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# floorvolstrip

Strip floorlet volatilities from flat floor volatilities

## Syntax

``````[FloorletVols,FloorletPaymentDates,FloorStrikes] = floorvolstrip(ZeroCurve,FloorSettle,FloorMaturity,FloorVolatility)``````
``````[FloorletVols,FloorletPaymentDates,FloorStrikes] = floorvolstrip(___,Name,Value)``````

## Description

example

``````[FloorletVols,FloorletPaymentDates,FloorStrikes] = floorvolstrip(ZeroCurve,FloorSettle,FloorMaturity,FloorVolatility)``` strips floorlet volatilities from the flat floor volatilities by using the bootstrapping method. The floor volatilities are interpolated on each floorlet payment date before stripping the floorlet volatilities.```

example

``````[FloorletVols,FloorletPaymentDates,FloorStrikes] = floorvolstrip(___,Name,Value)``` adds optional name-value pair arguments. The floor volatilities are interpolated on each floorlet payment date before stripping the floorlet volatilities.```

## Examples

collapse all

Compute the zero curve for discounting and projecting forward rates.

```ValuationDate = datenum('10-Aug-2015'); ZeroRates = [0.12 0.24 0.40 0.73 1.09 1.62]/100; CurveDates = datemnth(ValuationDate, [0.25 0.5 1 2 3 5]*12); ZeroCurve = IRDataCurve('Zero',ValuationDate,CurveDates,ZeroRates)```
```ZeroCurve = Type: Zero Settle: 736186 (10-Aug-2015) Compounding: 2 Basis: 0 (actual/actual) InterpMethod: linear Dates: [6x1 double] Data: [6x1 double] ```

Define the ATM floor volatility data.

```FloorSettle = datenum('12-Aug-2015'); FloorMaturity = datenum({'12-Aug-2016';'14-Aug-2017';'13-Aug-2018';... '12-Aug-2019',;'12-Aug-2020'}); FloorVolatility = [0.31;0.39;0.43;0.42;0.40];```

Strip floorlet volatilities from ATM floors.

```[FloorletVols, FloorletPaymentDates, ATMFloorStrikes] = floorvolstrip(ZeroCurve,... FloorSettle, FloorMaturity, FloorVolatility); PaymentDates = cellstr(datestr(FloorletPaymentDates)); format; table(PaymentDates, FloorletVols, ATMFloorStrikes)```
```ans=9x3 table null PaymentDates FloorletVols ATMFloorStrikes _____________ ____________ _______________ '12-Aug-2016' 0.31 0.0056551 '13-Feb-2017' 0.3646 0.0073508 '14-Aug-2017' 0.41948 0.0090028 '12-Feb-2018' 0.43152 0.010827 '13-Aug-2018' 0.46351 0.012617 '12-Feb-2019' 0.40407 0.013862 '12-Aug-2019' 0.39863 0.015105 '12-Feb-2020' 0.3674 0.016369 '12-Aug-2020' 0.35371 0.01762 ```

Compute the zero curve for discounting and projecting forward rates.

```ValuationDate = datenum('10-Jun-2015'); ZeroRates = [0.02 0.10 0.28 0.75 1.15 1.80]/100; CurveDates = datemnth(ValuationDate, [0.25 0.5 1 2 3 5]*12); ZeroCurve = IRDataCurve('Zero',ValuationDate,CurveDates,ZeroRates)```
```ZeroCurve = Type: Zero Settle: 736125 (10-Jun-2015) Compounding: 2 Basis: 0 (actual/actual) InterpMethod: linear Dates: [6x1 double] Data: [6x1 double] ```

Define the floor volatility data.

```FloorSettle = datenum('12-Jun-2015'); FloorMaturity = datenum({'13-Jun-2016';'12-Jun-2017';'12-Jun-2018';... '12-Jun-2019';'12-Jun-2020'}); FloorVolatility = [0.41;0.43;0.43;0.41;0.38]; FloorStrike = 0.015;```

