cbondbycrr
Price convertible bonds from CRR binomial tree
Syntax
Description
[
prices convertible bonds from a CRR binomial tree using a credit spread or incorporating
the risk of bond default.Price
,PriceTree
,EquityTree
,DebtTree
]
= cbondbycrr(___,Name,Value
)
To incorporate the risk in the form of credit spread (Tsiveriotis-Fernandes method),
use the optional name-value pair input argument Spread
. To
incorporate default risk into the algorithm, specify the optional name-value pair input
arguments DefaultProbability
and
RecoveryRate
.
Examples
Price Convertible Bond Using a CRR Tree
Price a convertible bond using the following data for the interest-rate term structure:
StartDates = datetime(2014,1,1); EndDates = datetime(2015,1,1); Rates = 0.1; Basis = 1;
Create the RateSpec
and StockSpec
.
Sigma = 0.3; Price = 50; RateSpec = intenvset('ValuationDate',StartDates,'StartDates',StartDates,'EndDates',EndDates,... 'Rates',Rates,'Compounding',-1,'Basis',Basis)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9048
Rates: 0.1000
EndTimes: 1
StartTimes: 0
EndDates: 735965
StartDates: 735600
ValuationDate: 735600
Basis: 1
EndMonthRule: 1
StockSpec = stockspec(Sigma,Price)
StockSpec = struct with fields:
FinObj: 'StockSpec'
Sigma: 0.3000
AssetPrice: 50
DividendType: []
DividendAmounts: 0
ExDividendDates: []
Create the CRR tree for the equity.
Settle = datetime(2014,1,1); Maturity = datetime(2014,10,1); NumSteps = 3; TimeSpec = crrtimespec(Settle,Maturity,NumSteps); CRRT = crrtree(StockSpec,RateSpec,TimeSpec)
CRRT = struct with fields:
FinObj: 'BinStockTree'
Method: 'CRR'
StockSpec: [1x1 struct]
TimeSpec: [1x1 struct]
RateSpec: [1x1 struct]
tObs: [0 0.2491 0.4982 0.7473]
dObs: [735600 735691 735782 735873]
STree: {[50] [58.0757 43.0472] [67.4558 50.0000 37.0613] [78.3509 58.0757 43.0472 31.9077]}
UpProbs: [0.5465 0.5465 0.5465]
Define and price the convertible bond.
CouponRate = 0; Period = 1; ConvRatio = 2; CallExDates = datetime(2014,10,1); CallStrike = 115; AmericanCall = 1; Spread = 0.05; [Price,PriceTree,EqtTree,DbtTree] = cbondbycrr(CRRT,CouponRate,Settle,Maturity,ConvRatio,... 'Period',Period,'Spread',Spread,'CallExDates',CallExDates,'CallStrike',CallStrike,'AmericanCall',AmericanCall)
Price = 104.9490
PriceTree = struct with fields:
FinObj: 'BinPriceTree'
PTree: {[104.9490] [116.1515 98.0892] [134.9117 105.6029 96.3327] [156.7019 116.1515 100.0000 100.0000]}
tObs: [0 0.2491 0.4982 0.7473]
dObs: [735600 735691 735782 735873]
EqtTree = struct with fields:
FinObj: 'BinPriceTree'
PTree: {[76.5211] [116.1515 33.0103] [134.9117 61.9209 0] [156.7019 116.1515 0 0]}
tObs: [0 0.2491 0.4982 0.7473]
dObs: [735600 735691 735782 735873]
DbtTree = struct with fields:
FinObj: 'BinPriceTree'
PTree: {[28.4278] [0 65.0790] [0 43.6821 96.3327] [0 0 100.0000 100.0000]}
tObs: [0 0.2491 0.4982 0.7473]
dObs: [735600 735691 735782 735873]
Price a Convertible Bond Using a CRR Tree and Incorporate Default Risk Using DefaultProbability
and RecoveryRate
Create the interest-rate term structure RateSpec
.
StartDates = 'Jan-1-2014'; EndDates = 'Jan-1-2016'; Rates = 0.025; Basis = 1; RateSpec = intenvset('ValuationDate',StartDates,'StartDates',... StartDates,'EndDates',EndDates,'Rates',Rates,'Compounding',-1,'Basis',Basis)
RateSpec = struct with fields:
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9512
Rates: 0.0250
EndTimes: 2
StartTimes: 0
EndDates: 736330
StartDates: 735600
ValuationDate: 735600
Basis: 1
EndMonthRule: 1
Create the StockSpec
.
AssetPrice = 110;
Sigma = 0.22;
Div = 0.02;
StockSpec = stockspec(Sigma,AssetPrice,'continuous',Div)
StockSpec = struct with fields:
FinObj: 'StockSpec'
Sigma: 0.2200
AssetPrice: 110
DividendType: {'continuous'}
DividendAmounts: 0.0200
ExDividendDates: []
Create the CRR tree for the equity.
