bkprice
Instrument prices from Black-Karasinski interest-rate tree
Description
[
computes arbitrage-free prices for instruments using an interest-rate tree created with
Price
,PriceTree
] = bkprice(BKTree
,InstSet
)bktree
. All instruments contained in a
financial instrument variable, InstSet
, are priced.
bkprice
handles instrument types: 'Bond'
,
'CashFlow'
, 'OptBond'
,
'OptEmBond'
, 'OptEmBond'
,
'OptFloat'
, 'OptEmFloat'
,
'Fixed'
, 'Float'
, 'Cap'
,
'Floor'
, 'RangeFloat'
, 'Swap'
.
See instadd
to construct defined types.
Examples
Price Cap and Bond Instruments in the Instrument Set
Load the BK tree and instruments from the data file deriv.mat
. Price the cap and bond instruments contained in the instrument set.
load deriv.mat; BKSubSet = instselect(BKInstSet,'Type', {'Bond', 'Cap'}); instdisp(BKSubSet)
Index Type CouponRate Settle Maturity Period Basis EndMonthRule IssueDate FirstCouponDate LastCouponDate StartDate Face Name Quantity 1 Bond 0.03 01-Jan-2004 01-Jan-2007 1 0 1 NaN NaN NaN NaN 100 3% bond 20 2 Bond 0.03 01-Jan-2004 01-Jan-2008 1 0 1 NaN NaN NaN NaN 100 3% bond 15 Index Type Strike Settle Maturity CapReset Basis Principal Name Quantity 3 Cap 0.04 01-Jan-2004 01-Jan-2008 1 0 100 4% Cap 10
[Price, PriceTree] = bkprice(BKTree, BKSubSet)
Price = 3×1
98.1096
95.6734
2.2706
PriceTree = struct with fields:
FinObj: 'BKPriceTree'
PTree: {[3x1 double] [3x3 double] [3x5 double] [3x5 double] [3x5 double]}
AITree: {[3x1 double] [3x3 double] [3x5 double] [3x5 double] [3x5 double]}
tObs: [0 1 2 3 4]
Connect: {[2] [2 3 4] [2 2 3 4 4]}
Probs: {[3x1 double] [3x3 double] [3x5 double]}
You can use treeviewer
to see the prices of these three instruments along the price tree.
Price Multi-Stepped Coupon Bonds
Price the following multi-stepped coupon bonds using the following data:
% The data for the interest rate term structure is as follows: Rates = [0.035; 0.042147; 0.047345; 0.052707]; ValuationDate = 'Jan-1-2010'; StartDates = ValuationDate; EndDates = {'Jan-1-2011'; 'Jan-1-2012'; 'Jan-1-2013'; 'Jan-1-2014'}; Compounding = 1; % Create RateSpec RS = intenvset('ValuationDate', ValuationDate, 'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding); % Create a portfolio of stepped coupon bonds with different maturities Settle = '01-Jan-2010'; Maturity = {'01-Jan-2011';'01-Jan-2012';'01-Jan-2013';'01-Jan-2014'}; CouponRate = {{'01-Jan-2011' .042;'01-Jan-2012' .05; '01-Jan-2013' .06; '01-Jan-2014' .07}}; ISet = instbond(CouponRate, Settle, Maturity, 1); instdisp(ISet)
Index Type CouponRate Settle Maturity Period Basis EndMonthRule IssueDate FirstCouponDate LastCouponDate StartDate Face 1 Bond [Cell] 01-Jan-2010 01-Jan-2011 1 0 1 NaN NaN NaN NaN 100 2 Bond [Cell] 01-Jan-2010 01-Jan-2012 1 0 1 NaN NaN NaN NaN 100 3 Bond [Cell] 01-Jan-2010 01-Jan-2013 1 0 1 NaN NaN NaN NaN 100 4 Bond [Cell] 01-Jan-2010 01-Jan-2014 1 0 1 NaN NaN NaN NaN 100
Build the BKTree
with the following data:
VolDates = ['1-Jan-2011'; '1-Jan-2012'; '1-Jan-2013'; '1-Jan-2014']; VolCurve = 0.01; AlphaDates = '01-01-2014'; AlphaCurve = 0.1; BKVolSpec = bkvolspec(RS.ValuationDate, VolDates, VolCurve,... AlphaDates, AlphaCurve); BKTimeSpec = bktimespec(RS.ValuationDate, VolDates, Compounding); BKT = bktree(BKVolSpec, RS, BKTimeSpec);
Compute the price of the stepped coupon bonds.
