# maxassetsensbystulz

Determine European rainbow option prices or sensitivities on maximum of two risky assets using Stulz pricing model

## Syntax

``PriceSens = maxassetsensbystulz(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr)``
``PriceSens = maxassetsensbystulz(___,Name,Value)``

## Description

example

````PriceSens = maxassetsensbystulz(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr)` computes option prices using the Stulz option pricing model.```

example

````PriceSens = maxassetsensbystulz(___,Name,Value)` specifies options using one or more optional name-value pair arguments in addition to the input arguments in the previous syntax.```

## Examples

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Consider a European rainbow option that gives the holder the right to buy either \$100,000 of an equity index at a strike price of 1000 (asset 1) or \$100,000 of a government bond (asset 2) with a strike price of 100% of face value, whichever is worth more at the end of 12 months. On January 15, 2008, the equity index is trading at 950, pays a dividend of 2% annually, and has a return volatility of 22%. Also on January 15, 2008, the government bond is trading at 98, pays a coupon yield of 6%, and has a return volatility of 15%. The risk-free rate is 5%. Using this data, calculate the price and sensitivity of the European rainbow option if the correlation between the rates of return is -0.5, 0, and 0.5.

Since the asset prices in this example are in different units, it is necessary to work in either index points (for asset 1) or in dollars (for asset 2). The European rainbow option allows the holder to buy the following: 100 units of the equity index at \$1000 each (for a total of \$100,000) or 1000 units of the government bonds at \$100 each (for a total of \$100,000). To convert the bond price (asset 2) to index units (asset 1), you must make the following adjustments:

• Multiply the strike price and current price of the government bond by 10 (1000/100).

• Multiply the option price by 100, considering that there are 100 equity index units in the option.

Once these adjustments are introduced, the strike price is the same for both assets (\$1000). First, create the `RateSpec`:

```Settle = 'Jan-15-2008'; Maturity = 'Jan-15-2009'; Rates = 0.05; Basis = 1; RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle,... 'EndDates', Maturity, 'Rates', Rates, 'Compounding', -1, 'Basis', Basis)```
```RateSpec = struct with fields: FinObj: 'RateSpec' Compounding: -1 Disc: 0.9512 Rates: 0.0500 EndTimes: 1 StartTimes: 0 EndDates: 733788 StartDates: 733422 ValuationDate: 733422 Basis: 1 EndMonthRule: 1 ```

Create the two `StockSpec` definitions.

```AssetPrice1 = 950; % Asset 1 => Equity index AssetPrice2 = 980; % Asset 2 => Government bond Sigma1 = 0.22; Sigma2 = 0.15; Div1 = 0.02; Div2 = 0.06; StockSpec1 = stockspec(Sigma1, AssetPrice1, 'continuous', Div1)```
```StockSpec1 = struct with fields: FinObj: 'StockSpec' Sigma: 0.2200 AssetPrice: 950 DividendType: {'continuous'} DividendAmounts: 0.0200 ExDividendDates: [] ```
`StockSpec2 = stockspec(Sigma2, AssetPrice2, 'continuous', Div2)`
```StockSpec2 = struct with fields: FinObj: 'StockSpec' Sigma: 0.1500 AssetPrice: 980 DividendType: {'continuous'} DividendAmounts: 0.0600 ExDividendDates: [] ```

Calculate the price and delta for different correlation levels.

```Strike = 1000 ; Corr = [-0.5; 0; 0.5]; OutSpec = {'price'; 'delta'}; OptSpec = 'call'; [Price, Delta] = maxassetsensbystulz(RateSpec, StockSpec1, StockSpec2,... Settle, Maturity, OptSpec, Strike, Corr,'OutSpec', OutSpec)```
```Price = 3×1 111.6683 103.7715 92.4412 ```
```Delta = 3×2 0.4594 0.3698 0.4292 0.3166 0.4053 0.2512 ```

The output `Delta` has two columns: the first column represents the `Delta` with respect to the equity index (asset 1), and the second column represents the `Delta` with respect to the government bond (asset 2). The value 0.4595 represents `Delta` with respect to one unit of the equity index. Since there are 100 units of the equity index, the overall `Delta` would be 45.94 (100 * 0.4594 ) for a correlation level of -0.5. To calculate the `Delta` with respect to the government bond, remember that an adjusted price of 980 was used instead of 98. Therefore, for example, the `Delta` with respect to government bond, for a correlation of 0.5 would be 251.2 (0.2512 * 100 * 10 ).

## Input Arguments

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Annualized, continuously compounded rate term structure, specified using `intenvset`.

Data Types: `structure`

Stock specification for asset 1, specified using `stockspec`.

Data Types: `structure`

Stock specification for asset 2, specified using `stockspec`.

Data Types: `structure`

Settlement or trade dates, specified as an `NINST`-by-`1` vector of numeric dates.

Data Types: `double`

Maturity dates, specified as an `NINST`-by-`1` vector.

Data Types: `double`

Option type, specified as an `NINST`-by-`1` cell array of character vectors with a value of `'call'` or `'put'`.

Data Types: `cell`

Strike prices, specified as an `NINST`-by-`1` vector.

Data Types: `double`

Correlation between the underlying asset prices, specified as an `NINST`-by-`1` vector.

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: ```[PriceSens] = maxassetsensbystulz(RateSpec, StockSpecA,StockSpecB,Settle,Maturity,OptSpec,Strike,Corr,'OutSpec',OutSpec)```

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: `cell`

## Output Arguments

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Expected prices or sensitivities, returned as an `NINST`-by-`1` or `NINST`-by-`2` vector.

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### Rainbow Option

A rainbow option payoff depends on the relative price performance of two or more assets.

A rainbow option gives the holder the right to buy or sell the best or worst of two securities, or options that pay the best or worst of two assets. Rainbow options are popular because of the lower premium cost of the structure relative to the purchase of two separate options. The lower cost reflects the fact that the payoff is generally lower than the payoff of the two separate options.

Financial Instruments Toolbox™ supports two types of rainbow options:

• Minimum of two assets — The option holder has the right to buy(sell) one of two risky assets, whichever one is worth less.

• Maximum of two assets — The option holder has the right to buy(sell) one of two risky assets, whichever one is worth more.