Price bond option from Black-Derman-Toy interest-rate tree
[ calculates
the price for a bond option from a Black-Derman-Toy interest-rate
tree.Price,PriceTree]
= optbndbybdt(BDTTree,OptSpec,Strike,ExerciseDates,AmericanOpt,CouponRate,Settle,Maturity)
[ adds
optional arguments.Price,PriceTree]
= optbndbybdt(___,Period,Basis,EndMonthRule,IssueDate,FirstCouponDate,LastCouponDate,StartDate,Face,Options)
Using the BDT interest-rate tree in the deriv.mat file, price a European call and put option on a 10% bond with a strike of 95. The exercise date for the option is Jan. 01, 2002. The settle date for the bond is Jan. 01, 2000, and the maturity date is Jan. 01, 2003.
Load the file deriv.mat, which provides BDTTree. The BDTTree structure contains the time and forward-rate information needed to price the bond.
load deriv.mat;Use optbondbybdt to compute the price of the 'Call' option.
[Price,PriceTree] = optbndbybdt(BDTTree,'Call',95,'01-Jan-2002',... 0,0.10,'01-Jan-2000','01-Jan-2003',1)
Price = 1.7657
PriceTree = struct with fields:
FinObj: 'BDTPriceTree'
tObs: [0 1 2 3 4]
PTree: {1x5 cell}
ExTree: {[0] [0 0] [1 1 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] = optbndbybdt(BDTTree,'Put',95,'01-Jan-2002',... 0,0.10,'01-Jan-2000','01-Jan-2003',1)
Price = 0.5740
PriceTree = struct with fields:
FinObj: 'BDTPriceTree'
tObs: [0 1 2 3 4]
PTree: {[0.5740] [0 1.2628] [0 0 2.8871] [0 0 0 0] [0 0 0 0]}
ExTree: {[0] [0 0] [0 0 1] [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.
BDTTree — Interest-rate tree structureInterest-rate tree structure, specified by using bdttree.
Data Types: struct
OptSpec — Definition of option '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 valuesOption 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 datesOption 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 type0 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 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 dateSettlement 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 BDT tree. The bond argument Settle is
ignored.
Data Types: double | char
Maturity — Maturity dateMaturity date, specified as an NINST-by-1 vector
of serial date numbers or date character vectors.
Data Types: double | char
Period — Coupons per year2 per year (default) | vector(Optional) Coupons per year, specified as an NINST-by-1 vector.
Data Types: double
Basis — Day-count basis0 (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 flag1 (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(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(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(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(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 value100 (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(Optional) Derivatives pricing options, specified as structure
that is created with derivset.
Data Types: struct
Price — Expected prices of bond option at time 0Expected 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 nodeStructure 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.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.
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.
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