Strip floorlet volatilities from floors with the same strike.

```[FloorletVols, FloorletPaymentDates, FloorStrikes] = floorvolstrip(ZeroCurve, ... FloorSettle, FloorMaturity, FloorVolatility, 'Strike', FloorStrike); PaymentDates = cellstr(datestr(FloorletPaymentDates)); format; table(PaymentDates, FloorletVols, FloorStrikes)```
```ans=9x3 table null PaymentDates FloorletVols FloorStrikes _____________ ____________ ____________ '13-Jun-2016' 0.41 0.015 '12-Dec-2016' 0.42 0.015 '12-Jun-2017' 0.43433 0.015 '12-Dec-2017' 0.43001 0.015 '12-Jun-2018' 0.43 0.015 '12-Dec-2018' 0.39173 0.015 '12-Jun-2019' 0.37244 0.015 '12-Dec-2019' 0.32056 0.015 '12-Jun-2020' 0.28308 0.015 ```

Compute the zero curve for discounting and projecting forward rates.

```ValuationDate = datenum('19-May-2015'); ZeroRates = [0.02 0.07 0.23 0.63 1.01 1.60]/100; CurveDates = datemnth(ValuationDate, [0.25 0.5 1 2 3 5]*12); ZeroCurve = IRDataCurve('Zero',ValuationDate,CurveDates,ZeroRates)```
```ZeroCurve = Type: Zero Settle: 736103 (19-May-2015) Compounding: 2 Basis: 0 (actual/actual) InterpMethod: linear Dates: [6x1 double] Data: [6x1 double] ```

Define the floor volatility data.

```FloorSettle = datenum('19-May-2015'); FloorMaturity = datenum({'19-May-2016';'19-May-2017';'21-May-2018'; ... '20-May-2019';'19-May-2020'}); FloorVolatility = [0.39;0.42;0.43;0.42;0.40]; FloorStrike = 0.010;```

Specify the quarterly and semiannual dates.

```FloorletDates = [cfdates(FloorSettle, '19-May-2016', 4)... cfdates('19-May-2016', '19-May-2020', 2)]'; FloorletDates(~isbusday(FloorletDates)) = ... busdate(FloorletDates(~isbusday(FloorletDates)), 'modifiedfollow');```

Strip floorlet volatilities using specified `FloorletDates`.

```[FloorletVols, FloorletPaymentDates, FloorStrikes] = floorvolstrip(ZeroCurve, ... FloorSettle, FloorMaturity, FloorVolatility, 'Strike', FloorStrike, ... 'FloorletDates', FloorletDates); PaymentDates = cellstr(datestr(FloorletPaymentDates)); format; table(PaymentDates, FloorletVols, FloorStrikes)```
```ans=11x3 table PaymentDates FloorletVols FloorStrikes _____________ ____________ ____________ '19-Nov-2015' 0.39 0.01 '19-Feb-2016' 0.39 0.01 '19-May-2016' 0.39 0.01 '21-Nov-2016' 0.4058 0.01 '19-May-2017' 0.4307 0.01 '20-Nov-2017' 0.43317 0.01 '21-May-2018' 0.44309 0.01 '19-Nov-2018' 0.40831 0.01 '20-May-2019' 0.39831 0.01 '19-Nov-2019' 0.3524 0.01 '19-May-2020' 0.32765 0.01 ```

Compute the zero curve for discounting and projecting forward rates.

```ValuationDate = datenum('3-May-2016'); ZeroRates = [-0.31 -0.21 -0.15 -0.10 0.009 0.19]/100; CurveDates = datemnth(ValuationDate, [0.25 0.5 1 2 3 5]*12); ZeroCurve = IRDataCurve('Zero',ValuationDate,CurveDates,ZeroRates) ```
```ZeroCurve = Type: Zero Settle: 736453 (03-May-2016) Compounding: 2 Basis: 0 (actual/actual) InterpMethod: linear Dates: [6x1 double] Data: [6x1 double] ```

Define the floor volatility (Shifted Black) data.

```FloorSettle = datenum('3-May-2016'); FloorMaturity = datenum({'3-May-2017';'3-May-2018';'3-May-2019'; ... '4-May-2020';'3-May-2021'}); FloorVolatility = [0.42;0.45;0.43;0.40;0.36]; % Shifted Black volatilities Shift = 0.01; % 1 percent shift. FloorStrike = -0.001; % -0.1 percent strike. ```

Strip floorlet volatilities from floors using the Shifted Black Model.

