# capvolstrip

Strip caplet volatilities from flat cap volatilities

## Syntax

``````[CapletVols,CapletPaymentDates,CapStrikes] = capvolstrip(ZeroCurve,CapSettle,CapMaturity,CapVolatility)``````
``````[CapletVols,CapletPaymentDates,CapStrikes] = capvolstrip(___,Name,Value)``````

## Description

example

``````[CapletVols,CapletPaymentDates,CapStrikes] = capvolstrip(ZeroCurve,CapSettle,CapMaturity,CapVolatility)``` strips caplet volatilities from the flat cap volatilities by using the bootstrapping method. The function interpolates the cap volatilities on each caplet payment date before stripping the caplet volatilities.```

example

``````[CapletVols,CapletPaymentDates,CapStrikes] = capvolstrip(___,Name,Value)``` specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax.```

## Examples

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Compute the zero curve for discounting and projecting forward rates.

```ValuationDate = datetime(2015,6,23); ZeroRates = [0.01 0.09 0.30 0.70 1.07 1.71]/100; CurveDates = datemnth(ValuationDate, [0.25 0.5 1 2 3 5]*12); ZeroCurve = IRDataCurve('Zero',ValuationDate,CurveDates,ZeroRates)```
```ZeroCurve = Type: Zero Settle: 736138 (23-Jun-2015) Compounding: 2 Basis: 0 (actual/actual) InterpMethod: linear Dates: [6x1 double] Data: [6x1 double] ```

Define the ATM cap volatility data.

```CapSettle = datetime(2015,6,25); CapMaturity = [datetime(2016,6,27) ; datetime(2017,6,26) ; datetime(2018,6,25) ; datetime(2019,6,25) ; datetime(2020,6,25)]; CapVolatility = [0.29;0.38;0.42;0.40;0.38];```

Strip caplet volatilities from ATM caps.

```[CapletVols, CapletPaymentDates, ATMCapStrikes] = capvolstrip(ZeroCurve, ... CapSettle, CapMaturity, CapVolatility); PaymentDates = cellstr(datestr(CapletPaymentDates)); format; table(PaymentDates, CapletVols, ATMCapStrikes)```
```ans=9×3 table PaymentDates CapletVols ATMCapStrikes _______________ __________ _____________ {'27-Jun-2016'} 0.29 0.0052014 {'27-Dec-2016'} 0.34657 0.0071594 {'26-Jun-2017'} 0.41404 0.0091175 {'26-Dec-2017'} 0.42114 0.010914 {'25-Jun-2018'} 0.45297 0.012698 {'26-Dec-2018'} 0.37257 0.014222 {'25-Jun-2019'} 0.36184 0.015731 {'26-Dec-2019'} 0.3498 0.017262 {'25-Jun-2020'} 0.33668 0.018774 ```

Compute the zero curve for discounting and projecting forward rates.

```ValuationDate = datetime(2015,2,17); ZeroRates = [0.02 0.07 0.25 0.70 1.10 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: 736012 (17-Feb-2015) Compounding: 2 Basis: 0 (actual/actual) InterpMethod: linear Dates: [6x1 double] Data: [6x1 double] ```

Define the cap volatility data.

```CapSettle = datetime(2015,2,19); CapMaturity = [datetime(2016,2,19) ; datetime(2017,2,21) ; datetime(2018,2,20) ; datetime(2019,2,19) ; datetime(2020,2,19)]; CapVolatility = [0.44;0.45;0.44;0.41;0.39]; CapStrike = 0.013;```

Strip caplet volatilities from caps with the same strike.

```[CapletVols, CapletPaymentDates, CapStrikes] = capvolstrip(ZeroCurve, ... CapSettle, CapMaturity, CapVolatility, 'Strike', CapStrike); PaymentDates = cellstr(datestr(CapletPaymentDates)); format; table(PaymentDates, CapletVols, CapStrikes)```
```ans=9×3 table PaymentDates CapletVols CapStrikes _______________ __________ __________ {'19-Feb-2016'} 0.44 0.013 {'19-Aug-2016'} 0.44495 0.013 {'21-Feb-2017'} 0.45256 0.013 {'21-Aug-2017'} 0.43835 0.013 {'20-Feb-2018'} 0.42887 0.013 {'20-Aug-2018'} 0.38157 0.013 {'19-Feb-2019'} 0.35237 0.013 {'19-Aug-2019'} 0.3525 0.013 {'19-Feb-2020'} 0.33136 0.013 ```

Compute the zero curve for discounting and projecting forward rates.