PBK = bkprice(BKT, ISet)
PBK = 4×1
100.6763
100.7368
100.9266
101.0115
Price a Portfolio of Stepped Callable Bonds and Stepped Vanilla Bonds
Price a portfolio of stepped callable bonds and stepped vanilla bonds using the following data:
% The data for the interest rate term structure is as follows: Rates = [0.035; 0.042147; 0.047345; 0.052707]; ValuationDate = 'Jan-1-2010'; StartDates = ValuationDate; EndDates = {'Jan-1-2011'; 'Jan-1-2012'; 'Jan-1-2013'; 'Jan-1-2014'}; Compounding = 1; % Create RateSpec RS = intenvset('ValuationDate', ValuationDate, 'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding); % Create an instrument portfolio of 3 stepped callable bonds and three % stepped vanilla bonds Settle = '01-Jan-2010'; Maturity = {'01-Jan-2012';'01-Jan-2013';'01-Jan-2014'}; CouponRate = {{'01-Jan-2011' .042;'01-Jan-2012' .05; '01-Jan-2013' .06; '01-Jan-2014' .07}}; OptSpec='call'; Strike=100; ExerciseDates='01-Jan-2011'; % Callable in one year % Bonds with embedded option ISet = instoptembnd(CouponRate, Settle, Maturity, OptSpec, Strike,... ExerciseDates, 'Period', 1); % Vanilla bonds ISet = instbond(ISet, CouponRate, Settle, Maturity, 1); % Display the instrument portfolio instdisp(ISet)
Index Type CouponRate Settle Maturity OptSpec Strike ExerciseDates Period Basis EndMonthRule IssueDate FirstCouponDate LastCouponDate StartDate Face AmericanOpt 1 OptEmBond [Cell] 01-Jan-2010 01-Jan-2012 call 100 01-Jan-2011 1 0 1 NaN NaN NaN NaN 100 0 2 OptEmBond [Cell] 01-Jan-2010 01-Jan-2013 call 100 01-Jan-2011 1 0 1 NaN NaN NaN NaN 100 0 3 OptEmBond [Cell] 01-Jan-2010 01-Jan-2014 call 100 01-Jan-2011 1 0 1 NaN NaN NaN NaN 100 0 Index Type CouponRate Settle Maturity Period Basis EndMonthRule IssueDate FirstCouponDate LastCouponDate StartDate Face 4 Bond [Cell] 01-Jan-2010 01-Jan-2012 1 0 1 NaN NaN NaN NaN 100 5 Bond [Cell] 01-Jan-2010 01-Jan-2013 1 0 1 NaN NaN NaN NaN 100 6 Bond [Cell] 01-Jan-2010 01-Jan-2014 1 0 1 NaN NaN NaN NaN 100
Build the BKTree
with the following data:
VolDates = ['1-Jan-2011'; '1-Jan-2012'; '1-Jan-2013'; '1-Jan-2014']; VolCurve = 0.01; AlphaDates = '01-01-2014'; AlphaCurve = 0.1; BKVolSpec = bkvolspec(RS.ValuationDate, VolDates, VolCurve,... AlphaDates, AlphaCurve); BKTimeSpec = bktimespec(RS.ValuationDate, VolDates, Compounding); BKT = bktree(BKVolSpec, RS, BKTimeSpec);
Compute the price, where the first three rows of the output corresponds to the price of the stepped callable bonds and the last three rows corresponds to the price of the stepped vanilla bonds.
