https://github.com/cran/fOptions
Tip revision: 6f9d6d063ca52f2494b9efd22298d2155e83de70 authored by Diethelm Wuertz on 08 August 1977, 00:00:00 UTC
version 220.10063
version 220.10063
Tip revision: 6f9d6d0
B5-BinaryOptions.Rd
\name{BinaryOptions}
\alias{BinaryOptions}
\alias{GapOption}
\alias{CashOrNothingOption}
\alias{TwoAssetCashOrNothingOption}
\alias{AssetOrNothingOption}
\alias{SuperShareOption}
\alias{BinaryBarrierOption}
\title{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.
\cr
The functions are:
\tabular{ll}{
\code{GapOption} \tab Gap Option, \cr
\code{CashOrNothingOption} \tab Cash Or Nothing Option, \cr
\code{TwoAssetCashOrNothingOption} \tab Two Asset Cash Or Nothing Option, \cr
\code{AssetOrNothingOption} \tab Asset Or Nothing Option, \cr
\code{SuperShareOption} \tab Super Share Option, \cr
\code{BinaryBarrierOption} \tab 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{
\item{b}{
the annualized cost-of-carry rate, a numeric value;
e.g. 0.1 means 10\% pa.
}
\item{b1, b2}{
[TwoAssetCashOrNothing*] - \cr
the annualized cost-of-carry rate for the first and second
asset, a numeric value.
}
\item{description}{
a character string which allows for a brief description.
}
\item{eta, phi}{
[BinaryBarrier*] - \cr
a set of parameters to price 28 different types of Binary
Barrier options:\cr
01: \code{eta=+1, phi=NA, [S>H]} down-and-in cash-at-hit-or-nothing, \cr
02: \code{eta=-1, phi=NA, [S<H]} up-and-in cash-at-hit-or-nothing, \cr
03: \code{eta=+1, phi=NA, [S>H]} down-and-in asset-at-hit-or-nothing, \cr
04: \code{eta=-1, phi=NA, [S<H]} up-and-in asset-at-hit-or-nothing, \cr
05: \code{eta=+1, phi=-1, [S>H]} down-and-in cash-at-expiry-or-nothing, \cr
06: \code{eta=-1, phi=+1, [S<H]} up-and-in cash-at-expiry-or-nothing, \cr
07: \code{eta=+1, phi=-1, [S>H]} down-and-in asset-at-expiry-or-nothing, \cr
08: \code{eta=-1, phi=+1, [S<H]} up-and-in asset-at-expiry-or-nothing, \cr
09: \code{eta=+1, phi=+1, [S>H]} down-and-out cash-or-nothing, \cr
10: \code{eta=-1, phi=-1, [S<H]} up-and-out cash-or-nothing, \cr
11: \code{eta=+1, phi=+1, [S>H]} down-and-out asset-or-nothing, \cr
12: \code{eta=-1, phi=-1, [S<H]} up-and-out asset-or-nothing, \cr
13: \code{eta=+1, phi=+1, [S>H]} down-and-in cash-or-nothing call, \cr
14: \code{eta=-1, phi=+1, [S<H]} up-and-in cash-or-nothing call, \cr
15: \code{eta=+1, phi=+1, [S>H]} down-and-in asset-or-nothing call, \cr
16: \code{eta=-1, phi=+1, [S<H]} up-and-in asset-or-nothing call, \cr
17: \code{eta=+1, phi=-1, [S>H]} down-and-in cash-or-nothing put, \cr
18: \code{eta=-1, phi=-1, [S<H]} up-and-in cash-or-nothing put, \cr
19: \code{eta=+1, phi=-1, [S>H]} down-and-in asset-or-nothing put, \cr
20: \code{eta=-1, phi=-1, [S<H]} up-and-in asset-or-nothing put, \cr
21: \code{eta=+1, phi=+1, [S>H]} down-and-out cash-or-nothing call, \cr
22: \code{eta=-1, phi=+1, [S<H]} up-and-out cash-or-nothing call, \cr
23: \code{eta=+1, phi=+1, [S>H]} down-and-out asset-or-nothing call, \cr
24: \code{eta=-1, phi=-1, [S<H]} up-and-out asset-or-nothing call, \cr
25: \code{eta=+1, phi=-1, [S>H]} down-and-out cash-or-nothing put, \cr
26: \code{eta=-1, phi=-1, [S<H]} up-and-out cash-or-nothing put, \cr
27: \code{eta=+1, phi=-1, [S>H]} down-and-out asset-or-nothing put, \cr
28: \code{eta=-1, phi=-1, [S<H]} up-and-out asset-or-nothing put.
