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

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

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

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

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

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

vector

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

Data Types: double

Maturity — Maturity dates

vector

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

Data Types: double

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 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] = maxassetsensbystulz(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.