```ValuationDate = datetime(2015,3,6); ZeroRates = [0.01 0.08 0.27 0.73 1.16 1.70]/100; CurveDates = datemnth(ValuationDate, [0.25 0.5 1 2 3 5]*12); ZeroCurve = IRDataCurve('Zero',ValuationDate,CurveDates,ZeroRates)```
```ZeroCurve = Type: Zero Settle: 736029 (06-Mar-2015) Compounding: 2 Basis: 0 (actual/actual) InterpMethod: linear Dates: [6x1 double] Data: [6x1 double] ```

Define the cap volatility data.

```CapSettle = datetime(2015,3,6); CapMaturity = [datetime(2016,3,7) ; datetime(2017,3,6) ; datetime(2018,3,6) ; datetime(2019,3,6) ; datetime(2020,3,6)]; CapVolatility = [0.43;0.44;0.44;0.43;0.41]; CapStrike = 0.011;```

Specify quarterly and semiannual dates.

```CapletDates = [cfdates(CapSettle, datetime(2016,3,6), 4) ... cfdates(datetime(2016,3,6), datetime(2020,3,6), 2)]'; CapletDates(~isbusday(CapletDates)) = ... busdate(CapletDates(~isbusday(CapletDates)), 'modifiedfollow');```

Strip caplet volatilities using specified `CapletDates`.

```[CapletVols, CapletPaymentDates, CapStrikes] = capvolstrip(ZeroCurve, ... CapSettle, CapMaturity, CapVolatility, 'Strike', CapStrike, ... 'CapletDates', CapletDates); PaymentDates = cellstr(datestr(CapletPaymentDates)); format; table(PaymentDates, CapletVols, CapStrikes)```
```ans=11×3 table PaymentDates CapletVols CapStrikes _______________ __________ __________ {'08-Sep-2015'} 0.43 0.011 {'07-Dec-2015'} 0.42999 0.011 {'07-Mar-2016'} 0.43 0.011 {'06-Sep-2016'} 0.43538 0.011 {'06-Mar-2017'} 0.44396 0.011 {'06-Sep-2017'} 0.43999 0.011 {'06-Mar-2018'} 0.44001 0.011 {'06-Sep-2018'} 0.41934 0.011 {'06-Mar-2019'} 0.40985 0.011 {'06-Sep-2019'} 0.36818 0.011 {'06-Mar-2020'} 0.34657 0.011 ```

Compute the zero curve for discounting and projecting forward rates.

```ValuationDate = datetime(2016,3,1); ZeroRates = [-0.38 -0.25 -0.21 -0.12 0.01 0.2]/100; CurveDates = datemnth(ValuationDate, [0.25 0.5 1 2 3 5]*12); ZeroCurve = IRDataCurve('Zero',ValuationDate,CurveDates,ZeroRates)```
```ZeroCurve = Type: Zero Settle: 736390 (01-Mar-2016) Compounding: 2 Basis: 0 (actual/actual) InterpMethod: linear Dates: [6x1 double] Data: [6x1 double] ```

Define the cap volatility (Shifted Black) data.

```CapSettle = datetime(2016,3,1); CapMaturity = [datetime(2017,3,1); datetime(2018,3,1) ; datetime(2019,3,1) ; datetime(2020,3,2) ; datetime(2021,3,1)]; CapVolatility = [0.35;0.40;0.37;0.34;0.32]; % Shifted Black volatilities Shift = 0.01; % 1 percent shift. CapStrike = -0.001; % -0.1 percent strike.```

Strip caplet volatilities from caps using the Shifted Black Model.

```[CapletVols, CapletPaymentDates, CapStrikes] = capvolstrip(ZeroCurve, ... CapSettle,CapMaturity,CapVolatility,'Strike',CapStrike,'Shift',Shift); PaymentDates = string(datestr(CapletPaymentDates)); format; table(PaymentDates,CapletVols,CapStrikes)```
```ans=9×3 table PaymentDates CapletVols CapStrikes _____________ __________ __________ "01-Mar-2017" 0.35 -0.001 "01-Sep-2017" 0.39129 -0.001 "01-Mar-2018" 0.4335 -0.001 "04-Sep-2018" 0.35284 -0.001 "01-Mar-2019" 0.3255 -0.001 "03-Sep-2019" 0.3011 -0.001 "02-Mar-2020" 0.27266 -0.001 "01-Sep-2020" 0.27698 -0.001 "01-Mar-2021" 0.25697 -0.001 ```

