barrierbystt

Price barrier options using standard trinomial tree

Syntax

[Price,PriceTree] = barrierbystt(STTTree,OptSpec,Strike,Settle,ExerciseDates,AmericanOpt,BarrierSpec,Barrier)

[Price,PriceTree] = barrierbystt(___,Name,Value)

Description

[Price,PriceTree] = barrierbystt(STTTree,OptSpec,Strike,Settle,ExerciseDates,AmericanOpt,BarrierSpec,Barrier) prices barrier options using a standard trinomial (STT) tree.

[Price,PriceTree] = barrierbystt(___,Name,Value) prices barrier options using a standard trinomial (STT) tree with an optional name-value pair argument for Rebate and Options.

Examples

Price a Barrier Option Using the Standard Trinomial Tree Model

Create a RateSpec.

StartDates = 'Jan-1-2009'; 
EndDates = 'Jan-1-2013'; 
Rates = 0.035; 
Basis = 1; 
Compounding = -1;
RateSpec = intenvset('ValuationDate', StartDates, 'StartDates', StartDates,...
'EndDates', EndDates, 'Rates', Rates,'Compounding', Compounding, 'Basis', Basis)

RateSpec = struct with fields:
           FinObj: 'RateSpec'
      Compounding: -1
             Disc: 0.8694
            Rates: 0.0350
         EndTimes: 4
       StartTimes: 0
         EndDates: 735235
       StartDates: 733774
    ValuationDate: 733774
            Basis: 1
     EndMonthRule: 1

Create a StockSpec.

AssetPrice = 85; 
Sigma = 0.15; 
StockSpec = stockspec(Sigma, AssetPrice)

StockSpec = struct with fields:
             FinObj: 'StockSpec'
              Sigma: 0.1500
         AssetPrice: 85
       DividendType: []
    DividendAmounts: 0
    ExDividendDates: []

Create an STTTree.

NumPeriods = 4;
TimeSpec = stttimespec(StartDates, EndDates, 4);
STTTree = stttree(StockSpec, RateSpec, TimeSpec)

STTTree = struct with fields:
       FinObj: 'STStockTree'
    StockSpec: [1x1 struct]
     TimeSpec: [1x1 struct]
     RateSpec: [1x1 struct]
         tObs: [0 1 2 3 4]
         dObs: [733774 734139 734504 734869 735235]
        STree: {1x5 cell}
        Probs: {[3x1 double]  [3x3 double]  [3x5 double]  [3x7 double]}

Define the barrier option and compute the price.

Settle = '1/1/09';
ExerciseDates = '1/1/12';
OptSpec =  'call';
Strike = 105;
AmericanOpt = 1;
BarrierSpec = 'UI';
Barrier = 115;

Price= barrierbystt(STTTree, OptSpec, Strike, Settle, ExerciseDates,...
AmericanOpt, BarrierSpec, Barrier)

Price = 3.7977

Input Arguments

STTTree — Stock tree structure for standard trinomial tree

structure

Stock tree structure for a standard trinomial tree, specified by using stttree.

Data Types: struct

OptSpec — Definition of option

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

Definition of option, specified as 'call' or 'put' using a character vector or a NINST-by-1 cell array of character vectors for 'call' or 'put'.

Data Types: char | cell

Strike — European or American option strike price value

nonnegative numeric

European or American option strike price value, specified with a nonnegative integer using a NINST-by-1 matrix of nonnegative numeric values. Each row is the schedule for one option. To compute the value of a floating-strike barrier option, Strike should be specified as NaN. Floating-strike barrier options are also known as average strike options.

Data Types: double

Settle — Settlement date or trade date

serial date number | date character vector

Settlement date or trade date for the barrier option, specified as a NINST-by-1 matrix of settlement or trade dates using serial date numbers or date character vectors.

Note

The Settle date for every barrier option is set to the ValuationDate of the stock tree. The barrier argument, Settle, is ignored.

Data Types: double | char | cell

ExerciseDates — Option exercise dates

serial date number | date character vector

Option exercise dates, specified as a serial date number or date character vector:

• For a European option, use aNINST-by-1 matrix of exercise dates. Each row is the schedule for one option. For a European option, there is only one ExerciseDates on the option expiry date.

• For an American option, use a NINST-by-2 vector of exercise date boundaries. The option can be exercised on any tree date between or including the pair of dates on that row. If only one non-NaN date is listed, or if ExerciseDates is a NINST-by-1 vector of serial date numbers or cell array of character vectors, the option can be exercised between ValuationDate of the stock tree and the single listed ExerciseDates.

Data Types: double | char | cell

AmericanOpt — Option type

scalar with values [0,1]

Option type, specified as an NINST-by-1 matrix of flags with values:

• 0 — European

• 1 — American

Data Types: double

BarrierSpec — Barrier option type

character vector with values: 'UI', 'UO', 'DI', 'DO' | cell array of character vectors with values: 'UI', 'UO', 'DI', 'DO'

Barrier option type, specified as a character vector or 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. Note, barrierbyfd does not support American knock-in barrier options.

• '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. Note, barrierbyfd does not support American knock-in barrier options.

• '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.

Option	Barrier Type	Payoff if Barrier Crossed	Payoff if Barrier not Crossed
Call/Put	Down Knock-out	Worthless	Standard Call/Put
Call/Put	Down Knock-in	Call/Put	Worthless
Call/Put	Up Knock-out	Worthless	Standard Call/Put
Call/Put	Up Knock-in	Standard Call/Put	Worthless

Data Types: char | cell

Barrier — Barrier levels

numeric

Barrier levels, 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 = barrierbystt(STTTree,OptSpec,Strike,Settle,ExerciseDates,1,'UI',115,'Rebate',25)

'Rebate' — Rebate values

0 (default) | numeric

Rebate values, specified as the comma-separated pair consisting of 'Rebate' and a 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 theBarrier is reached.

Data Types: double

'Options' — Derivatives pricing options

structure

Derivatives pricing options, specified as the comma-separated pair consisting of 'Options' and a structure that is created with derivset.

Data Types: struct

Output Arguments

Price — Expected prices for barrier options at time 0

matrix

Expected prices for barrier options at time 0, returned as a NINST-by-1 matrix.

PriceTree — Structure with vector of barrier option prices at each node

tree structure

Structure with a vector of barrier option prices at each node, returned as a tree structure.

PriceTree is a MATLAB^® structure of trees containing vectors of instrument prices and a vector of observation times for each node.

PriceTree.PTree contains the prices.

PriceTree.tObs contains the observation times.

PriceTree.dObs contains the observation dates.

More About

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] Derman, E., I. Kani, D. Ergener and I. Bardhan. “Enhanced Numerical Methods for Options with Barriers.” Financial Analysts Journal. (Nov.-Dec.), 1995, pp. 65–74.