capbyblk
Price caps using Black option pricing model
Syntax
Description
[
price caps using the Black option pricing model. CapPrice
,Caplets
]
= capbyblk(RateSpec
,Strike
,Settle
,Maturity
,Volatility
)capbyblk
computes prices
of vanilla caps and amortizing caps.
Note
Alternatively, you can use the Cap
object to price cap
instruments. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.
[
adds optional name-value pair arguments.CapPrice
,Caplets
]
= capbyblk(___,Name,Value
)
Examples
Price a Cap Using the Black Option Pricing Model
Consider an investor who gets into a contract that caps the interest rate on a $100,000 loan at 8% quarterly compounded for 3 months, starting on January 1, 2009. Assuming that on January 1, 2008 the zero rate is 6.9394% continuously compounded and the volatility is 20%, use this data to compute the cap price. First, calculate the RateSpec
:
ValuationDate = datetime(2008,1,1); EndDates = datetime(2010,4,1); Rates = 0.069394; Compounding = -1; Basis = 1; RateSpec = intenvset('ValuationDate', ValuationDate, ... 'StartDates', ValuationDate,'EndDates', EndDates, ... 'Rates', Rates,'Compounding', Compounding,'Basis', Basis)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.8554
Rates: 0.0694
EndTimes: 2.2500
StartTimes: 0
EndDates: 734229
StartDates: 733408
ValuationDate: 733408
Basis: 1
EndMonthRule: 1
Compute the price of the cap.
Settle = datetime(2009,1,1); % cap starts in a year Maturity = datetime(2009,4,1); Volatility = 0.20; CapRate = 0.08; CapReset = 4; Principal=100000; CapPrice = capbyblk(RateSpec, CapRate, Settle, Maturity, Volatility,... 'Reset',CapReset,'ValuationDate',ValuationDate,'Principal', Principal,... 'Basis', Basis)
CapPrice = 51.6125
Price a Cap Using a Different Curve to Generate the Future Forward Rates
Define the OIS and Libor rates.
Settle = datetime(2013,3,15); CurveDates = daysadd(Settle,360*[1/12 2/12 3/12 6/12 1 2 3 4 5 7 10],1); OISRates = [.0018 .0019 .0021 .0023 .0031 .006 .011 .017 .021 .026 .03]'; LiborRates = [.0045 .0047 .005 .0055 .0075 .0109 .0162 .0216 .0262 .0309 .0348]';
Create an associated RateSpec
for the OIS and Libor curves.
OISCurve = intenvset('Rates',OISRates,'StartDate',Settle,'EndDates',CurveDates,'Compounding',2,'Basis',1); LiborCurve = intenvset('Rates',LiborRates,'StartDate',Settle,'EndDates',CurveDates,'Compounding',2,'Basis',1);
Define the Cap instruments.
Maturity = [ datetime(2018,3,15) ; datetime(2020,3,15)]; Strike = [0.04;0.05]; BlackVol = 0.2;
Price the cap instruments using the term structure OISCurve
both for discounting the cash flows and generating future forward rates.
[Price, Caplets] = capbyblk(OISCurve, Strike, Settle, Maturity, BlackVol)
Price = 2×1
0.7472
0.9890
Caplets = 2×7
0 0.0000 0.0033 0.2996 0.4443 NaN NaN
0 0.0000 0.0003 0.1134 0.2112 0.2292 0.4349
Price the cap instruments using the term structure LiborCurve
to generate future forward rates. The term structure OISCurve
is used for discounting the cash flows.
[PriceLC, CapletsLC] = capbyblk(OISCurve, Strike, Settle, Maturity, BlackVol,'ProjectionCurve',LiborCurve)
PriceLC = 2×1
1.3293
1.6329
CapletsLC = 2×7
0 0.0000 0.0337 0.4250 0.8706 NaN NaN
0 0.0000 0.0052 0.1767 0.4849 0.3663 0.5998
Compute the Price of Two Amortizing Caps Using the Black Model
Define the RateSpec
.
Rates = [0.0358; 0.0421; 0.0473; 0.0527; 0.0543]; ValuationDate = datetime(2011,11,15); StartDates = ValuationDate; EndDates = [datetime(2012,11,15) ; datetime(2013,11,15) ; datetime(2014,11,15) ; datetime(2015,11,15) ; datetime(2016,11,15)]; Compounding = 1; RateSpec = intenvset('ValuationDate', ValuationDate,'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: 1
Disc: [5x1 double]
Rates: [5x1 double]
EndTimes: [5x1 double]
StartTimes: [5x1 double]
EndDates: [5x1 double]
StartDates: 734822
ValuationDate: 734822
Basis: 0
EndMonthRule: 1
Define the cap instruments.
Settle = datetime(2011,11,15); Maturity = datetime(2015,11,15); Strike = [0.03;0.035]; Reset = 1; Principal ={{datetime(2012,11,15) 100;datetime(2013,11,15) 70;datetime(2014,11,15) 40;datetime(2015,11,15) 10}};
Price the amortizing caps.
