capbybdt
Price cap instrument from Black-Derman-Toy interest-rate tree
Syntax
Description
[
computes the price of a cap instrument from a Black-Derman-Toy interest-rate tree.
Price
,PriceTree
]
= capbybdt(BDTTree
,Strike
,Settle
,Maturity
)capbybdt
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.
Examples
Price a 3% Cap Instrument Using a BDT Interest-Rate Tree
Load the file deriv.mat
, which provides BDTTree
. The BDTTree
structure contains the time and interest-rate information needed to price the cap instrument.
load deriv.mat;
Set the required values. Other arguments will use defaults.
Strike = 0.03; Settle = datetime(2000,1,1); Maturity = datetime(2004,1,1);
Use capbybdt
to compute the price of the cap instrument.
Price = capbybdt(BDTTree, Strike, Settle, Maturity)
Price = 28.4001
Price a 10% Cap Instrument Using a BDT Interest-Rate Tree
Set the required arguments for the three specifications required to create a BDT tree.
Compounding = 1; ValuationDate = datetime(2000,1,1); StartDate = ValuationDate; EndDates = [datetime(2001,1,1) ; datetime(2002,1,1) ; datetime(2003,1,1) ; datetime(2004,1,1) ; datetime(2005,1,1)]; Rates = [.1; .11; .12; .125; .13]; Volatility = [.2; .19; .18; .17; .16];
Create the specifications.
RateSpec = intenvset('Compounding', Compounding,... 'ValuationDate', ValuationDate,... 'StartDates', StartDate,... 'EndDates', EndDates,... 'Rates', Rates); BDTTimeSpec = bdttimespec(ValuationDate, EndDates, Compounding); BDTVolSpec = bdtvolspec(ValuationDate, EndDates, Volatility);
Create the BDT tree from the specifications.
BDTTree = bdttree(BDTVolSpec, RateSpec, BDTTimeSpec)
BDTTree = struct with fields:
FinObj: 'BDTFwdTree'
VolSpec: [1x1 struct]
TimeSpec: [1x1 struct]
RateSpec: [1x1 struct]
tObs: [0 1 2 3 4]
dObs: [730486 730852 731217 731582 731947]
TFwd: {[5x1 double] [4x1 double] [3x1 double] [2x1 double] [4]}
CFlowT: {[5x1 double] [4x1 double] [3x1 double] [2x1 double] [5]}
FwdTree: {[1.1000] [1.0979 1.1432] [1.0976 1.1377 1.1942] [1.0872 1.1183 1.1606 1.2179] [1.0865 1.1134 1.1486 1.1948 1.2552]}
Set the cap arguments. Remaining arguments will use defaults.
CapStrike = 0.10; Settlement = ValuationDate; Maturity = datetime(2002,1,1); CapReset = 1;
Use capbybdt
to find the price of the cap instrument.
Price= capbybdt(BDTTree, CapStrike, Settlement, Maturity,...
CapReset)
Price = 1.7169
Compute the Price of an Amortizing Cap Using the BDT Model
Define the RateSpec
.
Rates = [0.03583; 0.042147; 0.047345; 0.052707; 0.054302]; 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 instrument.
Settle = datetime(2011,11,15); Maturity = datetime(2015,11,15); Strike = 0.04; CapReset = 1; Principal ={{datetime(2012,11,15) 100;datetime(2013,11,15) 70;datetime(2014,11,15) 40;datetime(2015,11,15) 10}};
Build the BDT Tree.
BDTTimeSpec = bdttimespec(ValuationDate, EndDates); Volatility = 0.10; BDTVolSpec = bdtvolspec(ValuationDate, EndDates, Volatility*ones(1,length(EndDates))'); BDTTree = bdttree(BDTVolSpec, RateSpec, BDTTimeSpec)
BDTTree = struct with fields:
FinObj: 'BDTFwdTree'
VolSpec: [1x1 struct]
TimeSpec: [1x1 struct]
RateSpec: [1x1 struct]
tObs: [0 1 2 3 4]
dObs: [734822 735188 735553 735918 736283]
TFwd: {[5x1 double] [4x1 double] [3x1 double] [2x1 double] [4]}
CFlowT: {[5x1 double] [4x1 double] [3x1 double] [2x1 double] [5]}
FwdTree: {[1.0358] [1.0437 1.0534] [1.0469 1.0573 1.0700] [1.0505 1.0617 1.0754 1.0921] [1.0401 1.0490 1.0598 1.0731 1.0894]}
Price the amortizing cap.
Basis = 0; Price = capbybdt(BDTTree, Strike, Settle, Maturity, CapReset, Basis, Principal)
Price = 1.4042
Input Arguments
BDTTree
— Interest-rate tree structure
structure
Interest-rate tree structure, specified by using bdttree
.
Data Types: struct
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. The
Settle
date for every cap is set to the
ValuationDate
of the BDT tree. The cap argument
Settle
is ignored.
To support existing code, capbybdt
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, capbybdt
also
accepts serial date numbers as inputs, but they are not recommended.
CapReset
— Reset frequency payment per year
1
(default) | numeric
(Optional) Reset frequency payment per year, specified as a
NINST
-by-1
vector.
Data Types: double
Basis
— Day-count basis of instrument
0
(actual/actual) (default) | integer from 0
to 13
(Optional) 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
Principal
— Notional principal amount
100
(default) | numeric
(Optional) Notional principal amount, specified as a
NINST
-by-1
of notional principal amounts, or a
NINST
-by-1
cell array, where 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
Options
— Derivatives pricing options structure
structure
(Optional) Derivatives pricing options structure, specified using derivset
.
Data Types: struct
Output Arguments
Price
— Expected price of cap at time 0
vector
Expected price of the cap at time 0, returned as a NINST
-by-1
vector.
PriceTree
— Tree structure with values of cap at each node
vector
Tree structure with values of the cap at each node, returned as a MATLAB® structure of trees containing vectors of instrument prices and a vector of observation times for each node:
PriceTree.PTree
contains cap prices.PriceTree.tObs
contains the observation times.
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.
Version History
Introduced before R2006aR2022b: Serial date numbers not recommended
Although capbybdt
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
bdttree
| cfbybdt
| floorbybdt
| swapbybdt
| capbynormal
| Cap
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