Settle = '1-Jan-2014'; Maturity = '1-Oct-2014'; NumSteps = 3; TimeSpec = crrtimespec(Settle,Maturity,NumSteps); CRRT = crrtree(StockSpec,RateSpec,TimeSpec)
CRRT = struct with fields:
FinObj: 'BinStockTree'
Method: 'CRR'
StockSpec: [1x1 struct]
TimeSpec: [1x1 struct]
RateSpec: [1x1 struct]
tObs: [0 0.2491 0.4982 0.7473]
dObs: [735600 735691 735782 735873]
STree: {[110] [122.7658 98.5616] [137.0132 110.0000 88.3127] [152.9140 122.7658 98.5616 79.1295]}
UpProbs: [0.4782 0.4782 0.4782]
Define and price the convertible bond using the optional DefaultProbability
and RecoveryRate
arguments.
CouponRate = 0; Period = 1; ConvRatio = 2; CallExDates = '1-Oct-2014'; CallStrike = 115; AmericanCall = 1; DefaultProbability = .30; RecoveryRate = .82; [Price,PriceTree,EqtTree,DbtTree] = cbondbycrr(CRRT,CouponRate,Settle,Maturity,ConvRatio,... 'Period',Period,'CallExDates',CallExDates,'CallStrike',CallStrike,'AmericanCall',AmericanCall,... 'DefaultProbability',DefaultProbability,'RecoveryRate',RecoveryRate)
Price = 220
PriceTree = struct with fields:
FinObj: 'BinPriceTree'
PTree: {[220] [245.5317 197.1233] [274.0263 220.0000 176.6254] [305.8279 245.5317 197.1233 158.2589]}
tObs: [0 0.2491 0.4982 0.7473]
dObs: [735600 735691 735782 735873]
EqtTree = struct with fields:
FinObj: 'BinPriceTree'
PTree: {[220] [245.5317 197.1233] [274.0263 220.0000 176.6254] [305.8279 245.5317 197.1233 158.2589]}
tObs: [0 0.2491 0.4982 0.7473]
dObs: [735600 735691 735782 735873]
DbtTree = struct with fields:
FinObj: 'BinPriceTree'
PTree: {[0] [0 0] [0 0 0] [0 0 0 0]}
tObs: [0 0.2491 0.4982 0.7473]
dObs: [735600 735691 735782 735873]
Input Arguments
CRRTree
— Stock tree structure
structure
Stock tree structure, specified by using crrtree
.
Data Types: struct
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
datetime array | string array | date character vector
Settlement date, specified as an NINST
-by-1
vector using a datetime array, string array, or date character vectors.
Note
The Settle
date for every convertible bond is set to the
ValuationDate
of the CRR stock tree. The bond argument,
Settle
, is ignored.
To support existing code, cbondbycrr
also
accepts serial date numbers as inputs, but they are not recommended.
Maturity
— Maturity date
datetime array | string array | date character vector
Maturity date, specified as an NINST
-by-1
vector using a datetime array, string array, or date character vectors.
To support existing code, cbondbycrr
also
accepts serial date numbers as inputs, but they are not recommended.
ConvRatio
— Number of shares convertible to one bond
nonnegative number
Number of shares convertible to one bond, specified as an
NINST
-by-1
with a nonnegative number.
Data Types: double
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: [Price,PriceTree,EquityTree,DebtTree] =
cbondbycrr(CRRT,CouponRate,Settle, Maturity,
ConvRatio,'Spread',Spread,'CallExDates',CallExDates,'CallStrike',CallStrike,'AmericanCall',1)
Spread
— Number of basis points over the reference rate
0
(default) | vector
Number of basis points over the reference rate, specified as the comma-separated
pair consisting of 'Spread'
and a
NINST
-by-1
vector. For example, if the
reference rate is 2% and spread is 4%, then the Spread
value in
basis points would be 0.04
.
Note
To incorporate the risk in the form of credit spread (Tsiveriotis-Fernandes
method), use the optional input argument Spread
. To incorporate
default risk into the algorithm, specify the optional input arguments
DefaultProbability
and RecoveryRate
.
Do not use Spread
with DefaultProbability
and RecoveryRate
.
Data Types: double
Period
— Coupons per year
2
per year (default) | vector
Coupons per year, specified as the comma-separated pair consisting of
'Period'
and a NINST
-by-1
vector.
Data Types: double
IssueDate
— Bond issue date
datetime array | string array | date character vector
Bond issue date, specified as the comma-separated pair consisting of
'IssueDate'
and a
NINST
-by-1
vector using a datetime array,
string array, or date character vectors.
