swaptionbyblk
Price European swaption instrument using Black model
Syntax
Description
prices swaptions using the Black option pricing model. Price
= swaptionbyblk(RateSpec
,OptSpec
,Strike
,Settle
,ExerciseDates
,Maturity
,Volatility
)
Note
Alternatively, you can use the Swaption
object to price
swaption instruments. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.
adds optional name-value pair arguments. Price
= swaptionbyblk(___,Name,Value
)
Examples
Price a European swaption that gives the holder the right to enter in five years into a three-year paying swap where a fixed-rate of 6.2% is paid and floating is received. Assume that the yield curve is flat at 6% per annum with continuous compounding, the volatility of the swap rate is 20%, the principal is $100, and payments are exchanged semiannually.
Create the RateSpec
.
Rate = 0.06; Compounding = -1; ValuationDate = datetime(2010,1,1); EndDates = datetime(2020,1,1); Basis = 1; RateSpec = intenvset('ValuationDate', ValuationDate,'StartDates', ValuationDate, ... 'EndDates', EndDates, 'Rates', Rate, 'Compounding', Compounding, 'Basis', Basis);
Price the swaption using the Black model.
Settle = datetime(2011,1,1); ExerciseDates = datetime(2016,1,1); Maturity = datetime(2019,1,1); Reset = 2; Principal = 100; Strike = 0.062; Volatility = 0.2; OptSpec = 'call'; Price= swaptionbyblk(RateSpec, OptSpec, Strike, Settle, ExerciseDates, Maturity, ... Volatility, 'Reset', Reset, 'Principal', Principal, 'Basis', Basis)
Price = 2.0710
This example shows Price a European swaption with receiving and paying legs that gives the holder the right to enter in five years into a three-year paying swap where a fixed-rate of 6.2% is paid and floating is received. Assume that the yield curve is flat at 6% per annum with continuous compounding, the volatility of the swap rate is 20%, the principal is $100, and payments are exchanged semiannually.
Rate = 0.06; Compounding = -1; ValuationDate = datetime(2010,1,1); EndDates = datetime(2020,1,1); Basis = 1;
Define the RateSpec
.
RateSpec = intenvset('ValuationDate',ValuationDate,'StartDates',ValuationDate, ... 'EndDates',EndDates,'Rates',Rate,'Compounding',Compounding,'Basis',Basis);
Define the swaption arguments.
Settle = datetime(2011,1,1); ExerciseDates = datetime(2016,1,1); Maturity = datetime(2019,1,1); Reset = [2 4]; % 1st column represents receiving leg, 2nd column represents paying leg Principal = 100; Strike = 0.062; Volatility = 0.2; OptSpec = 'call'; Basis = [1 3]; % 1st column represents receiving leg, 2nd column represents paying leg
Price the swaption.
Price= swaptionbyblk(RateSpec,OptSpec,Strike,Settle,ExerciseDates,Maturity,Volatility, ... 'Reset',Reset,'Principal',Principal,'Basis',Basis)
Price = 1.6494
Price a European swaption that gives the holder the right to enter into a 5-year receiving swap in a year, where a fixed rate of 3% is received and floating is paid. Assume that the 1-year, 2-year, 3-year, 4-year and 5- year zero rates are 3%, 3.4%, 3.7%, 3.9% and 4% with continuous compounding. The swap rate volatility is 21%, the principal is $1000, and payments are exchanged semiannually.
Create the RateSpec
.
ValuationDate = datetime(2010,1,1); EndDates = [datetime(2011,1,1) ; datetime(2012,1,1) ; datetime(2013,1,1) ; datetime(2014,1,1) ; datetime(2015,1,1)]; Rates = [0.03; 0.034 ; 0.037; 0.039; 0.04;]; 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: [5×1 double]
Rates: [5×1 double]
EndTimes: [5×1 double]
StartTimes: [5×1 double]
EndDates: [5×1 double]
StartDates: 734139
ValuationDate: 734139
Basis: 1
EndMonthRule: 1
Price the swaption using the Black model.
Settle = datetime(2011,1,1); ExerciseDates = datetime(2012,1,1); Maturity = datetime(2017,1,1); Strike = 0.03; Volatility = 0.21; Principal =1000; Reset = 2; OptSpec = 'put'; Price = swaptionbyblk(RateSpec, OptSpec, Strike, Settle, ExerciseDates, ... Maturity, Volatility,'Basis', Basis, 'Reset', Reset,'Principal', Principal)
Price = 0.5771
Define the OIS and Libor curves.