Compute the zero curve for discounting and projecting forward rates.

```ValuationDate = datetime(2018,6,1); ZeroRates = [-0.38 -0.25 -0.21 -0.12 0.01 0.2]/100; CurveDates = datemnth(ValuationDate, [0.25 0.5 1 2 3 5]*12); ZeroCurve = IRDataCurve('Zero',ValuationDate,CurveDates,ZeroRates)```
```ZeroCurve = Type: Zero Settle: 737212 (01-Jun-2018) Compounding: 2 Basis: 0 (actual/actual) InterpMethod: linear Dates: [6x1 double] Data: [6x1 double] ```

Define the normal cap volatility data.

```CapSettle = datetime(2018,6,1); CapMaturity = [datetime(2019,6,3) ; datetime(2020,6,1) ; datetime(2021,6,1) ; datetime(2022,6,1) ; datetime(2023,6,1)]; CapVolatility = [0.0057;0.0059;0.0057;0.0053;0.0051]; % Normal volatilities CapStrike = -0.002; % -0.2 percent strike.```

Strip caplet volatilities from caps using the Normal (Bachelier) model.

```[CapletVols, CapletPaymentDates, CapStrikes] = capvolstrip(ZeroCurve, ... CapSettle,CapMaturity,CapVolatility,'Strike',CapStrike,'Model','normal'); PaymentDates = string(datestr(CapletPaymentDates)); format; table(PaymentDates,CapletVols,CapStrikes)```
```ans=9×3 table PaymentDates CapletVols CapStrikes _____________ __________ __________ "03-Jun-2019" 0.0057 -0.002 "02-Dec-2019" 0.0058686 -0.002 "01-Jun-2020" 0.0060472 -0.002 "01-Dec-2020" 0.0055705 -0.002 "01-Jun-2021" 0.0053912 -0.002 "01-Dec-2021" 0.0047404 -0.002 "01-Jun-2022" 0.004357 -0.002 "01-Dec-2022" 0.0046481 -0.002 "01-Jun-2023" 0.0044477 -0.002 ```

## 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. If you do not specify the optional argument `ProjectionCurve`, the function uses `ZeroCurve` to compute the underlying forward rates as well. The observation date of the `ZeroCurve` 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 cap settle date, specified as a scalar datetime, string, or date character vector. The `CapSettle` date cannot be earlier than the `ZeroCurve` valuation date.

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

Cap maturity dates, specified using a `NCap`-by-`1` vector using a datetime array, string array, or date character vectors.

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

Flat cap volatilities, specified as an `NCap`-by-`1` vector of positive decimals.

Data Types: `double`

### 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: ```[CapletVols,CapletPaymentDates,CapStrikes] = capvolstrip(ZeroCurve,CapSettle,CapMaturity,CapVolatility,'Strike',.2)```

Cap strike rate, specified as the comma-separated pair consisting of `'Strike'` and a scalar decimal value or an `NCapletVols`-by-`1` vector. Use `Strike` as a scalar to specify a single strike that applies equally to all caps. Or, specify an `NCapletVols`-by-`1` vector of strikes for the caps.

Data Types: `double`

Caplet reset and payment dates, specified as the comma-separated pair consisting of `'CapletDates'` and an `NCapletDates`-by-`1` vector using a datetime array, string array, or date character vectors.

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

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

If `CapletDates` is not specified, the default is to automatically generate periodic caplet dates after `CapSettle` based on the last `CapMaturity` date as the reference date, using the following optional inputs: `Reset`, `EndMonthRule`, `BusDayConvention`, and `Holidays`.

Frequency of periodic payments per year within a cap, specified as the comma-separated pair consisting of `'Reset'` and a positive scalar integer with values `1`,`2`, `3`, `4`, `6`, or `12`.

Note

If you specify `CapletDates`, the function ignores the input for `Reset`.

Data Types: `double`

End-of-month rule flag for generating caplet dates, specified as the comma-separated pair consisting of `'EndMonthRule'` and a scalar nonnegative integer [`0`, `1`].

• `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 as the comma-separated pair consisting of `'BusinessDayConvention'` and a character vector. Use this argument to specify how the function treats non-business days, which are days on which businesses are not open (such as weekends and statutory holidays).

• `'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`

Holidays used in computing business days, specified as the comma-separated pair consisting of `'Holidays'` and `NHolidays`-by-`1` vector of MATLAB dates.

Data Types: `datetime`

Rate curve for computing underlying forward rates, specified as the comma-separated pair consisting of `'ProjectionCurve'` and a `RateSpec` object or `IRDatCurve` object. For more information on creating a `RateSpec`, see `intenvset`. For more information on creating an `IRDataCurve` object, see `IRDataCurve`.

