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# optemfloatbyhjm

Price embedded option on floating-rate note HJM interest-rate tree

## Syntax

OptSpec,Strike,ExerciseDates)
example
Strike,ExerciseDates,AmericanOpt,Reset,Basis,Principal,
Options,EndMonthRule)
• [Price,PriceTree] = optemfloatbyhjm(___)
Strike,ExerciseDates)
• Price = optemfloatbyhjm(___,Name,Value)
• [Price,PriceTree] = optemfloatbyhjm(___)

## Description

example

OptSpec,Strike,ExerciseDates)
returns the embedded option price of the instrument(s) at the valuation date.

Strike,ExerciseDates,AmericanOpt,Reset,Basis,Principal,
Options,EndMonthRule)
returns the embedded option price of the instrument(s) at the valuation date using optional arguments.

[Price,PriceTree] = optemfloatbyhjm(___) returns the option price of the instrument(s) at the valuation date and a structure of trees containing vectors of option prices at each node.

Strike,ExerciseDates)
returns the embedded option price of the instrument(s) at the valuation date.

Price = optemfloatbyhjm(___,Name,Value) returns the embedded option price of the instrument(s) at the valuation date using optional name-value pair arguments.

[Price,PriceTree] = optemfloatbyhjm(___) returns the embedded option price of the instrument(s) at the valuation date and a structure of trees containing vectors of option prices at each node using optional name-value pair arguments.

## Examples

expand all

### Price European Callable Embedded Option for a Floating-Rate Note

Define the interest-rate term structure.

```Rates = [0.03;0.035;0.040;0.045];
ValuationDate = 'Jan-1-2012';
StartDates = {'Jan-1-2012'; 'Jan-1-2013'; 'Jan-1-2014'; 'Jan-1-2015'};
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 =

FinObj: 'RateSpec'
Compounding: 1
Disc: [4x1 double]
Rates: [4x1 double]
EndTimes: [4x1 double]
StartTimes: [4x1 double]
EndDates: [4x1 double]
StartDates: [4x1 double]
ValuationDate: 734869
Basis: 0
EndMonthRule: 1

```

Build the HJM tree.

```VolSpec = hjmvolspec('Constant', 0.01);
TimeSpec = hjmtimespec(RateSpec.ValuationDate, EndDates, Compounding);
HJMTree = hjmtree(VolSpec, RateSpec, TimeSpec)
```
```HJMTree =

FinObj: 'HJMFwdTree'
VolSpec: [1x1 struct]
TimeSpec: [1x1 struct]
RateSpec: [1x1 struct]
tObs: [0 1 2 3]
dObs: [734869 735235 735600 735965]
TFwd: {[4x1 double]  [3x1 double]  [2x1 double]  [3]}
CFlowT: {[4x1 double]  [3x1 double]  [2x1 double]  [4]}
FwdTree: {[4x1 double]  [3x1x2 double]  [2x2x2 double]  [1x4x2 double]}

```

The floater instrument has a spread of 15, a period of one year, and matures and is callable on Jan-1-2016.

```Spread = 15;
Settle = 'Jan-1-2012';
Maturity =  'Jan-1-2016';
Period = 1;
OptSpec = {'call'};
Strike = 95;
ExerciseDates = 'Jan-1-2016';
```

Compute the price of the floater with the embedded call.

```Price = optemfloatbyhjm(HJMTree, Spread, Settle, Maturity,...
OptSpec, Strike, ExerciseDates)
```
```Price =

96.2355

```

## Input Arguments

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### HJMTree — Interest-rate tree structurebinomial tree structure

Interest-rate tree, specified as a structure by using hjmtree.

Data Types: struct

### Spread — Number of basis points over 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 noteValuationDate of HW tree (default) | nonnegative integer | vector of nonnegative integers

Settlement dates of floating-rate note, specified as nonnegative integers using a NINST-by-1 vector of dates.

 Note:   The Settle date for every floating-rate note is set to the ValuationDate of the HW tree. The floating-rate note argument Settle is ignored.

Data Types: char

### OptSpec — Option definition string | cell array of strings

Option definition specified as a 'call' or 'put' using a NINST-by-1 cell array of strings.

Data Types: cell | char

### Strike — Option strike price valuesnonnegative integer | vector of nonnegative integers

Option strike price values, specified as nonnegative integers using a NINST-by-NSTRIKES vector of strike price values.

Data Types: single | double

### ExerciseDates — Exercise date for European, Bermuda, or American optionnonnegative integer | vector of nonnegative integers

Exercise date for European, Bermuda, or American option, specified as nonnegative integers using a NINST-by-NSTRIKES or NINST-by-2 vector of option exercise dates.

• For a European or Bermuda option, 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.

• For an American option, 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: char | cell

### AmericanOpt — (Optional) Option typescalar | vector of positive integers[0,1]

Option type, specified as NINST-by-1 positive integer flags with values:

• 0 — European/Bermuda

• 1 — American

Data Types: single | double

### Reset — (Optional) Frequency of payments per year1 (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 positive integers for the values [1,2,3,4,6,12] in a NINST-by-1 vector.

 Note:   Payments on floating-rate notes (FRNs) are determined by the effective interest-rate between reset dates. If the reset period for a 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 — (Optional) Day-count basis of instrument0 (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 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 (ISMA)

• 9 = actual/360 (ISMA)

• 10 = actual/365 (ISMA)

• 11 = 30/360E (ISMA)

• 12 = actual/365 (ISDA)

• 13 = BUS/252

Data Types: double

### Principal — (Optional) Principal values100 (default) | vector of nonnegative values | cell array of nonnegative values

Principal values specified as 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 — (Optional) Structure containing derivatives pricing optionsstructure

Structure containing derivatives pricing options, specified using derivset.

Data Types: struct

### EndMonthRule — (Optional) End-of-month rule flag1 (in effect) (default) | nonnegative integer [0,1]

End-of-month rule flag, specified as nonnegative integer 0 or 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

### 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.

### '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

### 'Reset' — Frequency of payments per year1 (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 positive integers for the values [1,2,3,4,6,12] in a NINST-by-1 vector.

 Note:   Payments on floating-rate notes (FRNs) are determined by the effective interest-rate between reset dates. If the reset period for a 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: char

### 'Basis' — Day-count basis of instrument0 (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 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 (ISMA)

• 9 = actual/360 (ISMA)

• 10 = actual/365 (ISMA)

• 11 = 30/360E (ISMA)

• 12 = actual/365 (ISDA)

• 13 = BUS/252

Data Types: char

### 'Principal' — (Optional) Principal values100 (default) | vector of nonnegative values | cell array of nonnegative values

Principal values specified as 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 using derivset.

Data Types: struct

### 'EndMonthRule' — End-of-month rule flag1 (in effect) (default) | nonnegative integer [0,1]

End-of-month rule flag, specified as 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: char

## Output Arguments

expand all

### Price — Expected prices of floating-rate note embedded option at time 0scalar | vector

Expected prices of the floating-rate note embedded option at time 0, returned as a scalar or a NINST-by-1 vector.

### PriceTree — Structure of trees containing vectors of embedded 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 embedded option prices.