cashsensbybls

Determine price or sensitivities of cash-or-nothing digital options using Black-Scholes model

Syntax

PriceSens = cashsensbybls(RateSpec,StockSpec,Settle,Maturity,OptSpec,Strike,Payoff)

PriceSens = cashsensbybls(___,Name,Value)

Description

PriceSens = cashsensbybls(RateSpec,StockSpec,Settle,Maturity,OptSpec,Strike,Payoff) computes the price or sensitivities for cash-or-nothing European digital options using the Black-Scholes option pricing model.

Note

Alternatively, you can use the Binary object to calculate price or sensitivities for digital options. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

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

Examples

Compute Cash-or-Nothing Option Prices and Sensitivities Using the Black-Scholes Option Pricing Model

Consider a European call and put cash-or-nothing options on a futures contract with an exercise price of $90, and a fixed payoff of $10 that expires on October 1, 2008. Assume that on January 1, 2008 the contract trades at $110, and has a volatility of 25% per annum and the risk-free rate is 4.5% per annum. Using this data, calculate the price and sensitivity of the call and put cash-or-nothing options on the futures contract. First, create the RateSpec:

Settle = datetime(2008,1,1);
Maturity = datetime(2008,10,1);
Rates = 0.045;
Compounding = -1;  
Basis = 1;
RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle,...
'EndDates', Maturity, 'Rates', Rates, 'Compounding', Compounding, 'Basis', Basis)

RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: -1
             Disc: 0.9668
            Rates: 0.0450
         EndTimes: 0.7500
       StartTimes: 0
         EndDates: 733682
       StartDates: 733408
    ValuationDate: 733408
            Basis: 1
     EndMonthRule: 1

Define the StockSpec.

AssetPrice = 110;
Sigma = .25;
DivType = 'Continuous';
DivAmount = Rates;
StockSpec = stockspec(Sigma, AssetPrice, DivType, DivAmount)

StockSpec = struct with fields:
             FinObj: 'StockSpec'
              Sigma: 0.2500
         AssetPrice: 110
       DividendType: {'continuous'}
    DividendAmounts: 0.0450
    ExDividendDates: []

Define the call and put options.

OptSpec = {'call'; 'put'};
Strike = 90;
Payoff = 10;

Compute the gamma, theta, and price.

OutSpec = { 'gamma';'theta';'price'};
[Gamma, Theta, Price] = cashsensbybls(RateSpec, StockSpec,...
Settle, Maturity, OptSpec, Strike, Payoff, 'OutSpec', OutSpec)

Gamma = 2×1

   -0.0050
    0.0050

Theta = 2×1

   -2.2489
    1.8139

Price = 2×1

    7.6716
    1.9965

Input Arguments

RateSpec — Interest-rate term structure

structure

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

StockSpec — Stock specification for underlying asset

structure

Stock specification for the underlying asset. For information on the stock specification, see stockspec.

stockspec handles several types of underlying assets. For example, for physical commodities the price is StockSpec.Asset, the volatility is StockSpec.Sigma, and the convenience yield is StockSpec.DividendAmounts.

Data Types: struct

Settle — Settlement or trade date

datetime array | string array | date character vector

Settlement or trade date for the basket option, specified as an NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, cashsensbybls also accepts serial date numbers as inputs, but they are not recommended.

Maturity — Maturity date

datetime array | string array | date character vector

Maturity date for the basket option, specified as an NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, cashsensbybls also accepts serial date numbers as inputs, but they are not recommended.

OptSpec — Definition of option

character vector with values 'call' or 'put' | cell array of character vectors with values 'call' or 'put'

Definition of the option as 'call' or 'put', specified as an NINST-by-1 vector.

Data Types: char | cell

Strike — Strike price value

vector

Strike price value, specified as an NINST-by-1 vector.

Data Types: double

Payoff — Payoff values

vector

Payoff values (or the amount to be paid at expiration), 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: [Gamma,Theta,Price] = cashsensbybls(RateSpec,StockSpec,Settle,Maturity,OptSpec,Strike,Payoff,'OutSpec',{'gamma';'theta';'price'})

OutSpec — Define outputs

{'Price'} (default) | character vector with values 'Price', 'Delta', 'Gamma', 'Vega', 'Lambda', 'Rho', 'Theta', and 'All' | 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 a 1-by-NOUT cell array of character vectors 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

Output Arguments

PriceSens — Expected prices or sensitivities for cash-or-nothing option

vector

Expected prices or sensitivities (defined using OutSpec) for cash-or-nothing option, returned as a NINST-by-1 vector.

More About

Cash-or-Nothing Digital Option

A cash-or-nothing digital option is a type of exotic option that provides a fixed cash payout if the underlying asset meets certain conditions at expiration, specifically if the asset's price is above (for a call option) or below (for a put option) a predetermined strike price.

If the condition is not met, the option pays nothing. This binary payoff structure distinguishes cash-or-nothing digital options from traditional options, which have payoffs that depend on the difference between the underlying asset's price and the strike price.

Version History

Introduced in R2009a

R2022b: Serial date numbers not recommended

Although cashsensbybls 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.