settle`

is ignored.

**Data Types: **`double`

| `cell`

| `char`

`Maturity`

— Floating-rate note maturity dateserial date number | vector of serial date numbers | date character vector | cell array of date character vectors

Floating-rate note maturity date specified as serial date numbers
or date character vectors using a `NINST`

-by-`1`

vector
of dates.

**Data Types: **`double`

| `cell`

| `char`

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`

.

```
[Price,PriceTree] =
optfloatbybk(BKTree,OptSpec,Strike,ExerciseDates,AmericanOpt,Spread,Settle,Maturity,'FloatReset',4,'Basis',7)
```

`'FloatReset'`

— Frequency of payments per year`1`

(default) | positive integer from the set`[1,2,3,4,6,12]`

| vector of positive integers from the set
`[1,2,3,4,6,12]`

Frequency of payments per year, specified as the comma-separated pair consisting
of `'FloatReset'`

and positive integers for the values
`[1,2,3,4,6,12]`

in a
`NINST`

-by-`1`

vector.

Payments on floating-rate notes (FRNs) are determined by the effective interest-rate between reset dates. If the reset period for an FRN spans more than one tree level, calculating the payment becomes impossible due to the recombining nature of the tree. That is, the tree path connecting the two consecutive reset dates cannot be uniquely determined because there will be more than one possible path for connecting the two payment dates.

**Data Types: **`double`

`'Basis'`

— Day-count basis of the instrument`0`

(actual/actual) (default) | positive integers of the set `[1...13]`

| vector of positive integers of the set `[1...13]`

Day-count basis of the instrument, specified as the comma-separated pair consisting of
`'Basis'`

and a positive integer using a
`NINST`

-by-`1`

vector. The
`Basis`

value represents the basis used when annualizing the input
forward-rate tree.

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`

`'Principal'`

— Principal values`100`

(default) | vector of nonnegative values | cell array of nonnegative valuesPrincipal values, specified as the comma-separated pair consisting of
`'Principal'`

and nonnegative values using a
`NINST`

-by-`1`

vector or
`NINST`

-by-`1`

cell array of notional principal
amounts. When using a `NINST`

-by-`1`

cell array,
each element is a `NumDates`

-by-`2`

cell array where
the first column is dates and the second column is associated principal amount. The
date indicates the last day that the principal value is valid.

**Data Types: **`double`

| `cell`

`'Options'`

— Structure containing derivatives pricing optionsstructure

Structure containing derivatives pricing options, specified as the comma-separated pair
consisting of `'Options'`

and a structure obtained from using
`derivset`

.

**Data Types: **`struct`

`'EndMonthRule'`

— End-of-month rule flag`1`

(in effect) (default) | nonnegative integer [0,1]End-of-month rule flag, specified as the comma-separated pair consisting of
`'EndMonthRule'`

and a nonnegative integer [`0`

,
`1`

] using a `NINST`

-by-`1`

vector. This rule applies only when `Maturity`

is an end-of-month
date for a month having 30 or fewer days.

`0`

= Ignore rule, meaning that a bond coupon payment date is always the same numerical day of the month.`1`

= Set rule on, meaning that a bond coupon payment date is always the last actual day of the month.

**Data Types: **`double`

`Price`

— Expected prices of the floating-rate note option at time 0scalar | vector

Expected prices of the floating-rate note option at time 0 is
returned as a scalar or an `NINST`

-by-`1`

vector.

`PriceTree`

— Structure of trees containing vectors of option prices at each node tree structure

Structure of trees containing vectors of instrument prices and accrued interest and a vector of observation times for each node returned as:

`PriceTree.PTree`

contains the clean prices.`PriceTree.AITree`

contains the accrued interest.`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.

A *floating-rate note option* is a put or call
option on a floating-rate note.

Financial Instruments Toolbox™ supports three types of put and call options on floating-rate notes:

American option — An option that you exercise any time until its expiration date.

European option — An option that you exercise only on its expiration date.

Bermuda option — A Bermuda option resembles a hybrid of American and European options; you can only exercise it on predetermined dates, usually monthly.

For more information, see Floating-Rate Note Options.

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