Price swap instrument from Black-Karasinski interest-rate tree
Price an interest-rate swap with a fixed receiving leg and a floating paying leg. Payments are made once a year, and the notional principal amount is $100. The values for the remaining arguments are:
Coupon rate for fixed leg: 0.06 (6%)
Spread for floating leg: 20 basis points
Swap settlement date: Jan. 01, 2004
Swap maturity date: Jan. 01, 2006
Based on the information above, set the required arguments and build the LegRate, LegType, and LegReset matrices:
Settle = '01-Jan-2004'; Maturity = '01-Jan-2006'; Basis = 0; Principal = 100; LegRate = [0.06 20]; % [CouponRate Spread] LegType = [1 0]; % [Fixed Float] LegReset = [1 1]; % Payments once per year
Price the swap using the BKTree included in the MAT-file deriv.mat. BKTree contains the time and forward-rate information needed to price the instrument.
load deriv.mat;Use swapbybk to price of the swap.
Price = swapbybk(BKTree, LegRate,... Settle, Maturity, LegReset, Basis, Principal, LegType)
Price = 5.0425
Using the previous data, calculate the swap rate, which is the coupon rate for the fixed leg, such that the swap price at time = 0 is zero.
LegRate = [NaN 20]; [Price, PriceTree, SwapRate] = swapbybk(BKTree, LegRate,... Settle, Maturity, LegReset, Basis, Principal, LegType)
Price = -2.8422e-14
PriceTree = struct with fields:
FinObj: 'BKPriceTree'
PTree: {1x5 cell}
tObs: [0 1 2 3 4]
Connect: {[2] [2 3 4] [2 2 3 4 4]}
Probs: {[3x1 double] [3x3 double] [3x5 double]}
SwapRate = 0.0336
Price an amortizing swap using the Principal input argument to define the amortization schedule.
Create the RateSpec.
Rates = 0.035; ValuationDate = '1-Jan-2011'; StartDates = ValuationDate; EndDates = '1-Jan-2017'; Compounding = 1; RateSpec = intenvset('ValuationDate', ValuationDate,'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: 1
Disc: 0.8135
Rates: 0.0350
EndTimes: 6
StartTimes: 0
EndDates: 736696
StartDates: 734504
ValuationDate: 734504
Basis: 0
EndMonthRule: 1
Create the swap instrument using the following data:
Settle ='1-Jan-2011'; Maturity = '1-Jan-2017'; Period = 1; LegRate = [0.04 10];
Define the swap amortizing schedule.
Principal ={{'1-Jan-2013' 100;'1-Jan-2014' 80;'1-Jan-2015' 60;'1-Jan-2016' 40; '1-Jan-2017' 20}};Build the BK tree and assume volatility is 10%.
MatDates = {'1-Jan-2012'; '1-Jan-2013';'1-Jan-2014';'1-Jan-2015';'1-Jan-2016';'1-Jan-2017'};
BKTimeSpec = bktimespec(ValuationDate, MatDates);
Volatility = 0.10;
AlphaDates = '01-01-2017';
AlphaCurve = 0.1;
BKVolSpec = bkvolspec(ValuationDate, MatDates, Volatility*ones(1,length(MatDates))',...
AlphaDates, AlphaCurve);
BKT = bktree(BKVolSpec, RateSpec, BKTimeSpec);Compute the price of the amortizing swap.
Price = swapbybk(BKT, LegRate, Settle, Maturity, 'Principal' , Principal)Price = 1.4574
Price a forward swap using the StartDate input argument to define the future starting date of the swap.
Create the RateSpec.
Rates = 0.0374; ValuationDate = '1-Jan-2012'; StartDates = ValuationDate; EndDates = '1-Jan-2018'; Compounding = 1; RateSpec = intenvset('ValuationDate', ValuationDate,'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: 1
Disc: 0.8023
Rates: 0.0374
EndTimes: 6
StartTimes: 0
EndDates: 737061
StartDates: 734869
ValuationDate: 734869
Basis: 0
EndMonthRule: 1
Build a BK tree.
