BinaryOptions: Valuation of Binary Options

In fExoticOptions: Rmetrics - Pricing and Evaluating Exotic Option

Valuation of Binary Options

Description

A collection and description of functions to valuate binary options. Binary options, also known as digital options, have discontinuous payoffs. They can be used as building blocks to develop options with more complicated payoffs. For example, a regular European call option is equivalent to a long position in an asset-or-nothing call and a short position in a cash-or-nothing call, where the both options have the same strike price and the cash payoff of the cash-or-nothing option equals the strike price. Unlike standard European style options, the payout for binary options does not depend on how much it is in-the-money but rather whether or not it is on the money. The option's payoff is fixed at the options inception and is based on the price of the underlying asset on the expiration date. Binary options may also incorporate barriers, as is the case with binary-barrier options.

The functions are:

GapOption	Gap Option,
CashOrNothingOption	Cash Or Nothing Option,
TwoAssetCashOrNothingOption	Two Asset Cash Or Nothing Option,
AssetOrNothingOption	Asset Or Nothing Option,
SuperShareOption	Super Share Option,
BinaryBarrierOption	Binary Barrier Option.

Usage

GapOption(TypeFlag, S, X1, X2, Time, r, b, sigma, title = NULL, 
    description = NULL)
CashOrNothingOption(TypeFlag, S, X, K, Time, r, b, sigma,
    title = NULL, description = NULL) 
TwoAssetCashOrNothingOption(TypeFlag, S1, S2, X1, X2, K, Time, r, 
    b1, b2, sigma1, sigma2, rho, title = NULL, description = NULL)
AssetOrNothingOption(TypeFlag, S, X, Time, r, b, sigma,
    title = NULL, description = NULL)
SuperShareOption(S, XL, XH, Time, r, b, sigma, title = NULL, 
    description = NULL)
BinaryBarrierOption(TypeFlag, S, X, H, K, Time, r, b, sigma, 
    eta, phi, title = NULL, description = NULL)

Arguments

b	the annualized cost-of-carry rate, a numeric value; e.g. 0.1 means 10% pa.
b1, b2	[TwoAssetCashOrNothing*] - the annualized cost-of-carry rate for the first and second asset, a numeric value.
description	a character string which allows for a brief description.
eta, phi	[BinaryBarrier*] - a set of parameters to price 28 different types of Binary Barrier options: 01: eta=+1, phi=NA, [S>H] down-and-in cash-at-hit-or-nothing, 02: eta=-1, phi=NA, [S<H] up-and-in cash-at-hit-or-nothing, 03: eta=+1, phi=NA, [S>H] down-and-in asset-at-hit-or-nothing, 04: eta=-1, phi=NA, [S<H] up-and-in asset-at-hit-or-nothing, 05: eta=+1, phi=-1, [S>H] down-and-in cash-at-expiry-or-nothing, 06: eta=-1, phi=+1, [S<H] up-and-in cash-at-expiry-or-nothing, 07: eta=+1, phi=-1, [S>H] down-and-in asset-at-expiry-or-nothing, 08: eta=-1, phi=+1, [S<H] up-and-in asset-at-expiry-or-nothing, 09: eta=+1, phi=+1, [S>H] down-and-out cash-or-nothing, 10: eta=-1, phi=-1, [S<H] up-and-out cash-or-nothing, 11: eta=+1, phi=+1, [S>H] down-and-out asset-or-nothing, 12: eta=-1, phi=-1, [S<H] up-and-out asset-or-nothing, 13: eta=+1, phi=+1, [S>H] down-and-in cash-or-nothing call, 14: eta=-1, phi=+1, [S<H] up-and-in cash-or-nothing call, 15: eta=+1, phi=+1, [S>H] down-and-in asset-or-nothing call, 16: eta=-1, phi=+1, [S<H] up-and-in asset-or-nothing call, 17: eta=+1, phi=-1, [S>H] down-and-in cash-or-nothing put, 18: eta=-1, phi=-1, [S<H] up-and-in cash-or-nothing put, 19: eta=+1, phi=-1, [S>H] down-and-in asset-or-nothing put, 20: eta=-1, phi=-1, [S<H] up-and-in asset-or-nothing put, 21: eta=+1, phi=+1, [S>H] down-and-out cash-or-nothing call, 22: eta=-1, phi=+1, [S<H] up-and-out cash-or-nothing call, 23: eta=+1, phi=+1, [S>H] down-and-out asset-or-nothing call, 24: eta=-1, phi=-1, [S<H] up-and-out asset-or-nothing call, 25: eta=+1, phi=-1, [S>H] down-and-out cash-or-nothing put, 26: eta=-1, phi=-1, [S<H] up-and-out cash-or-nothing put, 27: eta=+1, phi=-1, [S>H] down-and-out asset-or-nothing put, 28: eta=-1, phi=-1, [S<H] up-and-out asset-or-nothing put.
H	[BinaryBarrier*] - the barrier value, a numeric value.
K	[CashOrNothing*] - the cash amount at expiry if the option is in the money, a numerical value. [TwoAssetCashOrNothing*] - for the cash-or-nothing call the cash amount at expiry if asset S1 is above the strike X1 and asset S2 is above strike X2 at expiration, for the cash-or-nothing put the cash amount at expiry if asset S1 is below the strike X1 and asset S2 is below strike X2 at expiration, for the cash-or-nothing up-down the cash amount at expiry if asset S1 is above the strike X1 and asset S2 is below strike X2 at expiration, for the cash-or-nothing down-up the cash amount at expiry if asset S1 is below the strike X1 and asset S2 is above strike X2 at expiration. [BinaryBarrier*] - the prespecified cash amount, a numeric value.
r	the annualized rate of interest, a numeric value; e.g. 0.25 means 25% pa.
rho	[TwoAssetCashOrNothing*] - the correlation of the volatility between the first and second asset, a numeric value.
S	the asset price, a numeric value.
S1, S2	[TwoAssetCashOrNothing*] - the price of the first and second asset, a numeric value.
sigma	the annualized volatility of the underlying security, a numeric value; e.g. 0.3 means 30% volatility pa.
sigma1, sigma2	[TwoAssetCashOrNothing*] - the annualized volatility of the first and second underlying security, numeric values.
Time	the time to maturity measured in years, a numeric value; e.g. 0.5 means 6 months.
title	a character string which allows for a project title.
TypeFlag	a character string either "c" for a call option or a "p" for a put option. [TwoAssetCashOrNothing*] - a character string either "c" for a call option, or a "p" for a put option, or a "ud" for an up-down option, or a "du" for a down-up option. [BinaryBarrier*] - an integer between 1 and 28, selecting one of the 28 types, for a definition lookup the arguments eta and phi.
X	the exercise price, a numeric value.
X1, X2	[GapOption][TwoAssetCashOrNothing*] - the first and the second exercise price, a numeric value.
XL, XH	[SuperShare*] - the lower and upper boundary strike, a numeric value.

