optbndbybk

Price bond option from Black-Karasinski interest-rate tree

Syntax

[Price,PriceTree] = optbndbybk(BKTree,OptSpec,Strike,ExerciseDates,AmericanOpt,CouponRate,Settle,Maturity)

[Price,PriceTree] = optbndbybk(___,Period,Basis,EndMonthRule,IssueDate,FirstCouponDate,LastCouponDate,StartDate,Face,Options)

Description

[Price,PriceTree] = optbndbybk(BKTree,OptSpec,Strike,ExerciseDates,AmericanOpt,CouponRate,Settle,Maturity) calculates the price for a bond option from a Black-Karasinski interest-rate tree.

[Price,PriceTree] = optbndbybk(___,Period,Basis,EndMonthRule,IssueDate,FirstCouponDate,LastCouponDate,StartDate,Face,Options) adds optional arguments.

Examples

Price a European Call and Put Option on a Bond

Using the BK interest rate tree in the deriv.mat file, price a European call and put option on a 4% bond with a strike of 96. The exercise date for the option is Jan. 01, 2006. The settle date for the bond is Jan. 01, 2005, and the maturity date is Jan. 01, 2009.

Load the file deriv.mat, which provides BKTree. The BKTree structure contains the time and forward-rate information needed to price the bond.

load deriv.mat; 

Use optbondbybk to compute the price of the 'call' option.

[Price,PriceTree] = optbndbybk(BKTree,'Call',96,'01-Jan-2006',... 
0,0.04,'01-Jan-2005','01-Jan-2009')

Warning: OptBonds are valued at Tree ValuationDate rather than Settle. 
> In optbndbytrintree (line 40)
  In optbndbybk (line 92) 
Warning: Not all cash flows are aligned with the tree. Result will be approximated. 
> In optbndbytrintree (line 151)
  In optbndbybk (line 92) 

Price =

    0.1512


PriceTree = 

  struct with fields:

     FinObj: 'BKPriceTree'
      PTree: {[0.1512]  [0.0281 0.1481 0.3119]  [0 0 0.1329 0.3886 0.3086]  [0 0 0 0 0]  [0 0 0 0 0]}
       tObs: [0 1 2 3 4]
    Connect: {[2]  [2 3 4]  [2 2 3 4 4]}
      Probs: {[3×1 double]  [3×3 double]  [3×5 double]}
     ExTree: {[0]  [0 0 0]  [0 0 1 1 1]  [0 0 0 0 0]  [0 0 0 0 0]}

Now use optbndbybdt to compute the price of a 'put' option on the same bond.

[Price,PriceTree] = optbndbybk(BKTree,'Put',96,'01-Jan-2006',... 
0,0.04,'01-Jan-2005','01-Jan-2009')

Warning: OptBonds are valued at Tree ValuationDate rather than Settle. 
> In optbndbytrintree (line 40)
  In optbndbybk (line 92) 
Warning: Not all cash flows are aligned with the tree. Result will be approximated. 
> In optbndbytrintree (line 151)
  In optbndbybk (line 92) 

Price =

    0.0272


PriceTree = 

  struct with fields:

     FinObj: 'BKPriceTree'
      PTree: {[0.0272]  [0.0860 0.0204 0]  [0.0474 0.1266 0 0 0]  [0 0 0 0 0]  [0 0 0 0 0]}
       tObs: [0 1 2 3 4]
    Connect: {[2]  [2 3 4]  [2 2 3 4 4]}
      Probs: {[3×1 double]  [3×3 double]  [3×5 double]}
     ExTree: {[0]  [0 0 0]  [1 1 0 0 0]  [0 0 0 0 0]  [0 0 0 0 0]}

The PriceTree.ExTree output for the 'call' and 'put' option contains the exercise indicator arrays. Each element of the cell array is an array containing 1's where an option is exercised and 0's where it is not.

Input Arguments

BKTree — Interest-rate tree structure

structure

Interest-rate tree structure, specified by using bktree.

Data Types: struct

OptSpec — Definition of option

character vector with value 'call' or 'put' | cell array of character vectors with values 'call' or 'put'

Definition of option, specified as a NINST-by-1 cell array of character vectors.

Data Types: char

Strike — Option strike price values

nonnegative integer

Option strike price value, specified as a NINST-by-1 or NINST-by-NSTRIKES depending on the type of option:

• European option — NINST-by-1 vector of strike price values.

• Bermuda option — NINST by number of strikes (NSTRIKES) matrix of strike price values. Each row is the schedule for one option. If an option has fewer than NSTRIKES exercise opportunities, the end of the row is padded with NaNs.

• American option — NINST-by-1 vector of strike price values for each option.

Data Types: double

ExerciseDates — Option exercise dates

serial date number | date character vector

Option exercise dates, specified as a NINST-by-1, NINST-by-2, or NINST-by-NSTRIKES using serial date numbers or date character vectors, depending on the type of option:

• For a European option, use a NINST-by-1 vector of dates. For a European option, there is only one ExerciseDates on the option expiry date.

