Price barrier options using standard trinomial tree

`[`

prices
barrier options using a standard trinomial (STT) tree.`Price`

,`PriceTree`

]
= barrierbystt(`STTTree`

,`OptSpec`

,`Strike`

,`Settle`

,`ExerciseDates`

,`AmericanOpt`

,`BarrierSpec`

,`Barrier`

)

Create a `RateSpec`

.

StartDates = 'Jan-1-2009'; EndDates = 'Jan-1-2013'; Rates = 0.035; Basis = 1; Compounding = -1; RateSpec = intenvset('ValuationDate', StartDates, 'StartDates', StartDates,... 'EndDates', EndDates, 'Rates', Rates,'Compounding', Compounding, 'Basis', Basis)

`RateSpec = `*struct with fields:*
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.8694
Rates: 0.0350
EndTimes: 4
StartTimes: 0
EndDates: 735235
StartDates: 733774
ValuationDate: 733774
Basis: 1
EndMonthRule: 1

Create a `StockSpec`

.

AssetPrice = 85; Sigma = 0.15; StockSpec = stockspec(Sigma, AssetPrice)

`StockSpec = `*struct with fields:*
FinObj: 'StockSpec'
Sigma: 0.1500
AssetPrice: 85
DividendType: []
DividendAmounts: 0
ExDividendDates: []

Create an `STTTree`

.

NumPeriods = 4; TimeSpec = stttimespec(StartDates, EndDates, 4); STTTree = stttree(StockSpec, RateSpec, TimeSpec)

`STTTree = `*struct with fields:*
FinObj: 'STStockTree'
StockSpec: [1x1 struct]
TimeSpec: [1x1 struct]
RateSpec: [1x1 struct]
tObs: [0 1 2 3 4]
dObs: [733774 734139 734504 734869 735235]
STree: {1x5 cell}
Probs: {[3x1 double] [3x3 double] [3x5 double] [3x7 double]}

Define the barrier option and compute the price.

Settle = '1/1/09'; ExerciseDates = '1/1/12'; OptSpec = 'call'; Strike = 105; AmericanOpt = 1; BarrierSpec = 'UI'; Barrier = 115; Price= barrierbystt(STTTree, OptSpec, Strike, Settle, ExerciseDates,... AmericanOpt, BarrierSpec, Barrier)

Price = 3.7977

`STTTree`

— Stock tree structure for standard trinomial treestructure

Stock tree structure for a standard trinomial tree, specified
by using `stttree`

.

**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 `'call'`

or `'put'`

using
a character vector or a `NINST`

-by-`1`

cell
array of character vectors for `'call'`

or `'put'`

.

**Data Types: **`char`

| `cell`

`Strike`

— European or American option strike price valuenonnegative numeric

European or American option strike price value, specified with a nonnegative integer using a
`NINST`

-by-`1`

matrix of nonnegative numeric values. Each row is
the schedule for one option. To compute the value
of a floating-strike barrier option,
`Strike`

should be specified as
`NaN`

. Floating-strike barrier
options are also known as average strike
options.

**Data Types: **`double`

`Settle`

— Settlement date or trade dateserial date number | date character vector

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

-by-`1`

matrix of settlement
or trade dates using serial date numbers or date character vectors.

The `Settle`

date for every barrier option
is set to the `ValuationDate`

of the stock tree.
The barrier argument, `Settle`

, is ignored.

**Data Types: **`double`

| `char`

| `cell`

`ExerciseDates`

— Option exercise datesserial date number | date character vector

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

For a European option, use a

`NINST`

-by-`1`

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

on the option expiry date.For an American option, use a

`NINST`

-by-`2`

vector of exercise date boundaries. The option can be exercised on any tree 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 of serial date numbers or cell array of character vectors, the option can be exercised between`ValuationDate`

of the stock tree and the single listed`ExerciseDates`

.

**Data Types: **`double`

| `char`

| `cell`

`AmericanOpt`

— Option typescalar with values

`[0,1]`

Option type, specified as an `NINST`

-by-`1`

matrix of 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. Note,

`barrierbyfd`

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

— Up Knock-outThis option gives the option holder the right, but not the obligation, to buy or 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 levelsnumeric

Barrier levels, specified as an `NINST`

-by-`1`

matrix of
numeric values.

**Data Types: **`double`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`Price = barrierbystt(STTTree,OptSpec,Strike,Settle,ExerciseDates,1,'UI',115,'Rebate',25)`

`'Rebate'`

— Rebate values`0`

(default) | numericRebate values, specified as the comma-separated pair consisting of `'Rebate'`

and 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

Derivatives pricing options, specified as the comma-separated pair consisting of
`'Options'`

and a structure that is created with
`derivset`

.

**Data Types: **`struct`

`Price`

— Expected prices for barrier options at time `0`

matrix

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. For more information, see Barrier 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.

`derivset`

| `instbarrier`

| `sttprice`

| `sttsens`

| `stttimespec`

| `stttree`

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