dblbarrierbyfd

Calculate double barrier option price using finite difference method

Syntax

[Price,PriceGrid,AssetPrices,Times] = dblbarrierbyfd(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates,BarrierSpec,Barrier)

[Price,PriceGrid,AssetPrices,Times] = dblbarrierbyfd(___,Name,Value)

Description

[Price,PriceGrid,AssetPrices,Times] = dblbarrierbyfd(RateSpec,StockSpec,OptSpec,Strike,Settle,ExerciseDates,BarrierSpec,Barrier) calculates a European or American call or put double barrier option price on a single underlying asset using the finite difference method. dblbarrierbyfd assumes that the barrier is continuously monitored.

[Price,PriceGrid,AssetPrices,Times] = dblbarrierbyfd(___,Name,Value) specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax.

Examples

Price an American Double Knock-Out Call Option with Rebate

Compute the price of an American double barrier option for a double knock-out (down and out-up and out) call option with a rebate using the following data:

Rate = 0.05;
Settle = '01-Jun-2018';
Maturity = '01-Dec-2018';
Basis = 1;

Define a RateSpec.

RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle, 'EndDates', Maturity,'Rates', Rate, 'Compounding', -1, 'Basis', Basis);

Define a StockSpec.

AssetPrice = 100; 
Volatility = 0.25;
StockSpec = stockspec(Volatility, AssetPrice);

Define the double barrier option.

LBarrier = 80; 
UBarrier = 130; 
Barrier = [UBarrier LBarrier];
BarrierSpec = 'DKO';
OptSpec = 'Call';
Strike = 110;
Rebate = 1;

Compute the price of an American option using finite differences.

Price = dblbarrierbyfd(RateSpec, StockSpec, OptSpec, Strike, Settle, Maturity, BarrierSpec, Barrier,'Rebate', Rebate, 'AmericanOpt', 1)

Price = 4.0002

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, specified by the StockSpec obtained from 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

OptSpec — Definition of option

character vector with values 'call' or 'put' | string scalar with values "call" or "put"

Definition of an option, specified as a character vector with a value of 'call' or 'put', or a string scalar with values "call" or "put".

Data Types: char | string

Strike — Option strike price value

scalar numeric

Option strike price value, specified as a scalar numeric.

Data Types: double

Settle — Settlement or trade date

date character vector | serial date number | datetime object

Settlement or trade date for the barrier option, specified as a serial date number, a date character vector, or a datetime object.

Data Types: double | char | datetime

ExerciseDates — Option exercise dates

date character vector | serial date number | datetime object

Option exercise dates, specified as a date character vector, a serial date number, or datetime object.

• For a European option, the option expiry date has only one ExerciseDates value.

• For an American option, use a 1-by-2 vector of exercise date boundaries. The option can be exercised on any date between or including the pair of dates. If only one non-NaN date is listed, the option can be exercised between Settle and the single listed date in ExerciseDates.

Data Types: double | char | cell | datetime

BarrierSpec — Double barrier option type

character vector with value of 'DKI' or 'DKO' | scalar string with value of "DKI" or "DKO"

Double barrier option type, specified as a character vector or string with one of the following values:

• 'DKI' — Double Knock-in

  The 'DKI' option becomes effective when the price of the underlying asset reaches one of the barriers. It gives the option holder the right but not the obligation to buy or sell the underlying security at the strike price, if the underlying asset goes above or below the barrier levels during the life of the option.

• 'DKO' — Double Knock-out

  The 'DKO' option gives the option holder the right but not the obligation to buy or sell the underlying security at the strike price, as long as the underlying asset remains between the barrier levels during the life of the option. This option terminates when the price of the underlying asset passes one of the barriers.

