Price barrier option from Cox-Ross-Rubinstein binomial tree

```
[Price,PriceTree]
= barrierbycrr(CRRTree,OptSpec,Strike,Settle,AmericanOpt,ExerciseDates,BarrierSpec,Barrier)
```

```
[Price,PriceTree]
= barrierbycrr(___,Rebate,Options)
```

`[`

calculates
prices for barrier options using a Cox-Ross-Rubinstein binomial tree.`Price`

,`PriceTree`

]
= barrierbycrr(`CRRTree`

,`OptSpec`

,`Strike`

,`Settle`

,`AmericanOpt`

,`ExerciseDates`

,`BarrierSpec`

,`Barrier`

)

This example shows how to price a barrier option using a CRR binomial tree by loading the file deriv.mat, which provides CRRTree. The CRRTree structure contains the stock specification and time information needed to price the option.

load deriv.mat; OptSpec = 'Call'; Strike = 105; Settle = '01-Jan-2003'; ExerciseDates = '01-Jan-2006'; AmericanOpt = 1; BarrierSpec = 'UI'; Barrier = 102; Price = barrierbycrr(CRRTree, OptSpec, Strike, Settle, ... ExerciseDates, AmericanOpt, BarrierSpec, Barrier)

Price = 12.1272

`CRRTree`

— Stock tree structurestructure

Stock tree structure, specified by using `crrtree`

.

**Data Types: **`struct`

`OptSpec`

— Definition of option character vector with values

`'call'`

or `'put'`

| cell array of character vectors with values `'call'`

or `'put'`

Definition of an option as `'call'`

or `'put'`

,
specified as a `NINST`

-by-`1`

cell
array of character vector values.

**Data Types: **`char`

| `cell`

`Strike`

— Option strike price valueinteger

Option strike price value for a European or an American Option, specified as
`NINST`

-by-`1`

matrix of integers. Each row is the schedule for
one option.

**Data Types: **`double`

`Settle`

— Settlement or trade dateserial date number | date character vector

Settlement or trade date for the barrier option, specified as
a `NINST`

-by-`1`

matrix of serial
date numbers or date character vectors. The `Settle`

date
for every barrier is set to the `ValuationDate`

of
the stock tree. The barrier argument `Settle`

is
ignored.

**Data Types: **`double`

| `char`

`ExerciseDates`

— Option exercise datesserial date number | date character vector

Option exercise dates, specified as a serial date number or a date character vector:

For a European option, use a

`1`

-by-`1`

matrix of dates. Each row is the schedule for one option. For a European option, there is only one`ExerciseDates`

on the option expiry date which is the maturity of the instrument.For an American option, use a

`1`

-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`

, the option can be exercised between`ValuationDate`

of the stock tree and the single listed date in`ExerciseDates`

.

**Data Types: **`double`

| `char`

`AmericanOpt`

— Option typeOption type values

`0`

or
`1`

Option type, specified as `NINST`

-by-`1`

matrix of integer
flags with values:

`0`

— European`1`

— American

**Data Types: **`double`

`BarrierSpec`

— Barrier option typecharacter vector with values:

`'UI'`

, `'UO'`

, `'DI'`

, `'DO'`

| cell array of character vectors with values: `'UI'`

, `'UO'`

, `'DI'`

, `'DO'`

Barrier option type, specified as a character vector or an
`NINST`

-by-`1`

cell array of character vectors with the following
values:

`'UI'`

— Up Knock-inThis option becomes effective when the price of the underlying asset passes above the barrier level. It gives the option holder the right, but not the obligation, to buy or sell (call/put) the underlying security at the strike price if the underlying asset goes above the barrier level during the life of the option.

`'UO'`

— Up Knock-outThis option gives the option holder the right, but not the obligation, to buy/sell (call/put) the underlying security at the strike price as long as the underlying asset does not go above the barrier level during the life of the option. This option terminates when the price of the underlying asset passes above the barrier level. Usually, with an up-and-out option, the rebate is paid if the spot price of the underlying reaches or exceeds the barrier level.

`'DI'`

— Down Knock-inThis option becomes effective when the price of the underlying stock passes below the barrier level. It gives the option holder the right, but not the obligation, to buy or sell (call/put) the underlying security at the strike price if the underlying security goes below the barrier level during the life of the option. With a down-and-in option, the rebate is paid if the spot price of the underlying does not reach the barrier level during the life of the option. Note,

`barrierbyfd`

does not support American knock-in barrier options.`'DO'`

— Down Knock-upThis option gives the option holder the right, but not the obligation, to buy or sell (call/put) the underlying asset at the strike price as long as the underlying asset does not go below the barrier level during the life of the option. This option terminates when the price of the underlying security passes below the barrier level. Usually the option holder receives a rebate amount if the option expires worthless.

Option | Barrier Type | Payoff if Barrier Crossed | Payoff if Barrier not Crossed |
---|---|---|---|

Call/Put | Down Knock-out | Worthless | Standard Call/Put |

Call/Put | Down Knock-in | Call/Put | Worthless |

Call/Put | Up Knock-out | Worthless | Standard Call/Put |

Call/Put | Up Knock-in | Standard Call/Put | Worthless |

**Data Types: **`char`

| `cell`

`Barrier`

— Barrier levelnumeric

Barrier level, specified as a
`NINST`

-by-`1`

matrix of numeric values.

**Data Types: **`double`

`Rebate`

— Rebate value`0`

(default) | numeric(Optional) Rebate value, specified as a `NINST`

-by-`1`

matrix of numeric values. For Knock-in options,
the `Rebate`

is paid at expiry.
For Knock-out options, the
`Rebate`

is paid when the
`Barrier`

is reached.

**Data Types: **`double`

`Options`

— Derivatives pricing optionsstructure

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

.

**Data Types: **`struct`

`Price`

— Expected prices for barrier options at time 0matrix

Expected prices for barrier options at time 0, returned as a
`NINST`

-by-`1`

matrix.

`PriceTree`

— Structure with vector of barrier option prices at each nodetree structure

Structure with a vector of barrier option prices at each node, returned as a tree structure.

`PriceTree`

is a MATLAB^{®} structure of trees
containing vectors of instrument prices and a vector of observation
times for each node.

`PriceTree.PTree`

contains the prices.

`PriceTree.tObs`

contains the observation times.

`PriceTree.dObs`

contains the observation dates.

A *Barrier option* has
not only a strike price but also a barrier level and sometimes a rebate.

A rebate is a fixed amount that is paid if the option cannot be exercised because the barrier
level has been reached or not reached. The payoff for this type of option depends on
whether the underlying asset crosses the predetermined trigger value (barrier level),
indicated by `Barrier`

, during the life of the option.

[1] Derman, E., I. Kani, D. Ergener and I. Bardhan. “Enhanced
Numerical Methods for Options with Barriers.” *Financial
Analysts Journal.* (Nov.-Dec.), 1995, pp. 65–74.

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