# barrierbyfd

Calculate barrier option prices using finite difference method

## Syntax

## Description

`[`

calculates European and American barrier option prices on a single underlying asset
using the finite difference method. `Price`

,`PriceGrid`

,`AssetPrices`

,`Times`

]
= barrierbyfd(`RateSpec`

,`StockSpec`

,`OptSpec`

,`Strike`

,`Settle`

,`ExerciseDates`

,`BarrierSpec`

,`Barrier`

)`barrierbyfd`

assumes that the
barrier is continuously monitored.

`[`

adds optional name-value pair arguments. `Price`

,`PriceGrid`

,`AssetPrices`

,`Times`

]
= barrierbyfd(___,`Name,Value`

)`barrierbyfd`

assumes that
the barrier is continuously monitored.

## Examples

### Price a Barrier Down and Out Call Option Using Finite Difference Method

Create a `RateSpec`

.

AssetPrice = 50; Strike = 45; Rate = 0.035; Volatility = 0.30; Settle = '01-Jan-2015'; Maturity = '01-Jan-2016'; Basis = 1; RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle,... 'EndDates', Maturity,'Rates', Rate, 'Compounding', -1, 'Basis', Basis)

`RateSpec = `*struct with fields:*
FinObj: 'RateSpec'
Compounding: -1
Disc: 0.9656
Rates: 0.0350
EndTimes: 1
StartTimes: 0
EndDates: 736330
StartDates: 735965
ValuationDate: 735965
Basis: 1
EndMonthRule: 1

Create a `StockSpec`

.

StockSpec = stockspec(Volatility, AssetPrice)

`StockSpec = `*struct with fields:*
FinObj: 'StockSpec'
Sigma: 0.3000
AssetPrice: 50
DividendType: []
DividendAmounts: 0
ExDividendDates: []

Calculate the price of a European Down and Out call option using Finite Difference.

Barrier = 40; BarrierSpec = 'DO'; OptSpec = 'Call'; Price = barrierbyfd(RateSpec, StockSpec, OptSpec, Strike, Settle, Maturity,... BarrierSpec, Barrier)

Price = 8.5020

## Input Arguments

`StockSpec`

— Stock specification for underlying asset

structure

Stock specification for the underlying asset. For information
on the stock specification, see `stockspec`

.

`stockspec`

handles several
types of underlying assets. For example, for physical commodities
the price is `StockSpec.Asset`

, the volatility is `StockSpec.Sigma`

,
and the convenience yield is `StockSpec.DividendAmounts`

.

**Data Types: **`struct`

`OptSpec`

— Definition of option

character vector with values `'call'`

or
`'put'`

| string array with values `'call'`

or
`'put'`

Definition of an option as `'call'`

or `'put'`

, specified as
a character vector or string array with values `"call"`

or
`"put"`

.

**Data Types: **`char`

| `string`

`Strike`

— Option strike price value

numeric

Option strike price value, specified as a scalar numeric.

**Data Types: **`double`

`Settle`

— Settlement or trade date

serial date number | date character vector | datetime object

Settlement or trade date for the barrier option, specified as a serial date number, a date character vector, or a datetime object.

**Data Types: **`double`

| `char`

| `datetime`

`ExerciseDates`

— Option exercise dates

date character vector | serial date number | datetime object

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

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, the option can be exercised between`Settle`

and the single listed date in`ExerciseDates`

.

**Data Types: **`double`

| `char`

| `cell`

| `datetime`

`BarrierSpec`

— Barrier option type

character vector with values: `'UI'`

, `'UO'`

, `'DI'`

, `'DO'`

Barrier option type, specified as a character vector 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`

`Barrier`

— Barrier level

numeric

Barrier level, specified as a scalar numeric value.

**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 = barrierbyfd(RateSpec,StockSpec,OptSpec,Strike,Settle,Maturity,BarrierSpec,Barrier,Rebate,1000)`

`Rebate`

— Rebate value

`0`

(default) | numeric

Rebate value, specified as the comma-separated pair consisting of `'Rebate'`

and a scalar numeric. 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`

`AssetGridSize`

— Size of asset grid used for a finite difference grid

`400`

(default) | positive numeric

Size of the asset grid used for finite difference grid, specified as the comma-separated pair
consisting of `'AssetGridSize'`

and a scalar positive
numeric.

**Data Types: **`double`

`TimeGridSize`

— Size of time grid used for finite difference grid

`100`

(default) | positive numeric

Size of the time grid used for the finite difference grid, specified as the comma-separated
pair consisting of `'TimeGridSize'`

and a scalar
positive numeric.

**Data Types: **`double`

`AmericanOpt`

— Option type

`0`

European (default) | integer with values `0`

or
`1`

Option type, specified as the comma-separated pair consisting of
`'AmericanOpt'`

and a scalar flag with one of the
following values:

`0`

— European`1`

— American

**Data Types: **`logical`

## Output Arguments

`Price`

— Expected prices for barrier options

matrix

Expected prices for barrier options, returned as a `NINST`

-by-`1`

matrix.

`PriceGrid`

— Grid containing prices calculated by finite difference method

grid

Grid containing prices calculated by the finite difference method,
returned as a grid that is two-dimensional with size `PriceGridSize*length(Times)`

.
The number of columns does not have to be equal to the `TimeGridSize`

,
because ex-dividend dates in the `StockSpec`

are
added to the time grid. The price for `t = 0`

is
contained in `PriceGrid(:, end)`

.

`Times`

— Times corresponding to second dimension of `PriceGrid`

vector

Times corresponding to the second dimension of the `PriceGrid`

,
returned as a vector.

## More About

### Barrier Option

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.

## References

[1] Hull, J. *Options, Futures, and Other Derivatives.* Fourth
Edition. Prentice Hall. 2000, pp. 646–649.

[2] Aitsahlia, F., L. Imhof, and T.L. Lai. “Pricing and hedging of American knock-in
options.” *The Journal of Derivatives.* Vol. 11.3, 2004, pp.
44–50.

[3] Rubinstein M. and E. Reiner. “Breaking down the barriers.” *Risk.* Vol.
4(8), 1991, pp. 28–35.

## Version History

**Introduced in R2016b**

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