capvolstrip
Strip caplet volatilities from flat cap volatilities
Syntax
Description
[
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.CapletVols
,CapletPaymentDates
,CapStrikes
]
= capvolstrip(ZeroCurve
,CapSettle
,CapMaturity
,CapVolatility
)
[
specifies options using one or more name-value pair arguments in addition to the input
arguments in the previous syntax.CapletVols
,CapletPaymentDates
,CapStrikes
]
= capvolstrip(___,Name,Value
)
Examples
Stripping Caplet Volatilities from At-The-Money (ATM) Caps
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
Stripping Caplet Volatilities from Caps with the Same Strikes
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
Stripping Caplet Volatilities Using Manually Specified Caplet Dates
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
Stripping Caplet Volatilities from Caps Using the Shifted Black Model
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
Stripping Caplet Volatilities from Caps Using Normal Model
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
ZeroCurve
— Zero rate curve
ratecurve
object | RateSpec
object | IRDataCurve
object
Zero rate curve, specified using a ratecurve
, 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, see
the following:
To create an
ratecurve
object, seeratecurve
.To create a
RateSpec
, seeintenvset
.To create an
IRDataCurve
object, seeIRDataCurve
.
Data Types: struct
CapSettle
— Common cap settle date
datetime scalar | string scalar | date character vector
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.
CapMaturity
— Cap maturity dates
datetime array | string array | date character vector
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.
CapVolatility
— Flat cap volatilities
vector of positive decimals
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)
Strike
— Cap strike rate
If not specified, all caps are at-the-money and the function
computes the ATM strike for each cap maturing on each caplet payment date (default) | scalar decimal | vector
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
CapletDates
— Caplet reset and payment dates
if not specified, the default is to automatically generate
periodic caplet dates (default) | datetime array | string array | date character vector
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
.
Reset
— Frequency of periodic payments per year within a cap
2
(default) | positive scalar integer with values 1
,2
,
3
, 4
, 6
, or
12
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
EndMonthRule
— End-of-month rule flag for generating caplet dates
1
(in effect) (default) | scalar nonnegative integer [0,1]
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
BusinessDayConvention
— Business day conventions
'modifiedfollow'
(default) | character vector with values 'actual'
,
'follow'
, 'modifiedfollow'
,
'previous'
, 'modifiedprevious'
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
— Holidays used in computing business days
if not specified, the default is to use
holidays.m
(default) | vector of MATLAB® dates
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
ProjectionCurve
— Rate curve for computing underlying forward rates
if not specified, the default is to use the
ZeroCurve
input for computing the underlying forward rates (default) | RateSpec
object | IRDatCurve
object
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
MaturityInterpMethod
— Method for interpolating the cap volatilities on each caplet maturity date before stripping the caplet volatilities
'linear'
(default) | character vector with values: 'linear'
,
'nearest'
, 'next'
,
'previous'
, 'spline'
,
'pchip'
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
Limit
— Upper bound of implied volatility search interval
10
(or 1000% per annum) (default) | positive scalar decimal
Upper bound of implied volatility search interval, specified as the comma-separated pair
consisting of 'Limit'
and a positive scalar decimal.
Data Types: double
Tolerance
— Implied volatility search termination tolerance
1e-5
(default) | positive numeric scalar
Implied volatility search termination tolerance, specified as the comma-separated pair
consisting of 'Tolerance'
and a positive numeric scalar.
Data Types: double
OmitFirstCaplet
— Flag to omit the first caplet payment in the caps
true
(always omit the first caplet) (default) | logical
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
— Shift in decimals for shifted SABR model
0
(no shift) (default) | positive scalar decimal
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
— Model used for implied volatility
'lognormal'
(default) | character vector with value of 'lognormal'
or
'normal'
| string scalar with value of "lognormal"
or
"normal"
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' Value | Volatility Type |
---|---|---|
'lognormal' | Shift = 0 | Black |
'lognormal' | Shift > 0 | Shifted Black |
'normal' | Shift = 0 | Normal (Bachelier) |
Data Types: char
| string
Output Arguments
CapletVols
— Stripped caplet volatilities
vector of decimals
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.
CapletPaymentDates
— Payment dates
vector of date numbers
Payment dates (in date numbers), returned as an
NCapletVols
-by-1
vector of date numbers
corresponding to CapletVols
.
CapStrikes
— Cap strikes
vector of decimals
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.
More About
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:
For more information, see Cap.
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 R2016aR2024b: Support for ratecurve
object for ZeroCurve
argument
The ZeroCurve
input argument supports a ratecurve
object.
R2022b: Serial date numbers not recommended
Although capvolstrip
supports serial date numbers,
datetime
values are recommended instead. The
datetime
data type provides flexible date and time
formats, storage out to nanosecond precision, and properties to account for time
zones and daylight saving time.
To convert serial date numbers or text to datetime
values, use the datetime
function. For example:
t = datetime(738427.656845093,"ConvertFrom","datenum"); y = year(t)
y = 2021
There are no plans to remove support for serial date number inputs.
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