# dblbarrierbybls

Price European double barrier options using Black-Scholes option pricing model

## Syntax

## Description

calculates European double barrier option prices using the Black-Scholes option
pricing model and the Ikeda and Kunitomo approximation.`Price`

= dblbarrierbybls(`RateSpec`

,`StockSpec`

,`OptSpec`

,`Strike`

,`Settle`

,`ExerciseDates`

,`BarrierSpec`

,`Barrier`

)

**Note**

Alternatively, you can use the `DoubleBarrier`

object to price double barrier options. For more
information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

specifies options using one or more name-value pair arguments in addition to the
input arguments in the previous syntax.`Price`

= dblbarrierbybls(___,`Name,Value`

)

## Examples

### Price a European Double Knock-Out Call Option

Compute the price of a European for a double knock-out (down and out-up and out) call option using the following data:

Rate = 0.05; Settle = datetime(2018,6,1); Maturity = datetime(2018,12,1); Basis = 1;

Define a `RateSpec`

.

RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle, 'EndDates', Maturity,'Rates', Rate, 'Compounding', -1, 'Basis', Basis);

Define a `StockSpec`

.

AssetPrice = 100; Volatility = 0.25; StockSpec = stockspec(Volatility, AssetPrice);

Define the double barrier option.

LBarrier = 80; UBarrier = 130; Barrier = [UBarrier LBarrier]; BarrierSpec = 'DKO'; OptSpec = 'Call'; Strike = 110;

Compute price of option using flat boundaries.

PriceFlat = dblbarrierbybls(RateSpec, StockSpec, OptSpec, Strike, Settle, Maturity, BarrierSpec, Barrier)

PriceFlat = 1.1073

Compute price of option using two curved boundaries.

```
Curvature = [0.05 -0.05];
PriceCurved = dblbarrierbybls(RateSpec, StockSpec, OptSpec, Strike, Settle, Maturity, BarrierSpec, Barrier, 'Curvature', Curvature)
```

PriceCurved = 1.4548

## Input Arguments

`StockSpec`

— Stock specification for underlying asset

structure

Stock specification for the underlying asset, specified by the
`StockSpec`

obtained from `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

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

or
`'put'`

| string array with values `"call"`

or
`"put"`

Definition of the option as `'call'`

or
`'put'`

, specified as a
`NINST`

-by-`1`

cell array of character
vectors or string array with values `'call'`

or
`'put'`

or `"call"`

or
`"put"`

.

**Data Types: **`char`

| `cell`

| `string`

`Strike`

— Option strike price value

matrix

Option strike price value, specified as an
`NINST`

-by-`1`

matrix of numeric
values, where each row is the schedule for one option.

**Data Types: **`double`

`Settle`

— Settlement or trade date

datetime array | string array | date character vector

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

-by-`1`

vector using a datetime
array, string array, or date character vectors.

To support existing code, `dblbarrierbybls`

also
accepts serial date numbers as inputs, but they are not recommended.

`ExerciseDates`

— Option exercise dates

datetime array | string array | date character vector

Option exercise dates, specified as an
`NINST`

-by-`1`

vector using a datetime
array, string array, or date character vectors.

**Note**

For a European option, the option expiry date has only one
`ExerciseDates`

value, which is the maturity of
the instrument.

To support existing code, `dblbarrierbybls`

also
accepts serial date numbers as inputs, but they are not recommended.

`BarrierSpec`

— Double barrier option type

cell array of character vectors with values of `'DKI'`

or `'DKO'`

| string array with values of `"DKI"`

or
`"DKO"`

Double barrier option type, specified as an
`NINST`

-by-`1`

cell array of character
vectors or string array with the following values:

`'DKI'`

— Double Knock-InThe

`'DKI'`

option becomes effective when the price of the underlying asset reaches one of the barriers. It gives the option holder the right but not the obligation to buy or sell the underlying security at the strike price, if the underlying asset goes above or below the barrier levels during the life of the option.`'DKO'`

— Double Knock-OutThe

`'DKO'`

option gives the option holder the right but not the obligation to buy or sell the underlying security at the strike price, as long as the underlying asset remains between the barrier levels during the life of the option. This option terminates when the price of the underlying asset passes one of the barriers.

Option | Barrier Type | Payoff If Any Barrier Crossed | Payoff If Barriers Not Crossed |
---|---|---|---|

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

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

**Data Types: **`char`

| `cell`

| `string`

`Barrier`

— Double barrier value

numeric

Double barrier value, specified as
`NINST`

-by-`1`

matrix of numeric
values, where each element is a `1`

-by-`2`

vector where the first column is Barrier(1)(UB) and the second column is
Barrier(2)(LB). Barrier(1) must be greater than Barrier(2).

**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 =
dblbarrierbybls(RateSpec,StockSpec,OptSpec,Strike,Settle,Maturity,BarrierSpec,Barrier,'Curvature',[1,5])
```

`Curvature`

— Curvature levels of the upper and lower barriers

`[]`

(default) | matrix

Curvature levels of the upper and lower barriers, specified as the
comma-separated pair consisting of `'Curvature'`

and an
`NINST`

-by-`1`

matrix, where each
element is a `1`

-by-`2`

vector. The
first column is the upper barrier curvature (*d1*) and
the second column is the lower barrier curvature
(*d2*).

*d1*=*d2*=`0`

corresponds to two flat boundaries.*d1*< 0 <*d2*corresponds to an exponentially growing lower boundary and an exponentially decaying upper boundary.*d1*> 0 >*d2*corresponds to a convex downward lower boundary and a convex upward upper boundary.

**Data Types: **`double`

## Output Arguments

`Price`

— Expected prices for double barrier options

matrix

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

-by-`1`

matrix.

## More About

### Double Barrier Option

A *double barrier* option is similar to the
standard single barrier option except that it has two barrier levels: a lower
barrier (`LB`

) and an upper barrier (`UB`

).

The payoff for a double barrier option depends on whether the underlying asset remains between the barrier levels during the life of the option. Double barrier options are less expensive than single barrier options as they have a higher knock-out probability. Because of this, double barrier options allow investors to reduce option premiums and match an investor’s belief about the future movement of the underlying price process.

### Ikeda and Kunitomo Approximation

The analytical formulas of Ikeda and Kunitomo approach pricing as constrained by curved boundaries.

This approach has the advantage of covering barriers that are flat, have exponential growth or decay, or are concave. The Ikeda and Kunitomo model for pricing double barrier options focuses on calculating double knock-out barriers.

## References

[1] Hull, J. *Options, Futures, and Other Derivatives.* Fourth
Edition. Upper Saddle River, NJ: Prentice Hall, 2000.

[2] Kunitomo, N., and M. Ikeda. “Pricing Options with Curved
Boundaries.” *Mathematical Finance.* Vol. 2, Number 4,
1992.

[3] Rubinstein, M., and E. Reiner. “Breaking Down the Barriers.”
*Risk.* Vol. 4, Number 8, 1991, pp. 28–35.

## Version History

**Introduced in R2019a**

### R2022b: Serial date numbers not recommended

Although `dblbarrierbybls`

supports serial date numbers,
`datetime`

values are recommended instead. The
`datetime`

data type provides flexible date and time
formats, storage out to nanosecond precision, and properties to account for time
zones and daylight saving time.

To convert serial date numbers or text to `datetime`

values, use the `datetime`

function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum"); y = year(t)

y = 2021

There are no plans to remove support for serial date number inputs.

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