barriersensbybls

Calculate price or sensitivities for European barrier options using Black-Scholes option pricing model

Description

PriceSens = barriersensbybls(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates,BarrierSpec,Barrier) calculates European barrier option prices or sensitivities using the Black-Scholes option pricing model.

PriceSens = barriersensbybls(___,Name,Value) adds optional name-value pair arguments.

Examples

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Compute price of European barrier down out and down in call options using the following data:

Rates = 0.035;
Settle = '01-Jan-2015';
Maturity = '01-April-2015';
Compounding = -1;
Basis = 1;

Define a RateSpec.

 RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle, 'EndDates', Maturity, ...
     'Rates', Rates, 'Compounding', Compounding, 'Basis', Basis)
RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: -1
             Disc: 0.9913
            Rates: 0.0350
         EndTimes: 0.2500
       StartTimes: 0
         EndDates: 736055
       StartDates: 735965
    ValuationDate: 735965
            Basis: 1
     EndMonthRule: 1

Define a StockSpec.

AssetPrice = 19;
Volatility = 0.40;
DivType = 'Continuous'; 
DivAmount = 0.035;
StockSpec = stockspec(Volatility, AssetPrice, DivType, DivAmount)
StockSpec = struct with fields:
             FinObj: 'StockSpec'
              Sigma: 0.4000
         AssetPrice: 19
       DividendType: {'continuous'}
    DividendAmounts: 0.0350
    ExDividendDates: []

Calculate the price, delta, and gamma for European barrier down out and down in call options using the Black-Scholes option pricing model.

OptSpec = 'Call';
Strike = 20;
Barrier = 18;
BarrierSpec = {'DO';'DI'};
OutSpec = {'price', 'delta', 'gamma'};

[Price, Delta, Gamma] = barriersensbybls(RateSpec, StockSpec, OptSpec, Strike, Settle,...
Maturity,  BarrierSpec, Barrier,'OutSpec', OutSpec)
Price = 2×1

    0.6287
    0.4655

Delta = 2×1

    0.6376
   -0.2036

Gamma = 2×1

    0.0255
    0.0773

Input Arguments

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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

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

Definition of the option as 'call' or 'put', specified as an NINST-by-1 cell array of character vectors or string arrays with values "call" or "put".

Data Types: char | string | cell

Option strike price value, specified as an NINST-by-1 matrix of numeric values.

Data Types: double

Settlement or trade date for the barrier option, specified as an NINST-by-1 matrix using serial date numbers, date character vectors, or datetime objects.

Data Types: double | char | datetime

Option exercise dates, specified as an NINST-by-1 matrix of serial date numbers, date character vectors, or datetime objects.

Note

For a European option, there is only one ExerciseDates on the option expiry date which is the maturity of the instrument.

Data Types: double | char | datetime

Barrier option type, specified as an NINST-by-1 cell array of character vectors with the following values:

  • 'UI' — Up Knock-in

    This option becomes effective when the price of the underlying asset passes above the barrier level. It gives the option holder the right, but not the obligation, to buy or sell (call/put) the underlying security at the strike price if the underlying asset goes above the barrier level during the life of the option.

  • 'UO' — Up Knock-out

    This option gives the option holder the right, but not the obligation, to buy or sell (call/put) the underlying security at the strike price as long as the underlying asset does not go above the barrier level during the life of the option. This option terminates when the price of the underlying asset passes above the barrier level. Usually with an up-and-out option, the rebate is paid if the spot price of the underlying reaches or exceeds the barrier level.

  • 'DI' — Down Knock-in

    This option becomes effective when the price of the underlying stock passes below the barrier level. It gives the option holder the right, but not the obligation, to buy or sell (call/put) the underlying security at the strike price if the underlying security goes below the barrier level during the life of the option. With a down-and-in option, the rebate is paid if the spot price of the underlying does not reach the barrier level during the life of the option.

  • 'DO' — Down Knock-up

    This option gives the option holder the right, but not the obligation, to buy or sell (call/put) the underlying asset at the strike price as long as the underlying asset does not go below the barrier level during the life of the option. This option terminates when the price of the underlying security passes below the barrier level. Usually, the option holder receives a rebate amount if the option expires worthless.

OptionBarrier TypePayoff if Barrier CrossedPayoff if Barrier not Crossed
Call/PutDown Knock-outWorthlessStandard Call/Put
Call/PutDown Knock-inCall/PutWorthless
Call/PutUp Knock-outWorthlessStandard Call/Put
Call/PutUp Knock-inStandard Call/PutWorthless

Data Types: char | cell

Barrier level, specified as an NINST-by-1 matrix of numeric values.

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: Price = barriersensbybls(RateSpec,StockSpec,OptSpec,Strike,Settle,Maturity,BarrierSpec,Barrier,'Rebate',1000,'OutSpec','Delta')

Rebate values, specified as the comma-separated pair consisting of 'Rebate' and NINST-by-1 matrix of numeric values. For Knock-in options, the Rebate is paid at expiry. For Knock-out options, the Rebate is paid when the Barrier is reached.

Data Types: double

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

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Expected prices at time 0 or sensitivities (defined using OutSpec) for barrier options, returned as a NINST-by-1 matrix.

More About

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Barrier Option

A Barrier option has not only a strike price but also a barrier level and sometimes a rebate.

A rebate is a fixed amount that is paid if the option cannot be exercised because the barrier level has been reached or not reached. The payoff for this type of option depends on whether the underlying asset crosses the predetermined trigger value (barrier level), indicated by Barrier, during the life of the option. For more information, see Barrier Option.

References

[1] Hull, J. Options, Futures and Other Derivatives Fourth Edition. Prentice Hall, 2000, pp. 646–649.

[2] Aitsahlia, F., L. Imhof, and T.L. Lai. “Pricing and hedging of American knock-in options.” The Journal of Derivatives. Vol. 11.3, 2004, pp. 44–50.

[3] Rubinstein M. and E. Reiner. “Breaking down the barriers.” Risk. Vol. 4(8), 1991, pp. 28–35.

Introduced in R2016b