```[FloorletVols, FloorletPaymentDates, FloorStrikes] = floorvolstrip(ZeroCurve, ... FloorSettle,FloorMaturity,FloorVolatility,'Strike',FloorStrike,'Shift',Shift); PaymentDates = string(datestr(FloorletPaymentDates)); format; table(PaymentDates,FloorletVols,FloorStrikes) ```
```ans = 9x3 table PaymentDates FloorletVols FloorStrikes _____________ ____________ ____________ "03-May-2017" 0.42 -0.001 "03-Nov-2017" 0.44575 -0.001 "03-May-2018" 0.47092 -0.001 "05-Nov-2018" 0.41911 -0.001 "03-May-2019" 0.40197 -0.001 "04-Nov-2019" 0.36262 -0.001 "04-May-2020" 0.33615 -0.001 "03-Nov-2020" 0.27453 -0.001 "03-May-2021" 0.23045 -0.001 ```

## Input Arguments

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Zero rate curve, specified using a `RateSpec` or `IRDataCurve` object containing the zero rate curve for discounting according to its day count convention. `ZeroCurve` is also used for computing the underlying forward rates if the optional argument `ProjectionCurve` is not specified. Its observation date specifies the valuation date. For more information on creating a `RateSpec`, see `intenvset`. For more information on creating an `IRDataCurve` object, see `IRDataCurve`.

Data Types: `struct`

Common floor settle date, specified using a serial date number or date character vector. The `FloorSettle` date cannot be earlier than the `ZeroCurve` valuation date.

Data Types: `double` | `char`

Floor maturity dates, specified using serial date numbers or date character vectors as an `NFloor`-by-`1` vector.

Data Types: `double` | `char`

Flat floor volatilities, specified using positive decimals as a `NFloor`-by-`1` vector.

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`.

Example: ```[FloorletVols,FloorletPaymentDates,FloorStrikes] = floorvolstrip(ZeroCurve,FloorSettle,FloorMaturity,FloorVolatility,'Strike',.2)```

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Floor strike rate, specified as decimals. Use `Strike` to specify a single strike that is equally applied to all floors.

Data Types: `double`

Floorlet reset and payment dates, specified as serial date numbers or date character vectors using a `NFloorletDates`-by-`1` vector.

Use `FloorletDates` to manually specify all floorlet reset and payment dates. For example, some date intervals may be quarterly while others may be semiannual. All dates must be later than `FloorSettle` and cannot be later than the last `FloorMaturity` date. Dates are adjusted according to the `BusDayConvention` and `Holidays` inputs.

If `FloorletDates` is not specified, the default is to automatically generate periodic floorlet dates after `FloorSettle` based on the last `FloorMaturity` date as the reference date, using the following optional inputs: `Reset`, `EndMonthRule`, `BusDayConvention`, and `Holidays`

Data Types: `double` | `char`

Frequency of periodic payments per year within a floor, specified as a positive integer with values `1`,`2`, `3`, `4`, `6`, or `12`.

### Note

The input for `Reset` is ignored if `FloorletDates` is specified.

Data Types: `double`

End-of-month rule flag for generating floorlet dates is specified as nonnegative integer [`0`, `1`] using a `NINST`-by-`1` vector.

• `0` = Ignore rule, meaning that a payment date is always the same numerical day of the month.

• `1` = Set rule on, meaning that a payment date is always the last actual day of the month.

Data Types: `logical`

Business day conventions, specified by a character vector or `N`-by-`1` cell array of character vectors of business day conventions. The selection for business day convention determines how non-business days are treated. Non-business days are defined as weekends plus any other date that businesses are not open (e.g. statutory holidays). Values are:

• `actual` — Non-business days are effectively ignored. Cash flows that fall on non-business days are assumed to be distributed on the actual date.

• `follow` — Cash flows that fall on a non-business day are assumed to be distributed on the following business day.

• `modifiedfollow` — Cash flows that fall on a non-business day are assumed to be distributed on the following business day. However if the following business day is in a different month, the previous business day is adopted instead.

• `previous` — Cash flows that fall on a non-business day are assumed to be distributed on the previous business day.

• `modifiedprevious` — Cash flows that fall on a non-business day are assumed to be distributed on the previous business day. However if the previous business day is in a different month, the following business day is adopted instead.

Data Types: `char` | `cell`

Holidays used in computing business days, specified as MATLAB date numbers using an `NHolidays`-by-`1` vector.

Data Types: `double`

Rate curve for computing underlying forward rates, specified as a `RateSpec` or `IRDatCurve` object. For more information on creating a `RateSpec`, see `intenvset` and for more information on creating an `IRDataCurve` object, see `IRDataCurve`.

Data Types: `struct`

