# barrierbyfd

Calculate barrier option prices using finite difference method

## Syntax

``````[Price,PriceGrid,AssetPrices,Times] = barrierbyfd(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates,BarrierSpec,Barrier)``````
``````[Price,PriceGrid,AssetPrices,Times] = barrierbyfd(___,Name,Value)``````

## Description

example

``````[Price,PriceGrid,AssetPrices,Times] = barrierbyfd(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates,BarrierSpec,Barrier)``` calculates European and American barrier option prices on a single underlying asset using the finite difference method. `barrierbyfd` assumes that the barrier is continuously monitored. NoteAlternatively, you can use the `Barrier` object to price Barrier options. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments. ```

example

``````[Price,PriceGrid,AssetPrices,Times] = barrierbyfd(___,Name,Value)``` adds optional name-value pair arguments. `barrierbyfd` assumes that the barrier is continuously monitored. ```

## Examples

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Create a `RateSpec`.

```AssetPrice = 50; Strike = 45; Rate = 0.035; Volatility = 0.30; Settle = datetime(2015,1,1); Maturity = datetime(2016,1,1); 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

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Interest-rate term structure (annualized and continuously compounded), specified by the `RateSpec` obtained from `intenvset`. For information on the interest-rate specification, see `intenvset`.

Data Types: `struct`

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`

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`

Option strike price value, specified as a scalar numeric.

Data Types: `double`

Settlement or trade date for the barrier option, specified as a scalar datetime, string, or date character vector.

To support existing code, `barrierbyfd` also accepts serial date numbers as inputs, but they are not recommended.

Option exercise dates, specified as a datetime array, string array, or date character vectors:

• 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`.

To support existing code, `barrierbyfd` also accepts serial date numbers as inputs, but they are not recommended.

Barrier option type, specified as a character vector with the following values:

• `'UI'` — Up Knock-in

This 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-out

This 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-in

This 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-up

This 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.

OptionBarrier TypePayoff if Barrier CrossedPayoff if Barrier not Crossed
Call/PutDown Knock-outWorthlessStandard Call/Put
Call/PutDown Knock-inCall/PutWorthless
Call/PutUp Knock-outWorthlessStandard Call/Put
Call/PutUp Knock-inStandard Call/PutWorthless

Data Types: `char`

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 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`

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`

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`

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

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Expected prices for barrier options, returned as a `NINST`-by-`1` matrix.

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)`.

Prices of the asset defined by the `StockSpec` corresponding to the first dimension of `PriceGrid`, returned as a vector.

Times corresponding to the second dimension of the `PriceGrid`, returned as a vector.

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### 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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