# cbondbycrr

Price convertible bonds from CRR binomial tree

## Syntax

``Price = cbondbycrr(CRRTree,CouponRate,Settle,Maturity,ConvRatio)``
``````[Price,PriceTree] = cbondbycrr(CRRTree,CouponRate,Settle,Maturity,ConvRatio)``````
``````[Price,PriceTree,EquityTree,DebtTree] = cbondbycrr(___,Name,Value)``````

## Description

example

````Price = cbondbycrr(CRRTree,CouponRate,Settle,Maturity,ConvRatio)` prices convertible bonds from a CRR binomial tree using the Tsiveriotis and Fernandes method.```

example

``````[Price,PriceTree] = cbondbycrr(CRRTree,CouponRate,Settle,Maturity,ConvRatio)``` prices convertible bonds from a CRR binomial tree using the Tsiveriotis and Fernandes method.```

example

``````[Price,PriceTree,EquityTree,DebtTree] = cbondbycrr(___,Name,Value)``` prices convertible bonds from a CRR binomial tree using 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 input argument `Spread`. To incorporate default risk into the algorithm, specify the optional name-value pair input arguments `DefaultProbability` and `RecoveryRate`.```

## Examples

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Price a convertible bond using the following data for the interest-rate term structure:

```StartDates = 'Jan-1-2014'; EndDates = 'Jan-1-2015'; 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 = '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: {[50] [58.0757 43.0472] [67.4558 50.0000 37.0613] [78.3509 ... ]} UpProbs: [0.5465 0.5465 0.5465] ```

Define and price the convertible bond.

```CouponRate = 0; Period = 1; ConvRatio = 2; CallExDates = '1-Oct-2014'; 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: {1x4 cell} 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 ... ]} 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] ```

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: {1x4 cell} 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] [1x4 double]} 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] [1x4 double]} 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

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Stock tree structure, specified by using `crrtree`.

Data Types: `struct`

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`

Settlement date, specified as an `NINST`-by-`1` scalar using a serial date number or date character vector.

Note

The `Settle` date for every convertible bond is set to the `ValuationDate` of the CRR stock tree. The bond argument, `Settle`, is ignored.

Data Types: `double` | `char`

Maturity date, specified as an `NINST`-by-`1` scalar using a serial date number or date character vector.

Data Types: `double` | `char`

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)```

Number of basis points over the reference rate, specified as the comma-separated pair consisting of `'Spread'` and a `NINST`-by-`1` vector.

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`

Coupons per year, specified as the comma-separated pair consisting of `'Period'` and a `NINST`-by-`1` vector.

Data Types: `double`

Bond issue date, specified as the comma-separated pair consisting of `'IssueDate'` and a `NINST`-by-`1` vector using a serial date number or date character vector.

Data Types: `double` | `char`

Irregular first coupon date, specified as the comma-separated pair consisting of `'FirstCouponDate'` and a `NINST`-by-`1` vector using a serial date number or date character vector.

Data Types: `double` | `char`

Irregular last coupon date, specified as the comma-separated pair consisting of `'LastCouponDate'` and a `NINST`-by-`1` vector using a serial date number or date character vector.

Data Types: `char` | `double`

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`

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 integers

• For 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 than `NSTRIKES` exercise opportunities, the end of the row is padded with `NaN`s.

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

Data Types: `single` | `double`

Call exercise date for European, Bermuda, or American option, specified as the comma-separated pair consisting of `'CallExDates'` and one of the following values:

• For a European option — `NINST`-by-`1` vector of serial date numbers or 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 one `CallExDate` on the option expiry date.

• For an American option — `NINST`-by-`1` or `NINST`-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. If `CallExDates` is `NINST`-by-`1`, the option can be exercised between the `ValuationDate` of the CRR stock tree and the single listed `CallExDate`.

Data Types: `char` | `cell` | `double`

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` is `0` for each European or Bermuda option.

• For an American option — `AmericanCall` is `1` for each American option. The `AmericanCall` argument is required to invoke American exercise rules.

Data Types: `single` | `double`

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 integers

• For 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 than `NSTRIKES` exercise opportunities, the end of the row is padded with `NaN`s.

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

Data Types: `single` | `double`

Put exercise date for European, Bermuda, or American option, specified as the comma-separated pair consisting of `'PutExDates'` and one of the following values:

• For a European option — `NINST`-by-`1` vector of serial date numbers or 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 one `PutExDate` on the option expiry date.

• For an American option — `NINST`-by-`1` or `NINST`-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. If `PutExDates` is `NINST`-by-`1`, the put option can be exercised between the `ValuationDate` of the CRR stock tree and the single listed `PutExDate`.

Data Types: `double` | `char` | `cell`

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` is `0` for each European or Bermuda option.

• For an American option — `AmericanPut` is `1` for each American option. The `AmericanPut` argument is required to invoke American exercise rules.

Data Types: `single` | `double`

Convertible dates, specified as the comma-separated pair consisting of `'ConvDates'` and a `NINST`-by-`1` or `NINST`-by-`2` matrix of serial date numbers or date character vectors. If `ConvDates` is not specified, the bond is always convertible until maturity.

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`.

Data Types: `char` | `single` | `double`

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: `single` | `double`

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: `single` | `double`

## Output Arguments

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Expected price at time `0`, returned as an `NINST`-by-`1` array.

Structure with a vector of convertible bond prices at each node, returned as a tree structure.

Structure with a vector of convertible bond equity component at each node, returned as a tree structure.

Structure with a vector of convertible bond debt component at each node, returned as a tree structure.

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### 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 R2015a