PBK = bkprice(BKT, ISet)
PBK = 6×1
100.6729
100.6763
100.6763
100.7368
100.9266
101.0115
Price a Portfolio of Range Notes and Floating-Rate Notes
Price a portfolio of range notes and floating-rate notes using the following data:
% The data for the interest rate term structure is as follows: Rates = [0.035; 0.042147; 0.047345; 0.052707]; ValuationDate = 'Jan-1-2011'; StartDates = ValuationDate; EndDates = {'Jan-1-2012'; 'Jan-1-2013'; 'Jan-1-2014'; 'Jan-1-2015'}; Compounding = 1; % Create RateSpec RS = intenvset('ValuationDate', ValuationDate, 'StartDates',... StartDates, 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding); % Create an instrument portfolio with two range notes and a floating rate % note with the following data: Spread = 200; Settle = 'Jan-1-2011'; Maturity = 'Jan-1-2014'; % First Range Note: RateSched(1).Dates = {'Jan-1-2012'; 'Jan-1-2013' ; 'Jan-1-2014'}; RateSched(1).Rates = [0.045 0.055; 0.0525 0.0675; 0.06 0.08]; % Second Range Note: RateSched(2).Dates = {'Jan-1-2012'; 'Jan-1-2013' ; 'Jan-1-2014'}; RateSched(2).Rates = [0.048 0.059; 0.055 0.068 ; 0.07 0.09]; % Create InstSet InstSet = instadd('RangeFloat', Spread, Settle, Maturity, RateSched); % Add a floating-rate note InstSet = instadd(InstSet, 'Float', Spread, Settle, Maturity); % Display the portfolio instrument instdisp(InstSet)
Index Type Spread Settle Maturity RateSched FloatReset Basis Principal EndMonthRule 1 RangeFloat 200 01-Jan-2011 01-Jan-2014 [Struct] 1 0 100 1 2 RangeFloat 200 01-Jan-2011 01-Jan-2014 [Struct] 1 0 100 1 Index Type Spread Settle Maturity FloatReset Basis Principal EndMonthRule CapRate FloorRate 3 Float 200 01-Jan-2011 01-Jan-2014 1 0 100 1 Inf -Inf
Build the BKTree
with the following data:
VolDates = ['1-Jan-2012'; '1-Jan-2013'; '1-Jan-2014';'1-Jan-2015']; VolCurve = 0.01; AlphaDates = '01-01-2015'; AlphaCurve = 0.1; BKVS = bkvolspec(RS.ValuationDate, VolDates, VolCurve,... AlphaDates, AlphaCurve); BKTS = bktimespec(RS.ValuationDate, VolDates, Compounding); BKT = bktree(BKVS, RS, BKTS);
Price the portfolio.
Price = bkprice(BKT, InstSet)
Price = 3×1
105.5147
101.4805
105.5147
Input Arguments
BKTree
— Interest-rate tree structure
structure
Interest-rate tree structure, specified by using bktree
.
Data Types: struct
InstSet
— Instrument variable
structure
Instrument variable containing a collection of NINST
instruments,
specified using instadd
. Instruments are categorized by
type; each type can have different data fields. The stored data field is a row vector or
character vector for each instrument.
Data Types: struct
Options
— Derivatives pricing options structure
structure
(Optional) Derivatives pricing options structure, created using derivset
.
Data Types: struct
Output Arguments
Price
— Price for each instrument at time 0
vector
Price for each instrument at time 0, returned as a
NINST
-by-1
vector. The prices are computed by
backward dynamic programming on the interest-rate tree. If an instrument cannot be
priced, a NaN
is returned in that entry.
Related single-type pricing functions are:
bondbybk
— Price a bond from a Black-Karasinski tree.capbybk
— Price a cap from a Black-Karasinski tree.cfbybk
— Price an arbitrary set of cash flows from a Black-Karasinski tree.fixedbybk
— Price a fixed-rate note from a Black-Karasinski tree.floatbybk
— Price a floating-rate note from a Black-Karasinski tree.floorbybk
— Price a floor from a Black-Karasinski tree.optbndbybk
— Price a bond option from a Black-Karasinski tree.optembndbybk
— Price a bond with embedded option by a Black-Karasinski tree.optfloatbybk
— Price a floating-rate note with an option from a Black-Karasinski tree.optemfloatbybk
— Price a floating-rate note with an embedded option from a Black-Karasinski tree.rangefloatbybk
— Price range floating note from a Black-Karasinski tree.swapbybk
— Price a swap from a Black-Karasinski tree.swaptionbybk
— Price a swaption from a Black-Karasinski tree.
PriceTree
— Tree structure of instrument prices
structure
Tree structure of instrument prices, returned as a MATLAB® structure of trees containing vectors of instrument prices and accrued
interest, and a vector of observation times for each node. Within
PriceTree
:
PriceTree.PTree
contains the clean prices.PriceTree.AITree
contains the accrued interest.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 areNumNodes
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.
Version History
Introduced before R2006a
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