bkprice

Instrument prices from Black-Karasinski interest-rate tree

Syntax

[Price, PriceTree] = bkprice(BKTree, InstSet, Options)

Arguments

BKTree

Interest-rate tree structure created by bktree.

InstSet

Variable containing a collection of NINST instruments. Instruments are categorized by type. Each type can have different data fields. The stored data field is a row vector or string for each instrument.

Options

(Optional) Derivatives pricing options structure created with derivset.

Description

[Price, PriceTree] = bkprice(BKTree, InstSet, Options) computes arbitrage-free prices for instruments using an interest-rate tree created with bktree. All instruments contained in a financial instrument variable, InstSet, are priced.

Price is a number of instruments (NINST)-by-1 vector of prices for each instrument. The prices are computed by backward dynamic programming on the interest-rate tree. If an instrument cannot be priced, NaN is returned.

PriceTree is a MATLAB® structure of trees containing vectors of instrument prices and accrued interest, and a vector of observation times for each node.

PriceTree.PTree contains the clean prices.

PriceTree.AITree contains the accrued interest.

PriceTree.tObs contains the observation times.

bkprice handles instrument types: 'Bond', 'CashFlow', 'OptBond', 'OptEmBond', 'OptEmBond', 'OptFloat', 'OptEmFloat', 'Fixed', 'Float', 'Cap', 'Floor', 'RangeFloat', 'Swap'. See instadd to construct defined types.

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.

  • optfloatbybdt: Price a floating-rate note with an option from a Black-Karasinski tree.

  • optemfloatbybdt: 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.

Examples

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)
%Table of instrument portfolio partially displayed:
Index Type   CouponRate Settle      Maturity    Period ...  Name ... 
1     Bond   0.03       01-Jan-2004 01-Jan-2007 1      ... 3% bond
2     Bond   0.03       01-Jan-2004 01-Jan-2008 2      ...  3% bond
     
Index Type Strike Settle      Maturity     CapReset ...  Name ...
3     Cap  0.04   01-Jan-2004 01-Jan-2008  1        ...  4% Cap 
[Price, PriceTree] = bkprice(BKTree, BKSubSet);
Price =

   98.1096
   95.6734
    2.2706

You can use treeviewer to see the prices of these three instruments along the price tree.

treeviewer(PriceTree, BKSubSet)

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)
%Table of instrument portfolio partially displayed:
Index Type CouponRate Settle  Maturity  Period Basis EndMonthRule ... Face
1 Bond [Cell]  01-Jan-2010    01-Jan-2011   1   0     1           ... 100 
2 Bond [Cell]  01-Jan-2010    01-Jan-2012   1   0     1           ... 100 
3 Bond [Cell]  01-Jan-2010    01-Jan-2013   1   0     1           ... 100 
4 Bond [Cell]  01-Jan-2010    01-Jan-2014   1   0     1           ... 100 
% Build the tree 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 =

  100.6763
  100.7368
  100.9266
  101.0115

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)
%Table of instrument portfolio partially displayed:
Index Type CouponRate       Settle     Maturity    OptSpec Strike ExerciseDates ... AmericanOpt
1     OptEmBond [Cell]   01-Jan-2010   01-Jan-2012   call    100  01-Jan-2011   ... 0
2     OptEmBond [Cell]   01-Jan-2010   01-Jan-2013   call    100  01-Jan-2011   ... 0
3     OptEmBond [Cell]   01-Jan-2010   01-Jan-2014   call    100  01-Jan-2011   ... 0
 
Index Type CouponRate Settle         Maturity       Period Basis EndMonthRule ... Face
4     Bond [Cell]     01-Jan-2010    01-Jan-2012    1      0     1            ... 100 
5     Bond [Cell]     01-Jan-2010    01-Jan-2013    1      0     1            ... 100 
6     Bond [Cell]     01-Jan-2010    01-Jan-2014    1      0     1            ... 100 
% Build the tree 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);

% The first three rows 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 =

  100.6735
  100.6763
  100.6763
  100.7368
  100.9266
  101.0115

Compute the price of a portfolio 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
3     Float 200    01-Jan-2011    01-Jan-2014    1          0     100       1           
% The data to build the tree is as follows:
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 =

  105.5147
  101.4740
  105.5147

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