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 swaption instruments.
ExerciseDate = datetime(2018,3,15);
Maturity = [datetime(2020,3,15) ; datetime(2023,3,15)];
OptSpec = 'call';
Strike = 0.04;
BlackVol = 0.2;
Price the swaption instruments using the term structure OISCurve
both for discounting the cash flows and generating the future forward rates.
Price = swaptionbyblk(OISCurve, OptSpec, Strike, Settle, ExerciseDate, Maturity, BlackVol,'Reset',1)
Price = 2×1
1.0956
2.6944
Price the swaption instruments using the term structure LiborCurve
to generate the future forward rates. The term structure OISCurve
is used for discounting the cash flows.
PriceLC = swaptionbyblk(OISCurve, OptSpec, Strike, Settle, ExerciseDate, Maturity, BlackVol,'ProjectionCurve',LiborCurve,'Reset',1)
PriceLC = 2×1
1.5346
3.8142
Create the RateSpec
.
ValuationDate = datetime(2016,1,1); EndDates = [datetime(2017,1,1) ; datetime(2018,1,1) ; datetime(2019,1,1) ; datetime(2020,1,1) ; datetime(2021,1,1)]; Rates = [-0.02; 0.024 ; 0.047; 0.090; 0.12;]/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: [5×1 double]
Rates: [5×1 double]
EndTimes: [5×1 double]
StartTimes: [5×1 double]
EndDates: [5×1 double]
StartDates: 736330
ValuationDate: 736330
Basis: 1
EndMonthRule: 1
Price the swaption with a negative strike using the Shifted Black model.
Settle = datetime(2016,1,1); ExerciseDates = datetime(2017,1,1); Maturity = 'Jan-1-2020'; Strike = -0.003; % Set -0.3 percent strike. ShiftedBlackVolatility = 0.31; Principal = 1000; Reset = 1; OptSpec = 'call'; Shift = 0.008; % Set 0.8 percent shift. Price = swaptionbyblk(RateSpec,OptSpec,Strike,Settle,ExerciseDates, ... Maturity,ShiftedBlackVolatility,'Basis',Basis,'Reset',Reset,... 'Principal',Principal,'Shift',Shift)
Price = 12.8301
Create the RateSpec
.
ValuationDate = datetime(2016,1,1); EndDates = [datetime(2017,1,1) ; datetime(2018,1,1) ; datetime(2019,1,1) ; datetime(2020,1,1) ; datetime(2021,1,1)]; Rates = [-0.02; 0.024 ; 0.047; 0.090; 0.12;]/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: [5×1 double]
Rates: [5×1 double]
EndTimes: [5×1 double]
StartTimes: [5×1 double]
EndDates: [5×1 double]
StartDates: 736330
ValuationDate: 736330
Basis: 1
EndMonthRule: 1
Price the swaptions with using the Shifted Black model.
Settle = datetime(2016,1,1); ExerciseDates = datetime(2017,1,1); Maturities = [datetime(2018,1,1) ; datetime(2019,1,1) ; datetime(2020,1,1)]; Strikes = [-0.0034;-0.0032;-0.003]; ShiftedBlackVolatilities = [0.33;0.32;0.31]; % A vector of volatilities. Principal = 1000; Reset = 1; OptSpec = 'call'; Shifts = [0.0085;0.0082;0.008]; % A vector of shifts. Prices = swaptionbyblk(RateSpec,OptSpec,Strikes,Settle,ExerciseDates, ... Maturities,ShiftedBlackVolatilities,'Basis',Basis,'Reset',Reset, ... 'Principal',Principal,'Shift',Shifts)
Prices = 3×1
4.1117
8.0577
12.8301
Input Arguments
Interest-rate term structure (annualized and continuously compounded),
specified by the RateSpec
obtained from intenvset
. For information on the interest-rate
specification, see intenvset
.
If the paying leg is different than the receiving leg, the RateSpec
can
be a NINST
-by-2
input variable
of RateSpec
s, with the second input being the discount
curve for the paying leg. If only one curve is specified, then it
is used to discount both legs.
Data Types: struct
Definition of the option as 'call'
or 'put'
,
specified as a NINST
-by-1
cell
array of character vectors.
A 'call'
swaption, or Payer
swaption, allows the option buyer to enter into an interest-rate
swap in which the buyer of the option pays the fixed rate and receives
the floating rate.