Details

Gap Options:

The payoff on a gap option depends on the usual factors of a plain option, but is also affected by a "gap" amount of exercise prices, which may be positive or negative. Note, that a gap call (put) option is equivalent to being long (short) an asset-or-nothing call (put) and short (long) a cash-or-nothing call (put). The option price is calculated analytically according to Reiner and Rubinstein (1991).
[Haug's Book, Chapter 2.11.1]

Cash-or-Nothing Options:

For this option a predetermined amount is paid at expiration if the asset is above for a call or below for a put some strike level. The amount independent of the path taken. These options require no payment of an exercise price. The exercise price determines whether or not the option returns a payoff. The value of a cash-or-nothing call (put) option is the present value of the fixed cash payoff multiplied by the probability that the terminal price will be greater than (less than) the exercise price. The option price is calculated analytically according to Reiner and Rubinstein (1991).
[Haug's Book, Chapter 2.11.2]

Two-Asset-Cash-Or-Nothing Options:

These options are building blocks for constructing more complex exotic options. There are four types of two-asset cash-or-nothing options, the first two situationsa are: A two-asset-cash-or-nothing call pays out a fixed cash amount if the price of the first asset is above (below) the strike price of the first asset and the price of the second asset is also above (below) the strike price of the second asset at expiration. The other two situations arise under the following conditions: A two-asset cash-or-nothing down-up pays out a fixed cash amount if the price of the first asset is is below (above) the strike price of the first asset and the price of the second asset is above (below) the strike price of the second asset at expiration. The option price is calculated analytically according to Heynen and Kat (1996).
[Haug's Book, Chapter 2.11.3]

Asset-Or-Nothing Options:

In this option a predetermined asset value is paid if the asset is, at expiration, above for a call or below for a put some strike level, independent of the path taken. For a call (put) the terminal price is greater than (less than) the exercise price, the call (put) expires worthless. The exercise price is never paid. Instead, the value of the asset relative to the exercise price determines whether or not the option returns a payoff. The value of an asset-or-nothing call (put) option is the present value of the asset multiplied by the probability that the terminal price will be greater than (less than) the exercise price. The option price is calculated analytically according to Cox and Rubinstein (1985).
[Haug's Book, Chapter 2.11.4]

Supershare Options:

These options represents a contingent claim on a fraction of the underlying portfolio. The contingency is that the value of the portfolio must lie between a lower and an upper bound at expiration. If the value lies within these boundaries, the supershare is worth a proportion of the assets underlying the portfolio, else the supershare expires worthless. A supershare has a payoff that is basically like a spread of two asset-or-nothing calls, in which the owner of a supershare purchases an asset-or-nothing call with an strike price of the lower strike and sells an asset-or-nothing call with an strike price of the upper strike. The option price is calculated analytically according to Hakansson (1976).
[Haug's Book, Chapter 2.11.5]