VolDates = {'1-Jan-2013'; '1-Jan-2014';'1-Jan-2015';'1-Jan-2016';'1-Jan-2017';'1-Jan-2018'};
VolCurve = 0.1;
AlphaDates = '01-01-2018';
AlphaCurve = 0.1;
BKVolSpec = bkvolspec(RateSpec.ValuationDate, VolDates, VolCurve,...
AlphaDates, AlphaCurve);
BKTimeSpec = bktimespec(RateSpec.ValuationDate, VolDates, Compounding);
BKT = bktree(BKVolSpec, RateSpec, BKTimeSpec);Compute the price of a forward swap that starts in a year (Jan 1, 2013) and matures in four years with a forward swap rate of 4.25%.
Settle ='1-Jan-2012'; Maturity = '1-Jan-2017'; StartDate = '1-Jan-2013'; LegRate = [0.0425 10]; Price = swapbybk(BKT, LegRate, Settle, Maturity, 'StartDate', StartDate)
Price = 1.4434
Using the previous data, compute the forward swap rate, the coupon rate for the fixed leg, such that the forward swap price at time = 0 is zero.
LegRate = [NaN 10];
[Price, ~,SwapRate] = swapbybk(BKT, LegRate, Settle, Maturity, 'StartDate', StartDate)Price = 1.4211e-14
SwapRate = 0.0384
BKTree — Interest-rate structureInterest-rate tree structure, created by bktree
Data Types: struct
LegRate — Leg rateLeg rate, specified as a NINST-by-2 matrix,
with each row defined as one of the following:
[CouponRate Spread] (fixed-float)
[Spread CouponRate] (float-fixed)
[CouponRate CouponRate] (fixed-fixed)
[Spread Spread] (float-float)
CouponRate is the decimal annual rate.
Spread is the number of basis points over the reference rate. The
first column represents the receiving leg, while the second column represents the
paying leg.
Data Types: double
Settle — Settlement dateSettlement date, specified either as a scalar or NINST-by-1 vector
of serial date numbers or date character vectors.
The Settle date for every swap is set to the
ValuationDate of the BK tree. The swap argument
Settle is ignored.
Data Types: char | double
Maturity — Maturity dateMaturity date, specified as a NINST-by-1 vector
of serial date numbers or date character vectors representing the
maturity date for each swap.
Data Types: char | 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.
[Price,PriceTree,CFTree,SwapRate] = swapbybk(BKTree,LegRate,Settle,Maturity,LegReset,Basis,Principal,LegType)'LegReset' — Reset frequency per year for each swap[1 1] (default) | vectorReset frequency per year for each swap, specified as the comma-separated pair consisting of
'LegReset' and a NINST-by-2
vector.
Data Types: double
'Basis' — Day-count basis representing the basis for each leg0 (actual/actual) (default) | integer from 0 to 13Day-count basis representing the basis for each leg, specified as the comma-separated pair
consisting of 'Basis' and a
NINST-by-1 array (or
NINST-by-2 if Basis is
different for each leg).
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 amounts or principal value schedules100 (default) | vector or cell arrayNotional principal amounts or principal value schedules, specified as the comma-separated pair
consisting of 'Principal' and a vector or cell array.
Principal accepts a NINST-by-1 vector
or NINST-by-1 cell array (or NINST-by-2 if Principal is
different for each leg) of the notional principal amounts or principal
value schedules. For schedules, each element of the cell array is
a NumDates-by-2 array where
the first column is dates and the second column is its associated
notional principal value. The date indicates the last day that the
principal value is valid.
Data Types: cell | double
'LegType' — Leg type[1 0] for each instrument (default) | matrix with values [1 1] (fixed-fixed), [1
0] (fixed-float), [0 1] (float-fixed),
or [0 0] (float-float)Leg type, specified as the comma-separated pair consisting of 'LegType' and
a NINST-by-2 matrix with values [1
1] (fixed-fixed), [1 0] (fixed-float), [0
1] (float-fixed), or [0 0] (float-float). Each row
represents an instrument. Each column indicates if the corresponding leg is fixed
(1) or floating (0). This matrix defines the
interpretation of the values entered in LegRate.