Volatility = 0.10; Price = capbyblk (RateSpec, Strike, Settle, Maturity, Volatility,... 'Reset', Reset,'Principal', Principal)
Price = 2×1
3.0339
2.0141
Price a Cap Using the Shifted Black Model
Create the RateSpec
.
ValuationDate = datetime(2016,3,1); EndDates = [datetime(2017,3,1) ; datetime(2018,3,1) ; datetime(2019,3,1) ; datetime(2020,3,1) ; datetime(2021,3,1)]; Rates = [-0.21; -0.12; 0.01; 0.10; 0.20]/100; Compounding = 1; Basis = 1; RateSpec = intenvset('ValuationDate',ValuationDate,'StartDates',ValuationDate, ... 'EndDates',EndDates,'Rates',Rates,'Compounding',Compounding,'Basis',Basis)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: 1
Disc: [5x1 double]
Rates: [5x1 double]
EndTimes: [5x1 double]
StartTimes: [5x1 double]
EndDates: [5x1 double]
StartDates: 736390
ValuationDate: 736390
Basis: 1
EndMonthRule: 1
Price the cap with a negative strike using the Shifted Black model.
Settle = datetime(2016,6,1); % Cap starts in 3 months. Maturity = datetime(2016,9,1); ShiftedBlackVolatility = 0.31; CapRate = -0.003; % -0.3 percent strike. CapReset = 4; Principal = 100000; Shift = 0.01; % 1 percent shift. CapPrice = capbyblk(RateSpec,CapRate,Settle,Maturity,ShiftedBlackVolatility,... 'Reset',CapReset,'ValuationDate',ValuationDate,'Principal',Principal,... 'Basis',Basis,'Shift',Shift)
CapPrice = 26.0733
Input Arguments
Strike
— Rate at which cap is exercised
decimal
Rate at which cap is exercised, specified as a NINST
-by-1
vector
of decimal values.
Data Types: double
Settle
— Settlement date for cap
datetime array | string array | date character vector
Settlement date for the cap, specified as a NINST
-by-1
vector using a datetime array, string array, or date character vectors.
To support existing code, capbyblk
also
accepts serial date numbers as inputs, but they are not recommended.
Maturity
— Maturity date for cap
datetime array | string array | date character vector
Maturity date for the cap, specified as a NINST
-by-1
vector using a datetime array, string array, or date character vectors.
To support existing code, capbyblk
also
accepts serial date numbers as inputs, but they are not recommended.
Volatility
— Volatilities values
numeric
Volatilities values, specified as a NINST
-by-1
vector
of numeric values.
The Volatility
input is not intended for
volatility surfaces or cubes. If you specify a matrix for the Volatility
input, capbyblk
internally
converts it into a vector. capbyblk
assumes that
the volatilities specified in the Volatility
input
are flat volatilities, which are applied equally to each of the caplets.
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: [CapPrice,Caplets] =
capbyblk(RateSpec,Strike,Settle,Maturity,Volatility,'Reset',CapReset,'Principal',100000,'Basis',7)
Reset
— Reset frequency payment per year
1
(default) | numeric
Reset frequency payment per year, specified as a
NINST
-by-1
vector.
Data Types: double
Principal
— Notional principal amount
100
(default) | numeric
Notional principal amount, specified as a NINST
-by-1
vector
or a NINST
-by-1
cell array.
When Principal
is 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.
Use Principal
to pass a schedule to compute
the price for an amortizing cap.
Data Types: double
| cell
Basis
— Day-count basis of instrument
0
(actual/actual) (default) | integer from 0
to 13
Day-count basis representing the basis used when annualizing
the input forward rate, specified as a NINST
-by-1
vector
of integers.
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
ProjectionCurve
— Rate curve used in generating future forward rates
if ProjectionCurve
is not
specified, then RateSpec
is used both for discounting
cash flows and projecting future forward rates (default) | structure
The rate curve to be used in generating the future forward rates.
This structure must be created using intenvset
.
Use this optional input if the forward curve is different from the
discount curve.
Data Types: struct
Shift
— Shift in decimals for shifted Black model
0
(no shift) (default) | positive decimal
Shift in decimals for the shifted Black model, specified using
a scalar or NINST
-by-1
vector
of rate shifts in positive decimals. Set this parameter to a positive
rate 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. For example, a Shift
of 0.01
is
equal to a 1% shift.
Data Types: double
Output Arguments
CapPrice
— Expected price of cap
vector
Expected price of the cap, returned as a NINST
-by-1
vector.
Caplets
— Caplets
array
Caplets, returned as a NINST
-by-NCF
array
of caplets, padded with NaN
s.
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.
Shifted Black
The Shifted Black model is essentially the same as the Black’s model, except that it models the movements of (F + Shift) as the underlying asset, instead of F (which is the forward rate in the case of caplets).
This model allows negative rates, with a fixed negative lower bound defined by the amount of shift; that is, the zero lower bound of Black’s model has been shifted.
Algorithms
Black Model
Where F is the forward value and K is the strike.
Shifted Black Model
Where F+Shift is the forward value and K+Shift is the strike for the shifted version.
Version History
Introduced in R2009aR2022b: Serial date numbers not recommended
Although capbyblk
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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