Price caps using Normal or Bachelier pricing model
[
adds optional name-value pair arguments.CapPrice
,Caplets
]
= capbynormal(___,Name,Value
)
Consider an investor who gets into a contract that caps the interest rate on a $100,000 loan at –.08% quarterly compounded for 3 months, starting on January 1, 2009. Assuming that on January 1, 2008 the zero rate is .069394% continuously compounded and the volatility is 20%, use this data to compute the cap price. First, calculate the RateSpec
, and then use capbynormal
to compute the CapPrice
.
ValuationDate = 'Jan-01-2008'; EndDates ='April-01-2010'; Rates = 0.0069394; Compounding = -1; Basis = 1; RateSpec = intenvset('ValuationDate', ValuationDate, ... 'StartDates', ValuationDate,'EndDates', EndDates, ... 'Rates', Rates,'Compounding', Compounding,'Basis', Basis); Settle = 'Jan-01-2009'; % cap starts in a year Maturity = 'April-01-2009'; Volatility = 0.20; CapRate = -0.008; CapReset = 4; Principal=100000; CapPrice = capbynormal(RateSpec, CapRate, Settle, Maturity, Volatility,... 'Reset',CapReset,'ValuationDate',ValuationDate,'Principal', Principal,... 'Basis', Basis)
CapPrice = 2.1682e+03
capbynormal
and Compare to capbyblk
Define the RateSpec
.
Settle = datenum('20-Jan-2016'); ZeroTimes = [.5 1 2 3 4 5 7 10 20 30]'; ZeroRates = [0.0052 0.0055 0.0061 0.0073 0.0094 0.0119 0.0168 0.0222 0.0293 0.0307]'; ZeroDates = datemnth(Settle,12*ZeroTimes); RateSpec = intenvset('StartDate',Settle,'EndDates',ZeroDates,'Rates',ZeroRates)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: 2
Disc: [10x1 double]
Rates: [10x1 double]
EndTimes: [10x1 double]
StartTimes: [10x1 double]
EndDates: [10x1 double]
StartDates: 736349
ValuationDate: 736349
Basis: 0
EndMonthRule: 1
Define the cap instrument and price with capbyblk
.
ExerciseDate = datenum('20-Jan-2026');
[~,ParSwapRate] = swapbyzero(RateSpec,[NaN 0],Settle,ExerciseDate)
ParSwapRate = 0.0216
Strike = .01; BlackVol = .3; NormalVol = BlackVol*ParSwapRate; Price = capbyblk(RateSpec,Strike,Settle,ExerciseDate,BlackVol)
Price = 11.8693
Price the cap instrument using capbynormal
.
Price_Normal = capbynormal(RateSpec,Strike,Settle,ExerciseDate,NormalVol)
Price_Normal = 12.5495
Price the cap instrument using capbynormal
for a negative strike.
Price_Normal = capbynormal(RateSpec,-.005,Settle,ExerciseDate,NormalVol)
Price_Normal = 24.4816
Strike
— Rate at which cap is exercisedRate at which cap is exercised, specified as a NINST
-by-1
vector
of decimal values.
Data Types: double
Settle
— Settlement date for capSettlement date for the cap, specified as a NINST
-by-1
vector
of serial date numbers, date character vectors, datetime objects,
or string objects.
Data Types: double
| char
| datetime
| string
Maturity
— Maturity date for capMaturity date for the cap, specified as a NINST
-by-1
vector
of serial date numbers, date character vectors, datetime objects,
or string objects.
Data Types: double
| char
| datetime
| string
Volatility
— Normal volatilities valuesNormal volatilities values, specified as a NINST
-by-1
vector
of numeric values.
For more information on the Normal model, see Work with Negative Interest Rates.
Data Types: double
Specify optional
comma-separated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.
[CapPrice,Caplets] = capbynormal(RateSpec,Strike,Settle,Maturity,Volatility,'Reset',CapReset,'Principal',100000,'Basis',7)
'Reset'
— Reset frequency payment per year 1
(default) | numericReset frequency payment per year, specified as the comma-separated
pair consisting of 'Reset'
and a NINST
-by-1
vector.
Data Types: double
'Principal'
— Notional principal amount100
(default) | numericNotional principal amount, specified as the comma-separated
pair consisting of 'Principal'
and a NINST
-by-1
of
notional principal amounts, or a NINST
-by-1
cell
array. Each element in the NINST
-by-1
cell
array is a NumDates
-by-2
cell
array, where the first column is dates, and the second column is the
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 instrument0
(actual/actual) (default) | integer from 0
to 13
Day-count basis of instrument representing the basis used when
annualizing the input forward rate, specified as the comma-separated
pair consisting of 'Basis'
and a NINST
-by-1
vector
of integers. Values are:
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
'ValuationDate'
— Observation date of investment horizonValuationDate
is not specified,
then Settle
is used (default) | serial date number | date character vector | datetime object | string objectObservation date of the investment horizon, specified as the comma-separated pair consisting
of 'ValuationDate'
and a serial date number, date character
vector, datetime object, or string array.
Data Types: double
| char
| datetime
| string
'ProjectionCurve'
— Rate curve used in generating future cash flowsProjectionCurve
is not
specified, then RateSpec
is used both for discounting
cash flows and projecting future cash flows (default) | structureThe rate curve to be used in projecting the future cash flows,
specified as the comma-separated pair consisting of 'ProjectionCurve'
and
rate curve structure. This structure must be created using intenvset
. Use this optional input if
the forward curve is different from the discount curve.
Data Types: struct
CapPrice
— Expected price of capExpected price of the cap, returned as a NINST
-by-1
vector.
Caplets
— CapletsCaplets, returned as a NINST
-by-NCF
array
of caplets, padded with NaN
s.
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.
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