To support existing code, cbondbycrr
also
accepts serial date numbers as inputs, but they are not recommended.
FirstCouponDate
— Irregular first coupon date
datetime array | string array | date character vector
Irregular first coupon date, specified as the comma-separated pair consisting of
'FirstCouponDate'
and a
NINST
-by-1
vector using a datetime array,
string array, or date character vectors.
To support existing code, cbondbycrr
also
accepts serial date numbers as inputs, but they are not recommended.
LastCouponDate
— Irregular last coupon date
datetime array | string array | date character vector
Irregular last coupon date, specified as the comma-separated pair consisting of
'LastCouponDate'
and a
NINST
-by-1
vector using a datetime array,
string array, or date character vectors.
To support existing code, cbondbycrr
also
accepts serial date numbers as inputs, but they are not recommended.
Face
— Face value
100
(default) | nonnegative value | cell array of nonnegative values
Face value, specified as the comma-separated pair consisting of
'Face'
and a NINST
-by-1
vector of nonnegative face values or a
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 the associated face value. The date indicates the last day that the
face value is valid.
Data Types: cell
| double
CallStrike
— Call strike price for European, Bermuda, or American option
nonnegative integer | vector of nonnegative integers
Call strike price for European, Bermuda, or American option, specified as the
comma-separated pair consisting of 'CallStrike'
and one of the
following values:
For a European call option —
NINST
-by-1
vector of nonnegative integersFor a Bermuda call option —
NINST
-by-NSTRIKES
matrix of call strike price values, where each row is the schedule for one call option. If a call option has fewer thanNSTRIKES
exercise opportunities, the end of the row is padded withNaN
s.For an American call option —
NINST
-by-1
vector of strike price values for each option.
Data Types: single
| double
CallExDates
— Call exercise date for European, Bermuda, or American option
datetime array | string array | date character vector
Call exercise date for European, Bermuda, or American option, specified as the
comma-separated pair consisting of 'CallExDates'
and a datetime
array, string array, or date character vectors for the following values:
For a European option —
NINST
-by-1
vector of date character vectors.For a Bermuda option —
NINST
-by-NSTRIKES
matrix of exercise dates, where each row is the schedule for one option. For a European option, there is only oneCallExDate
on the option expiry date.For an American option —
NINST
-by-1
orNINST
-by-2
matrix of exercise date boundaries. For each instrument, the call option can be exercised on any tree date between or including the pair of dates on that row. IfCallExDates
isNINST
-by-1
, the option can be exercised between theValuationDate
of the CRR stock tree and the single listedCallExDate
.
To support existing code, cbondbycrr
also
accepts serial date numbers as inputs, but they are not recommended.
AmericanCall
— Call option type indicator
0
if AmericanCall
is
NaN
or not entered (default) | scalar | vector of positive integers[0,1]
Call option type, specified as the comma-separated pair consisting of
'AmericanCall'
and a
NINST
-by-1
vector of positive integer flags
with values 0
or 1
.
For a European or Bermuda option —
AmericanCall
is0
for each European or Bermuda option.For an American option —
AmericanCall
is1
for each American option. TheAmericanCall
argument is required to invoke American exercise rules.
Data Types: single
| double
PutStrike
— Put strike values for European, Bermuda, or American option
positive integer | vector of positive integers
Put strike values for European, Bermuda, or American option, specified as the
comma-separated pair consisting of 'PutStrike'
and one of the
following values:
For a European put option —
NINST
-by-1
vector of nonnegative integersFor a Bermuda put option —
NINST
-by-NSTRIKES
matrix of strike price values where each row is the schedule for one option. If a put option has fewer thanNSTRIKES
exercise opportunities, the end of the row is padded withNaN
s.For an American put option —
NINST
-by-1
vector of strike price values for each option.
Data Types: single
| double
PutExDates
— Put exercise date for European, Bermuda, or American option
datetime array | string array | date character vector
Put exercise date for European, Bermuda, or American option, specified as the
comma-separated pair consisting of 'PutExDates'
and a datetime
array, string array, or date character vectors for the following values:
For a European option —
NINST
-by-1
vector of date character vectors.For a Bermuda option —
NINST
-by-NSTRIKES
matrix of exercise dates where each row is the schedule for one option. For a European option, there is only onePutExDate
on the option expiry date.For an American option —
NINST
-by-1
orNINST
-by-2
matrix of exercise date boundaries. For each instrument, the put option can be exercised on any tree date between or including the pair of dates on that row. IfPutExDates
isNINST
-by-1
, the put option can be exercised between theValuationDate
of the CRR stock tree and the single listedPutExDate
.
To support existing code, cbondbycrr
also
accepts serial date numbers as inputs, but they are not recommended.