LegType allows [1 1] (fixed-fixed),
[1 0] (fixed-float), [0 1] (float-fixed), or
[0 0] (float-float) swaps
Data Types: double
'Options' — Derivatives pricing options structureDerivatives pricing options structure, specified as the comma-separated pair consisting of
'Options' and a structure obtained from using derivset.
Data Types: struct
'EndMonthRule' — End-of-month rule flag for generating dates when Maturity is end-of-month date for month having 30 or fewer days1 (in effect) (default) | nonnegative integer [0,1]End-of-month rule flag for generating dates when Maturity is an
end-of-month date for a month having 30 or fewer days, specified as the
comma-separated pair consisting of 'EndMonthRule' and a nonnegative
integer [0, 1] using a
NINST-by-1 (or
NINST-by-2 if EndMonthRule
is different for each leg).
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
'AdjustCashFlowsBasis' — Flag to adjust cash flows based on actual period day countfalse (default) | value of 0 (false) or 1 (true)Flag to adjust cash flows based on actual period day count, specified as the comma-separated
pair consisting of 'AdjustCashFlowsBasis' and a
NINST-by-1 (or
NINST-by-2 if
AdjustCashFlowsBasis is different for each leg) of logicals with
values of 0 (false) or 1 (true).
Data Types: logical
'BusinessDayConvention' — Business day conventionsactual
(default) | character vector | cell array of character vectorsBusiness day conventions, specified as the comma-separated pair consisting of
'BusinessDayConvention' and a character vector or a
N-by-1 (or
NINST-by-2 if
BusinessDayConvention is different for each leg) cell array of
character vectors of business day conventions. The selection for business day
convention determines how non-business days are treated. Non-business days are defined
as weekends plus any other date that businesses are not open (e.g. statutory
holidays). Values are:
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 | cell
'Holidays' — Holidays used in computing business daysholidays.m (default) | MATLAB® date numbersHolidays used in computing business days, specified as the comma-separated pair consisting of
'Holidays' and MATLAB date numbers using a
NHolidays-by-1 vector.
Data Types: double
'StartDate' — Date swap actually startsSettle date (default) | serial date number | character vectorDate swap actually starts, specified as the comma-separated pair consisting of
'StartDate' and a
NINST-by-1 vector of dates using a serial date
number or a character vector.
Use this argument to price forward swaps, that is, swaps that start in a future date
Data Types: char | double
Price — Expected swap prices at time 0Expected swap prices at time 0, returned as a NINST-by-1 vector.
PriceTree — Tree structure of instrument pricesTree structure of instrument prices, returned as a MATLAB structure
of trees containing vectors of swaption instrument prices and a vector
of observation times for each node. Within PriceTree:
PriceTree.PTree contains the clean
prices.
PriceTree.tObs contains the observation
times.
PriceTree.Connect contains the connectivity vectors. Each element in the
cell array describes how nodes in that level connect to the next. For a given
tree level, there are NumNodes elements in the vector, and
they contain the index of the node at the next level that the middle branch
connects to. Subtracting 1 from that value indicates where the up-branch
connects to, and adding 1 indicated where the down branch connects to.
PriceTree.Probs contains the probability
arrays. Each element of the cell array contains the up, middle, and
down transition probabilities for each node of the level.
CFTree — Swap cash flowsSwap cash flows, returned as a tree structure with a vector
of the swap cash flows at each node. This structure contains only NaNs
because with binomial recombining trees, cash flows cannot be computed
accurately at each node of a tree.
SwapRate — Rates applicable to fixed legRates applicable to the fixed leg, returned as a NINST-by-1 vector
of rates applicable to the fixed leg such that the swaps’ values
are zero at time 0. This rate is used in calculating the swaps’
prices when the rate specified for the fixed leg in LegRate is NaN.
The SwapRate output is padded with NaN for
those instruments in which CouponRate is not set
to NaN.
In an amortizing swap, the notional principal decreases periodically because it is tied to an underlying financial instrument with a declining (amortizing) principal balance, such as a mortgage.
Agreement to enter into an interest-rate swap arrangement on a fixed date in future.
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