A 'put'
swaption, or Receiver
swaption, allows the option buyer to enter into an interest-rate
swap in which the buyer of the option receives the fixed rate and
pays the floating rate.
Data Types: char
| cell
Strike swap rate values, specified as a NINST
-by-1
vector
of decimal values.
Data Types: double
Settlement date (representing the settle date for each swaption), specified as a
NINST
-by-1
vector using a datetime array, string
array, or date character vectors. Settle
must not be later than
ExerciseDates
.
To support existing code, swaptionbyblk
also
accepts serial date numbers as inputs, but they are not recommended.
The Settle
date input for swaptionbyblk
is
the valuation date on which the swaption (an option to enter into
a swap) is priced. The swaption buyer pays this price on this date
to hold the swaption.
Dates, specified as a vector using a datetime array, string array, or date character vectors on which the swaption expires and the underlying swap starts. The swaption holder can choose to enter into the swap on this date if the situation is favorable.
To support existing code, swaptionbyblk
also
accepts serial date numbers as inputs, but they are not recommended.
For a European option, ExerciseDates
are
a NINST
-by-1
vector of exercise
dates. Each row is the schedule for one option. When using a European
option, there is only one ExerciseDate
on the option
expiry date.
Maturity date for each forward swap, specified as a
NINST
-by-1
vector using a datetime array, string
array, or date character vectors.
To support existing code, swaptionbyblk
also
accepts serial date numbers as inputs, but they are not recommended.
Annual volatilities values, specified as a
NINST
-by-1
vector of numeric values.
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: Price = swaptionbyblk(OISCurve,OptSpec,Strike,Settle,ExerciseDate,Maturity,BlackVol,'Reset',1,'Shift',.5)
Day-count basis of the instrument, specified as the comma-separated pair
consisting of 'Basis'
and a
NINST
-by-1
vector or
NINST
-by-2
matrix representing the basis for
each leg. If Basis
is
NINST
-by-2
, the first column represents the
receiving leg, while the second column represents the paying leg. Default is
0
(actual/actual).
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
Notional principal amount, specified as the comma-separated pair consisting of
'Principal'
and a
NINST
-by-1
vector.
Data Types: double
Reset frequency per year for the underlying forward swap, specified as the comma-separated
pair consisting of 'Reset'
and a
NINST
-by-1
vector or
NINST
-by-2
matrix representing the reset
frequency per year for each leg. If Reset
is
NINST
-by-2
, the first column represents the
receiving leg, while the second column represents the paying leg.
Data Types: double
The rate curve to be used in generating the future forward rates, specified as the
comma-separated pair consisting of 'ProjectionCurve'
and a
structure created using intenvset
. Use this optional input
if the forward curve is different from the discount curve.
Data Types: struct
Shift in decimals for the shifted Black model, specified as the comma-separated pair
consisting of 'Shift'
and 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 swap rate and strike, which effectively sets a negative lower
bound for the forward swap rate and strike. For example, a Shift
of
0.01
is equal to a 1% shift.
Data Types: double
Output Arguments
Prices for the swaptions at time 0, returned as a NINST
-by-1
vector
of prices.
More About
A swaption (swap option) is a financial derivative that gives the holder the right, but not the obligation, to enter into an interest-rate swap agreement at a specified future date and under predetermined terms.
Swaptions are used by investors and institutions to hedge against interest rate fluctuations or to speculate on future changes in interest rates.
A call swaption or payer swaption allows the option buyer to enter into an interest-rate swap in which the buyer of the option pays the fixed rate and receives the floating rate.
A put swaption or receiver swaption allows the option buyer to enter into an interest-rate swap in which the buyer of the option receives the fixed rate and pays the floating rate.
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 swap rate in the case of swaptions).
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
Where F is the forward value and K is the strike.
Where F+Shift is the forward value and K+Shift is the strike for the shifted version.
Version History
Introduced before R2006aAlthough swaptionbyblk
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.
See Also
bondbyzero
| cfbyzero
| fixedbyzero
| floatbyzero
| blackvolbysabr
| intenvset
| swaptionbynormal
| capbyblk
| floorbyblk
| Swaption
Topics
- Calibrate the SABR Model
- Price a Swaption Using the SABR Model
- Price Swaptions with Negative Strikes Using the Shifted SABR Model
- Price a Swaption Using SABR Model and Analytic Pricer
- Swaption
- Work with Negative Interest Rates Using Functions
- Supported Interest-Rate Instrument Functions
- Mapping Financial Instruments Toolbox Functions for Interest-Rate Instrument Objects
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