Binary Barrier Options:

These options combine characteristics of both binary and barrier options. They are path dependent with a discontinuous payoff. Similar to barrier options, the payoff depends on whether or not the asset price crosses a predetermined barrier. There are 28 different types of binary barrier options, which can be divided into two main categories: Cash-or-nothing and Asset-or-nothing barrier options. Cash-or-nothing barrier options pay out a predetermined cash amount or nothing, depending on whether the asset price has hit the barrier. Asset-or-nothing barrier options pay out the value of the asset or nothing, depending on whether the asset price has crossed the barrier. The barrier monitoring frequency can be adjusted to account for discrete monitoring using an approximation developed by Broadie, Glasserman, and Kou (1995). Binary-barrier options can be priced analytically using a model introduced by Reiner and Rubinstein (1991).
[Haug's Book, Chapter 2.11.6]

Value

The option price, a numeric value.

Note

The functions implement the algorithms to valuate plain vanilla options as described in Chapter 2.11 of Haug's Book (1997).

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

References

Cox J.C., Rubinstein M. (1985); Innovations in Option Markets, Prentice-Hall, New Jersey.

Hakkansson N.H. (1976); The Purchasing Power Fund: A New Kind of Financial Intermediary, Financial Analysts Journal 32, 49–59.

Haug E.G. (1997); The complete Guide to Option Pricing Formulas, Chapter 2.11, McGraw-Hill, New York.

Heinen R.C., Kat H.M. (1996); Brick by Brick, Risk Magazine 9, 6.

Reiner E., Rubinstein M. (1991); Unscrambling the Binary Code; Risk Magazine 4, 9.

Examples

## Examples from Chapter 2.11 in E.G. Haug's Option Guide (1997)

## Gap Option [2.11.1]:
   GapOption(TypeFlag = "c", S = 50, X1 = 50, X2 = 57, Time = 0.5, 
     r = 0.09, b = 0.09, sigma = 0.20)

## Cash Or Nothing Option [2.11.2]:
   CashOrNothingOption(TypeFlag = "p", S = 100, X = 80, K = 10, 
     Time = 9/12, r = 0.06, b = 0, sigma = 0.35) 

## Two Asset Cash Or Nothing Option [2.11.3]:
   # Type 1 - call:
   TwoAssetCashOrNothingOption(TypeFlag = "c", S1 = 100, S2 = 100, 
     X1 = 110, X2 = 90, K = 10, Time = 0.5, r = 0.10, b1 = 0.05, 
     b2 = 0.06, sigma1 = 0.20, sigma2 = 0.25, rho = 0.5)
   # Type 2 - put:
   TwoAssetCashOrNothingOption(TypeFlag = "p", S1 = 100, S2 = 100, 
     X1 = 110, X2 = 90, K = 10, Time = 0.5, r = 0.10, b1 = 0.05, 
     b2 = 0.06, sigma1 = 0.20, sigma2 = 0.25, rho = -0.5)
   # Type 3 - down-up:
   TwoAssetCashOrNothingOption(TypeFlag = "ud", S1 = 100, S2 = 100, 
     X1 = 110, X2 = 90, K = 10, Time = 1, r = 0.10, b1 = 0.05, 
     b2 = 0.06, sigma1 = 0.20, sigma2 = 0.25, rho = 0)
   # Type 4 - up-down:
   TwoAssetCashOrNothingOption(TypeFlag = "du", S1 = 100, S2 = 100, 
     X1 = 110, X2 = 90, K = 10, Time = 1, r = 0.10, b1 = 0.05, 
     b2 = 0.06, sigma1 = 0.20, sigma2 = 0.25, rho = 0)

## Asset Or Nothing Option [2.11.4]: 
   AssetOrNothingOption(TypeFlag = "p", S = 70, X = 65, Time = 0.5, 
     r = 0.07, b = 0.07 - 0.05, sigma = 0.27)

## Super Share Option [2.11.5]:  
   SuperShareOption(S = 100, XL = 90, XH = 110, Time = 0.25, r = 0.10, 
     b = 0, sigma = 0.20)

## Binary Barrier Option [2.11.6]: 
   BinaryBarrierOption(TypeFlag = "6", S = 95, X=102, H = 100, 
     K = 15, Time = 0.5, r = 0.1, b = 0.1, sigma = 0.20)
   BinaryBarrierOption(TypeFlag = "12", S = 95, X = 98, H = 100, 
     K = 15, Time = 0.5, r = 0.1, b = 0.1, sigma = 0.20)
     

fExoticOptions documentation built on Sept. 9, 2022, 3:01 p.m.