Data Types: `struct`

Method for interpolating the cap volatilities on each caplet maturity date before stripping the caplet volatilities, specified as the comma-separated pair consisting of `'MaturityInterpMethod'` and a character vector with values: `'linear'`, `'nearest'`, `'next'`, `'previous'`, `'spline'`, or `'pchip'`.

• `'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

The function uses constant extrapolation to calculate volatilities falling outside the range of user-supplied data.

Data Types: `char`

Upper bound of implied volatility search interval, specified as the comma-separated pair consisting of `'Limit'` and a positive scalar decimal.

Data Types: `double`

Implied volatility search termination tolerance, specified as the comma-separated pair consisting of `'Tolerance'` and a positive numeric scalar.

Data Types: `double`

Flag to omit the first caplet payment in the caps, specified as the comma-separated pair consisting of `'OmitFirstCaplet'` and a scalar logical.

If the caps are spot-starting, the first caplet payment is omitted. If the caps are forward-starting, the first caplet payment is included. Regardless of the status of the caps, if you set this logical to `false`, then the function includes the first caplet payment.

In general, “spot lag” is the delay between the fixing date and the effective date for LIBOR-like indices. "Spot lag" determines whether a cap is spot-starting or forward-starting (Corb, 2012). Caps 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 caplet is omitted if caps 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 as the comma-separated pair consisting of `'Shift'` and a positive scalar 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`

Model used for the implied volatility calculation, specified as the comma-separated pair consisting of `'Model'` and a scalar character vector or string scalar with one of the following values:

• `'lognormal'` - Implied Black (no shift) or Shifted Black volatility.

• `'normal'` - Implied Normal (Bachelier) volatility. If you specify `'normal'`, `Shift` must be zero.

The `capvolstrip` function supports three volatility types.

'Model' Value'Shift' ValueVolatility Type
`'lognormal'``Shift` = `0`Black
`'lognormal'``Shift` > `0`Shifted Black
`'normal'``Shift` = `0`Normal (Bachelier)

Data Types: `char` | `string`

## Output Arguments

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Stripped caplet volatilities, returned as an `NCapletVols`-by-`1` vector of decimals.

Note

`capvolstrip` can output `NaN`s for some caplet volatilities. You might encounter this output if no volatility matches the caplet price implied by the user-supplied cap data.

Payment dates (in date numbers), returned as an `NCapletVols`-by-`1` vector of date numbers corresponding to `CapletVols`.

Cap strikes, returned as an `NCapletVols`-by-`1` vector of strikes in decimals for caps maturing on the corresponding `CapletPaymentDates`. `CapStrikes` are the same as the strikes of the corresponding caplets that have been stripped.

## Limitations

When bootstrapping the caplet volatilities from ATM caps, the function reuses the caplet volatilities stripped from the shorter maturity caps in the longer maturity caps without adjusting for the difference in strike. `capvolstrip` follows the simplified approach described in Gatarek, 2006.

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### Cap

A cap is a contract that includes a guarantee that sets the maximum interest rate to be paid by the holder, based on an otherwise floating interest rate.

The payoff for a cap is:

$\mathrm{max}\left(CurrentRate-CapRate,0\right)$

### At-The-Money

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

The forward swap rate 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 equal to the forward swap rate. So, to be precise, the individual caplets in an ATM cap have slightly different moneyness and are only approximately ATM (Alexander, 2003).

In addition, the swap rate changes with swap maturity. Similarly, the ATM cap strike also changes with cap maturity, so the ATM cap strikes are 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 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. Chichester, UK: Wiley, 2006.

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

## Version History

Introduced in R2016a

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