Method used when interpolating the floor volatilities on each floorlet maturity date before stripping the floorlet volatilities, specified using a character vector with values: `linear`, `nearest`, `next`, `previous`, `spline`, or `pchip`. The definitions of the methods are:

• `linear` — Linear interpolation. The interpolated value at a query point is based on linear interpolation of the values at neighboring grid points in each respective dimension. This is the default interpolation method.

• `nearest` — Nearest neighbor interpolation. The interpolated value at a query point is the value at the nearest sample grid point.

• `next` — Next neighbor interpolation. The interpolated value at a query point is the value at the next sample grid point.

• `previous` — Previous neighbor interpolation. The interpolated value at a query point is the value at the previous sample grid point.

• `spline` — Spline interpolation using not-a-knot end conditions. The interpolated value at a query point is based on a cubic interpolation of the values at neighboring grid points in each respective dimension.

• `pchip` — Shape-preserving piecewise cubic interpolation. The interpolated value at a query point is based on a shape-preserving piecewise cubic interpolation of the values at neighboring grid points.

For more information on interpolation methods, see `interp1`.

### Note

Constant extrapolation is used for volatilities falling outside the range of user-supplied data.

Data Types: `char`

Upper bound of implied volatility search interval, specified as a positive scalar decimal.

Data Types: `double`

Implied volatility search termination tolerance, specified as a positive scalar.

Data Types: `double`

Flag indicating whether to omit the first floorlet payment in the floors, specified as a scalar logical. For example, the first floorlet payment is omitted in spot-starting floors, while it is included in forward-starting floors. Setting this logical to `false` means to always include the first floorlet.

In general, “spot lag” is the delay between the fixing date and the effective date for LIBOR-like indices. It also determines whether a floor is spot-starting or forward-starting (Corb, 2012). Floors are considered to be spot-starting if they settle within “spot lag” business days after the valuation date. Those that settle later are considered to be forward-starting. The first floorlet is omitted if floors are spot-starting, while it is included if they are forward-starting (Tuckman, 2012).

Data Types: `logical`

Shift in decimals for the shifted SABR model (to be used with the Shifted Black model), specified using a scalar positive decimal value. Set this parameter to a positive shift in decimals to add a positive shift to the forward rate and strike, which effectively sets a negative lower bound for the forward rate and strike. For example, a `Shift` value of 0.01 is equal to a 1% shift.

Data Types: `double`

## Output Arguments

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Stripped floorlet volatilities, returned as a `NFloorletVols`-by-`1` vector in decimals.

### Note

`floorvolstrip` may output `NaN`s for some floorlet volatilities. This could be the case if no volatility matches the floorlet price implied by the user-supplied floor data.

Payment dates (in date numbers), returned as a `NFloorletVols`-by-`1` vector corresponding to `FloorletVols`.

Floor strikes, returned as a `NFloorletVols`-by-`1` vector of strikes in decimals for floors maturing on corresponding `FloorletPaymentDates`.

## Limitations

When bootstrapping the floorlet volatilities from ATM floors, the floorlet volatilities stripped from the shorter maturity floors are reused in the longer maturity floors without adjusting for the difference in strike. `floorvolstrip` follows the simplified approach described in Gatarek, 2006.

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### ATM

A cap or floor is at-the-money (ATM) if its strike is equal to the forward swap rate.

This is the fixed rate of a swap that makes the present value of the floating leg equal to that of the fixed leg. In comparison, a caplet or floorlet is ATM if its strike is equal to the forward rate (not the forward swap rate). In general (except over a single period), the forward rate is not necessarily equal to the forward swap rate. So, to be precise, the individual caplets in an ATM cap have slightly different moneyness and are actually only approximately ATM (Alexander, 2003). In addition, note that swap rate changes with swap maturity. Similarly, the ATM cap strike also changes with cap maturity, so the ATM cap strikes need to be computed for each cap maturity before stripping the caplet volatilities. As a result, when stripping the caplet volatilities from the ATM caps with increasing maturities, the ATM strikes of the consecutive caps are different.

## References

[1] Alexander, C. “Common Correlation and Calibrating the Lognormal Forward Rate Model.” Wilmott Magazine, 2003.

[2] Corb, H. “Interest Rate Swaps and Other Derivatives.” Columbia Business School Publishing, 2012.

[3] Gatarek, D.P., Bachert, and R. Maksymiuk. The LIBOR Market Model in Practice. Wiley, 2006.

[4] Tuckman, B., Serrat, A. Fixed Income Securities: Tools for Today’s Markets. Wiley Finance, 2012.