AmericanPut
— Put option type indicator
0
if AmericanPut
is
NaN
or not entered (default) | positive integer [0,1]
| vector of positive integers [0,1]
Put option type, specified as the comma-separated pair consisting of
'AmericanPut'
and a
NINST
-by-1
vector of positive integer flags
with values 0
or 1
.
For a European or Bermuda option —
AmericanPut
is0
for each European or Bermuda option.For an American option —
AmericanPut
is1
for each American option. TheAmericanPut
argument is required to invoke American exercise rules.
Data Types: double
ConvDates
— Convertible dates
MaturityDate
(default) | datetime array | string array | date character vector
Convertible dates, specified as the comma-separated pair consisting of
'ConvDates'
and a
NINST
-by-1
or
NINST
-by-2
vector using a datetime array,
string array, or date character vectors. If ConvDates
is not
specified, the bond is always convertible until maturity.
To support existing code, cbondbycrr
also
accepts serial date numbers as inputs, but they are not recommended.
For each instrument, the bond can be converted on any tree date between or including the pair of dates on that row.
If ConvDates
is
NINST
-by-1
, the bond can be converted between
the ValuationDate
of the CRR stock tree and the single listed
ConvDates
.
DefaultProbability
— Annual probability of default rate
0
(default) | nonnegative decimal
Annual probability of default rate, specified as the comma-separated pair
consisting of 'DefaultProbability'
and a
NINST
-by-1
nonnegative decimal value.
Note
To incorporate default risk into the algorithm, specify the optional input
arguments DefaultProbability
and
RecoveryRate
. To incorporate the risk in the form of credit
spread (Tsiveriotis-Fernandes method), use the optional input argument
Spread
. Do not use DefaultProbability
and
RecoveryRate
with Spread
.
Data Types: double
RecoveryRate
— Recovery rate
1
(default) | nonnegative decimal
Recovery rate, specified as the comma-separated pair consisting of
'RecoveryRate'
and a
NINST
-by-1
nonnegative decimal.
Note
To incorporate default risk into the algorithm, specify the optional input
arguments DefaultProbability
and
RecoveryRate
. To incorporate the risk in the form of credit
spread (Tsiveriotis-Fernandes method), use the optional input argument
Spread
. Do not use DefaultProbability
and RecoveryRate
with Spread
.
Data Types: double
Output Arguments
Price
— Expected price at time 0
array
Expected price at time 0
, returned as an
NINST
-by-1
array.
PriceTree
— Structure with vector of convertible bond prices at each node
tree structure
Structure with a vector of convertible bond prices at each node, returned as a tree structure.
EquityTree
— Structure with vector of convertible bond equity component at each node
tree structure
Structure with a vector of convertible bond equity component at each node, returned as a tree structure.
DebtTree
— Structure with vector of convertible bond debt component at each node
tree structure
Structure with a vector of convertible bond debt component at each node, returned as a tree structure.
More About
Callable Convertible
A convertible bond that is callable by the issuer. The issuer of the bond forces conversion, removing the advantage that conversion is at the discretion of the bondholder.
Upon call, the bondholder can either convert the bond or redeem at the call price. This option enables the issuer to control the price of the convertible bond and, if necessary, refinance the debt with a new cheaper one.
Puttable Convertible
A convertible bond with a put feature allows the bondholder to sell back the bond at a premium on a specific date.
This option protects the holder against rising interest rates by reducing the year to maturity.
Algorithms
cbondbycrr
, cbondbyeqp
, cbondbyitt
, and cbondbystt
return price information in the form of a price vector and a price
tree. These functions implement the risk in the form of either a credit spread or
incorporating the risk of bond default. To incorporate the risk in the form of credit spread
(Tsiveriotis-Fernandes method), use the optional name-value pair argument
Spread
. To incorporate default risk into the algorithm, specify the
optional name-value pair arguments DefaultProbability
and
RecoveryRate
.
References
[1] Tsiveriotis, K., and C. Fernandes. “Valuing Convertible Bonds with Credit Risk.” Journal of Fixed Income. Vol. 8, 1998, pp. 95–102.
[2] Hull, J. Options, Futures and Other Derivatives. Fourth Edition. Prentice Hall, 2000, pp. 646–649.
Version History
Introduced in R2015aR2022b: Serial date numbers not recommended
Although cbondbycrr
supports serial date numbers,
datetime
values are recommended instead. The
datetime
data type provides flexible date and time
formats, storage out to nanosecond precision, and properties to account for time
zones and daylight saving time.
To convert serial date numbers or text to datetime
values, use the datetime
function. For example:
t = datetime(738427.656845093,"ConvertFrom","datenum"); y = year(t)
y = 2021
There are no plans to remove support for serial date number inputs.
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