• For a Bermuda option, use a NINST-by-NSTRIKES vector of dates.

• For an American option, use a NINST-by-2 vector of exercise date boundaries. The option can be exercised on any date between or including the pair of dates on that row. If only one non-NaN date is listed, or if ExerciseDates is a NINST-by-1 vector, the option can be exercised between ValuationDate of the stock tree and the single listed ExerciseDates.

Data Types: double | char

AmericanOpt — Option type

0 European/Bermuda (default) | integer with values 0 or 1

(Optional) Option type, specified as NINST-by-1 positive integer flags with values:

• 0 — European/Bermuda

• 1 — American

Data Types: double

CouponRate — Bond coupon rate

positive decimal value

Bond coupon rate, specified as an NINST-by-1 decimal annual rate or NINST-by-1 cell array, where each element is a NumDates-by-2 cell array. The first column of the NumDates-by-2 cell array is dates and the second column is associated rates. The date indicates the last day that the coupon rate is valid.

Data Types: double | cell

Settle — Settlement date

serial date number | date character vector

Settlement date for the bond option, specified as a NINST-by-1 vector of serial date numbers or date character vectors.

Note

The Settle date for every bond is set to the ValuationDate of the BK tree. The bond argument Settle is ignored.

Data Types: double | char

Maturity — Maturity date

serial date number | date character vector

Maturity date, specified as an NINST-by-1 vector of serial date numbers or date character vectors.

Data Types: double | char

Period — Coupons per year

2 per year (default) | vector

(Optional) Coupons per year, specified as an NINST-by-1 vector.

Data Types: double

Basis — Day-count basis

0 (actual/actual) (default) | integer from 0 to 13

(Optional) Day-count basis, 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

EndMonthRule — End-of-month rule flag

1 (in effect) (default) | nonnegative integer with values 0 or 1

(Optional) End-of-month rule flag is specified as a nonnegative integer using a NINST-by-1 vector. This rule applies only when Maturity is an end-of-month date for a month having 30 or fewer days.

• 0 = Ignore rule, meaning that a bond coupon payment date is always the same numerical day of the month.

• 1 = Set rule on, meaning that a bond coupon payment date is always the last actual day of the month.

Data Types: double

IssueDate — Bond issue date

serial date number | date character vector

(Optional) Bond issue date, specified as an NINST-by-1 vector using serial date numbers or date character vectors.

Data Types: double | char

FirstCouponDate — Irregular first coupon date

serial date number | date character vector

(Optional) Irregular first coupon date, specified as an NINST-by-1 vector using serial date numbers date or date character vectors.

When FirstCouponDate and LastCouponDate are both specified, FirstCouponDate takes precedence in determining the coupon payment structure. If you do not specify a FirstCouponDate, the cash flow payment dates are determined from other inputs.

Data Types: double | char

LastCouponDate — Irregular last coupon date

serial date number | date character vector

(Optional) Irregular last coupon date, specified as a NINST-by-1 vector using serial date numbers or date character vectors.

In the absence of a specified FirstCouponDate, a specified LastCouponDate determines the coupon structure of the bond. The coupon structure of a bond is truncated at the LastCouponDate, regardless of where it falls, and is followed only by the bond's maturity cash flow date. If you do not specify a LastCouponDate, the cash flow payment dates are determined from other inputs.

Data Types: char | double

StartDate — Forward starting date of payments

serial date number | date character vector

(Optional) Forward starting date of payments (the date from which a bond cash flow is considered), specified as a NINST-by-1 vector using serial date numbers or date character vectors.

If you do not specify StartDate, the effective start date is the Settle date.

Data Types: char | double

Face — Face value

100 (default) | nonnegative value | cell array of nonnegative values

(Optional) Face or par value, specified as anNINST-by-1 vector.

Data Types: double

Options — Derivatives pricing options

structure

(Optional) Derivatives pricing options, specified as structure that is created with derivset.

Data Types: struct

Output Arguments

Price — Expected prices of bond option at time 0

matrix

Expected price of the bond option at time 0, returned as a NINST-by-1 matrix.

PriceTree — Structure containing trees of vectors of instrument prices and accrued interest for each node

structure

Structure containing trees of vectors of instrument prices and accrued interest, and a vector of observation times for each node. Values are:

• 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.

• PriceTree.ExTree contains the exercise indicator arrays. Each element of the cell array is an array containing 1's where an option is exercised and 0's where it isn't.

More About

Bond Option

A bond option gives the holder the right to sell a bond back to the issuer (put) or to redeem a bond from its current owner (call) at a specific price and on a specific date.

Financial Instruments Toolbox™ supports three types of put and call options on bonds:

• American option: An option that you exercise any time until its expiration date.

• European option: An option that you exercise only on its expiration date.

• Bermuda option: A Bermuda option resembles a hybrid of American and European options. You can exercise it on predetermined dates only, usually monthly.

For more information, see Bond Options.