minassetsensbystulz

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

Syntax

PriceSens = minassetsensbystulz(RateSpec,StockSpec1,StockSpec2,Settle,Maturity,OptSpec,Strike,Corr)

PriceSens = minassetsensbystulz(___,Name,Value)

Description

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

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

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.