Documentation

# asiansensbyhhm

Calculate price and sensitivities of European discrete arithmetic fixed Asian options using Haug, Haug, Margrabe model

## Syntax

``PriceSens = asiansensbyhhm(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates)``
``PriceSens = asiansensbyhhm(___,Name,Value)``

## Description

example

````PriceSens = asiansensbyhhm(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates)` calculates prices and sensitivities for European discrete arithmetic fixed Asian options using the Haug, Haug, Margrabe model.```

example

````PriceSens = asiansensbyhhm(___,Name,Value)` adds optional name-value pair arguments.```

## Examples

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Define the Asian option parameters.

```AssetPrice = 100; Strike = 95; Rates = 0.1; Sigma = 0.15; Settle = 'Apr-1-2013'; Maturity = 'Oct-1-2013';```

Create a `RateSpec` using the `intenvset` function.

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

Create a `StockSpec` for the underlying asset using the `stockspec` function.

```DividendType = 'Continuous'; DividendAmounts = 0.05; StockSpec = stockspec(Sigma, AssetPrice, DividendType, DividendAmounts);```

Calculate the price and sensitivities of the Asian option using the Haug, Haug, Margrabe approximation. Assume that the averaging period has started before the `Settle` date.

```OptSpec = 'Call'; ExerciseDates = 'Oct-1-2013'; NumFixings = 12; AvgDate = 'Jan-1-2013'; AvgPrice = 100; OutSpec = {'Price','Delta','Gamma'}; [Price,Delta,Gamma] = asiansensbyhhm(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates, ... 'NumFixings',NumFixings,'AvgDate',AvgDate,'AvgPrice',AvgPrice,'OutSpec',OutSpec)```
```Price = 5.8216 ```
```Delta = 0.5907 ```
```Gamma = 0.0143 ```

Define the Asian option parameters.

```AssetPrice = 100; Strike = 95; Rates = 0.1; Sigma = 0.15; Settle = 'Apr-1-2013'; Maturity = 'Oct-1-2013';```

Create a `RateSpec` using the `intenvset` function.

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

Create a `StockSpec` for the underlying asset using the `stockspec` function.

```DividendType = 'Continuous'; DividendAmounts = 0.05; StockSpec = stockspec(Sigma, AssetPrice, DividendType, DividendAmounts);```

Calculate the price and sensitivities of the Asian option using the Haug, Haug, Margrabe approximation. Assume that the averaging period started after the `Settle` date.

```OptSpec = 'Call'; ExerciseDates = 'Oct-1-2013'; NumFixings = 15; AvgDate = 'Jan-1-2013'; OutSpec = {'Price','Delta','Gamma'}; [Price,Delta,Gamma] = asiansensbyhhm(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates, ... 'NumFixings',NumFixings,'AvgDate',AvgDate,'OutSpec',OutSpec)```
```Price = 1.3785e-07 ```
```Delta = 1.1438e-07 ```
```Gamma = 9.0830e-08 ```

## 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 underlying asset, specified using `StockSpec` obtained from `stockspec`. For information on the stock specification, see `stockspec`.

`stockspec` can handle other types of underlying assets. For example, stocks, stock indices, and commodities. If dividends are not specified in `StockSpec`, dividends are assumed to be `0`.

Data Types: `struct`

Definition of option, specified as `'call'` or `'put'` using a character vector, cell array of character vectors, or string array.

Data Types: `char` | `cell` | `string`

Option strike price value, specified with a nonnegative integer using a `NINST`-by-`1` vector of strike price values.

Data Types: `double`

Settlement date or trade date for the Asian option, specified as a `NINST`-by-`1` vector using serial date numbers, date character vectors, datetimes, or string arrays.

Data Types: `double` | `char` | `datetime` | `string`

European option exercise dates, specified as a `NINST`-by-`1` vector using serial date numbers, date character vectors, datetimes, or string arrays.

### Note

For a European option, there is only one `ExerciseDates` on the option expiry date.

Data Types: `double` | `char` | `datetime` | `string`

### Name-Value Pair Arguments

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`.

Example: ```PriceSens = asiansensbyhhm(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates,'OutSpec',{'All'},'NumFixings',15)```

Define outputs, specified as the comma-separated pair consisting of `'OutSpec'` and a `NOUT`- by-`1` or `1`-by-`NOUT` cell array of character vectors or string array with possible values of `'Price'`, `'Delta'`, `'Gamma'`, `'Vega'`, `'Lambda'`, `'Rho'`, `'Theta'`, and `'All'`.

`OutSpec = {'All'}` specifies that the output is `Delta`, `Gamma`, `Vega`, `Lambda`, `Rho`, `Theta`, and `Price`, in that order. This is the same as specifying `OutSpec` to include each sensitivity:

Example: ```OutSpec = {'delta','gamma','vega','lambda','rho','theta','price'}```

Data Types: `char` | `cell` | `string`

Average price of underlying asset at the `Settle` date, specified as the comma-separated pair consisting of `'AvgPrice'` and a `NINST`-by-`1` vector.

### Note

Use the `AvgPrice` argument when `AvgDate` < `Settle`.

Data Types: `double`

Date averaging period begins, specified as the comma-separated pair consisting of `'AvgDate'` and a `NINST`-by-`1` vector using character vectors, serial date numbers, datetimes, or string arrays.

Data Types: `char` | `double` | `datetime` | `string`

Total number of fixings or averaging points, specified as the comma-separated pair consisting of `'NumFixings'` and a `NINST`-by-`1` vector.

Data Types: `double`

## Output Arguments

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Expected prices or sensitivities for fixed Asian options, returned as a `NINST`-by-`1` vector. `asianbyhhm` calculates prices of European arithmetic fixed (average price) Asian options with discretely monitoring.

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### Asian Option

An Asian option is a path-dependent option with a payoff linked to the average value of the underlying asset during the life (or some part of the life) of the option.

Asian options are similar to lookback options in that there are two types of Asian options: fixed (average price option) and floating (average strike option). Fixed Asian options have a specified strike, while floating Asian options have a strike equal to the average value of the underlying asset over the life of the option. For more information, see Asian Option.

 Haug, E. G. The Complete Guide to Option Pricing Formulas. McGraw-Hill Education, 2007.