Option	Barrier Type	Payoff If Any Barrier Crossed	Payoff If Barriers Not Crossed
Call/Put	Double Knock-in	Standard Call/Put	Worthless
Call/Put	Double Knock-out	Worthless	Standard Call/Put

Data Types: char | string

Barrier — Barrier level

vector

Barrier level, specified as a 1-by-2 vector of numeric values, where the first column is the upper barrier (1)(UB) and the second column is the lower barrier (2)(LB). Barrier(1) must be greater than Barrier(2).

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 = dblbarrierbyfd(RateSpec,StockSpec,OptSpec,Strike,Settle,Maturity,BarrierSpec,Barrier,'Rebate',[100,100])

'Rebate' — Rebate value

[0 0] for Double Knock-out or 0 for Double Knock-in (default) | vector | scalar numeric

Rebate value, specified as the comma-separated pair consisting of 'Rebate' and one of the following:

• For a Double Knock-out option, use a 1-by-2 vector of rebate values where the first column is the payout if the upper barrier(1)(UB) is hit and the second column is payout if the lower barrier(2)(LB) is hit. The rebate is paid when the barrier is reached.

• For a Double Knock-in option, use a scalar rebate value. The rebate is paid at expiry.

Data Types: double

'AssetGridSize' — Size of asset grid used for a finite difference grid

400 (default) | positive scalar numeric

Size of the asset grid used for the finite difference grid, specified as the comma-separated pair consisting of 'AssetGridSize' and a positive scalar numeric.

Data Types: double

'TimeGridSize' — Size of time grid used for finite difference grid

100 (default) | positive scalar numeric

Size of the time grid used for the finite difference grid, specified as the comma-separated pair consisting of 'TimeGridSize' and a positive scalar numeric.

Note

The actual time grid may have a larger size because exercise and ex-dividend dates might be added to the grid from StockSpec.

Data Types: double

'AmericanOpt' — Option type

0 European (default) | integer with values 0 or 1

Option type, specified as the comma-separated pair consisting of 'AmericanOpt' and a scalar flag with one of the following values:

• 0 — European

• 1 — American

Data Types: logical

Output Arguments

Price — Expected prices for double barrier options

matrix

Expected prices for double barrier options, returned as a 1-by-1 matrix.

PriceGrid — Grid containing prices

grid

Grid containing prices calculated by the finite difference method, returned as a two-dimensional grid with the size AssetGridSize*TimeGridSize. The number of columns does not have to be equal to the TimeGridSize, because exercise and ex-dividend dates in StockSpec are added to the time grid. PriceGrid(:, end) contains the price for t = 0.

AssetPrices — Prices of asset defined by StockSpec

vector

Prices of the asset defined by the StockSpec corresponding to the first dimension of PriceGrid, returned as a vector.

Times — Times corresponding to second dimension of PriceGrid

vector

Times corresponding to the second dimension of PriceGrid, returned as a vector.

More About

Double Barrier Option

A double barrier option is similar to the standard single barrier option except that it has two barrier levels: a lower barrier (LB) and an upper barrier (UB).

The payoff for a double barrier option depends on whether the underlying asset remains between the barrier levels during the life of the option. Double barrier options are less expensive than single barrier options as they have a higher knock-out probability. Because of this, double barrier options allow investors to reduce option premiums and match an investor’s belief about the future movement of the underlying price process.

References

[1] Boyle, P., and Y. Tian. “An Explicit Finite Difference Approach to the Pricing of Barrier Options.” Applied Mathematical Finance. Vol. 5, Number 1, 1998, pp. 17–43.

[2] Hull, J. Options, Futures, and Other Derivatives. Fourth Edition. Upper Saddle River, NJ: Prentice Hall, 2000, pp. 646–649.

[3] Rubinstein, M., and E. Reiner. “Breaking Down the Barriers.” Risk. Vol. 4, Number 8, 1991, pp. 28–35.

[4] Zvan, R., P. A. Forsyth and K. R. Vetzal. “PDE Methods for Pricing Barrier Options.” Journal of Economic Dynamics and Control. Vol. 24, Number 11-12, 2000, pp. 1563–1590.