# minassetsensbystulz

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

## Syntax

## Description

computes option prices using the Stulz option
pricing model.`PriceSens`

= minassetsensbystulz(`RateSpec`

,`StockSpec1`

,`StockSpec2`

,`Settle`

,`Maturity`

,`OptSpec`

,`Strike`

,`Corr`

)

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

= minassetsensbystulz(___,`Name,Value`

)

## Examples

### Compute Rainbow Option Prices and Sensitivities Using the Stulz Option Pricing Model

Consider a European rainbow put option that gives the holder the right to sell either stock A or stock B at a strike of 50.25, whichever has the lower value on the expiration date May 15, 2009. On November 15, 2008, stock A is trading at 49.75 with a continuous annual dividend yield of 4.5% and has a return volatility of 11%. Stock B is trading at 51 with a continuous dividend yield of 5% and has a return volatility of 16%. The risk-free rate is 4.5%. Using this data, if the correlation between the rates of return is -0.5, 0, and 0.5, calculate the price and sensitivity of the minimum of two assets that are European rainbow put options. First, create the `RateSpec`

:

Settle = datetime(2000,11,15); Maturity = datetime(2009,5,15); Rates = 0.045; 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: 0.6822
Rates: 0.0450
EndTimes: 8.5000
StartTimes: 0
EndDates: 733908
StartDates: 730805
ValuationDate: 730805
Basis: 1
EndMonthRule: 1

Create the two `StockSpec`

definitions.

```
AssetPriceA = 49.75;
AssetPriceB = 51;
SigmaA = 0.11;
SigmaB = 0.16;
DivA = 0.045;
DivB = 0.05;
StockSpecA = stockspec(SigmaA, AssetPriceA, 'continuous', DivA)
```

`StockSpecA = `*struct with fields:*
FinObj: 'StockSpec'
Sigma: 0.1100
AssetPrice: 49.7500
DividendType: {'continuous'}
DividendAmounts: 0.0450
ExDividendDates: []

`StockSpecB = stockspec(SigmaB, AssetPriceB, 'continuous', DivB)`

`StockSpecB = `*struct with fields:*
FinObj: 'StockSpec'
Sigma: 0.1600
AssetPrice: 51
DividendType: {'continuous'}
DividendAmounts: 0.0500
ExDividendDates: []

Calculate price and delta for different correlation levels.

Strike = 50.25; Corr = [-0.5;0;0.5]; OptSpec = 'put'; OutSpec = {'Price'; 'delta'}; [P, D] = minassetsensbystulz(RateSpec, StockSpecA, StockSpecB,... Settle, Maturity, OptSpec, Strike, Corr, 'OutSpec', OutSpec)

`P = `*3×1*
10.0002
9.1433
8.1622

`D = `*3×2*
-0.2037 -0.2192
-0.1774 -0.2101
-0.1452 -0.2075

The output `Delta`

has two columns: the first column represents the `Delta`

with respect to the stock A (asset 1), and the second column represents the `Delta`

with respect to the stock B (asset 2). The value 0.4183 represents `Delta`

with respect to the stock A for a correlation level of -0.5. The `Delta`

with respect to stock B, for a correlation of zero is -0.3189.

## Input Arguments

`RateSpec`

— Annualized, continuously compounded rate term structure

structure

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

.

**Data Types: **`structure`

`StockSpec1`

— Stock specification for asset 1

structure

Stock specification for asset 1, specified
using `stockspec`

.

**Data Types: **`structure`

`StockSpec2`

— Stock specification for asset 2

structure

Stock specification for asset 2, specified
using `stockspec`

.

**Data Types: **`structure`

`Settle`

— Settlement or trade dates

datetime array | string array | date character vector

Settlement or trade dates, specified as an
`NINST`

-by-`1`

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

To support existing code, `minassetsensbystulz`

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

`Maturity`

— Maturity dates

datetime array | string array | date character vector

Maturity dates, specified as an
`NINST`

-by-`1`

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

To support existing code, `minassetsensbystulz`

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

`OptSpec`

— Option type

cell array of character vectors with a
value of `'call'`

or
`'put'`

Option type, specified as an
`NINST`

-by-`1`

cell array of character vectors with a value of
`'call'`

or
`'put'`

.

**Data Types: **`cell`

`Strike`

— Strike prices

vector

Strike prices, specified as an
`NINST`

-by-`1`

vector.

**Data Types: **`double`

`Corr`

— Correlation between the underlying asset prices

vector

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] =
minassetsensbystulz(RateSpec,
StockSpecA,StockSpecB,Settle,Maturity,OptSpec,Strike,Corr,'OutSpec',OutSpec)
```

`OutSpec`

— Define outputs

`{'Price'}`

(default) | cell array of character vectors with values
`'Price'`

,
`'Delta'`

,
`'Gamma'`

,
`'Vega'`

,
`'Lambda'`

,
`'Rho'`

,
`'Theta'`

, and
`'All'`

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

`PriceSens`

— Expected prices or sensitivities

vector

Expected prices or sensitivities, returned
as an
`NINST`

-by-`1`

or
`NINST`

-by-`2`

vector.

## More About

### 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.

For more information, see Rainbow Option.

## Version History

**Introduced in R2009a**

### R2022b: Serial date numbers not recommended

Although `minassetsensbystulz`

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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