}
\item{H}{
[BinaryBarrier*] - \cr
the barrier value, a numeric value.
}
\item{K}{
[CashOrNothing*] - \cr
the cash amount at expiry if the option is in the money,
a numerical value.\cr
[TwoAssetCashOrNothing*] - \cr
for the cash-or-nothing call the cash amount at expiry if
asset \code{S1} is above the strike \code{X1} and asset
\code{S2} is above strike \code{X2} at expiration,
\cr
for the cash-or-nothing put the cash amount at expiry if
asset \code{S1} is below the strike \code{X1} and asset
\code{S2} is below strike \code{X2} at expiration,
\cr
for the cash-or-nothing up-down the cash amount at expiry if
asset \code{S1} is above the strike \code{X1} and asset
\code{S2} is below strike \code{X2} at expiration,
\cr
for the cash-or-nothing down-up the cash amount at expiry if
asset \code{S1} is below the strike \code{X1} and asset
\code{S2} is above strike \code{X2} at expiration.\cr
[BinaryBarrier*] - \cr
the prespecified cash amount, a numeric value.
}
\item{r}{
the annualized rate of interest, a numeric value;
e.g. 0.25 means 25\% pa.
}
\item{rho}{
[TwoAssetCashOrNothing*] - \cr
the correlation of the volatility between the first and
second asset, a numeric value.
}
\item{S}{
the asset price, a numeric value.
}
\item{S1, S2}{
[TwoAssetCashOrNothing*] - \cr
the price of the first and second asset, a numeric value.
}
\item{sigma}{
the annualized volatility of the underlying security,
a numeric value; e.g. 0.3 means 30\% volatility pa.
}
\item{sigma1, sigma2}{
[TwoAssetCashOrNothing*] - \cr
the annualized volatility of the first and second underlying
security, numeric values.
}
\item{Time}{
the time to maturity measured in years, a numeric value;
e.g. 0.5 means 6 months.
}
\item{title}{
a character string which allows for a project title.
}
\item{TypeFlag}{
a character string either \code{"c"} for a call option or
a \code{"p"} for a put option. \cr
[TwoAssetCashOrNothing*] - \cr
a character string either \code{"c"} for a call option, or
a \code{"p"} for a put option, or a \code{"ud"} for an
up-down option, or a \code{"du"} for a down-up option.\cr
[BinaryBarrier*] - \cr
an integer between 1 and 28, selecting one of the 28 types,
for a definition lookup the arguments \code{eta} and
\code{phi}.
}
\item{X}{
the exercise price, a numeric value.
}
\item{X1, X2}{
[GapOption][TwoAssetCashOrNothing*] -
the first and the second exercise price, a numeric value.
}
\item{XL, XH}{
[SuperShare*] -
the lower and upper boundary strike, a numeric value.
}
}
\details{
\bold{Gap Options:}
\cr\cr
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).
\cr
[Haug's Book, Chapter 2.11.1]
\cr
\bold{Cash-or-Nothing Options:}
\cr\cr
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).
\cr
[Haug's Book, Chapter 2.11.2]
\cr
\bold{Two-Asset-Cash-Or-Nothing Options:}
\cr\cr
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).
\cr
[Haug's Book, Chapter 2.11.3]
\cr
\bold{Asset-Or-Nothing Options:}
\cr\cr
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).
\cr
[Haug's Book, Chapter 2.11.4]
\cr
\bold{Supershare Options:}
\cr\cr
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).
\cr
[Haug's Book, Chapter 2.11.5]
\cr
\bold{Binary Barrier Options:}
\cr\cr
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).
\cr
[Haug's Book, Chapter 2.11.6]
}
\note{
The functions implement the algorithms to valuate plain vanilla
options as described in Chapter 2.11 of Haug's Book (1997).
}
\value{
The option price, a numeric value.
}
\references{
Cox J.C., Rubinstein M. (1985);
\emph{Innovations in Option Markets},
Prentice-Hall, New Jersey.
Hakkansson N.H. (1976);
\emph{The Purchasing Power Fund: A New Kind of Financial
Intermediary},
Financial Analysts Journal 32, 49--59.
Haug E.G. (1997);
\emph{The complete Guide to Option Pricing Formulas},
Chapter 2.11, McGraw-Hill, New York.
Heinen R.C., Kat H.M. (1996);
\emph{Brick by Brick},
Risk Magazine 9, 6.
Reiner E., Rubinstein M. (1991);
\emph{Unscrambling the Binary Code};
Risk Magazine 4, 9.
}
\author{
Diethelm Wuertz for the Rmetrics \R-port.
}
\examples{
## SOURCE("fOptions.B5-BinaryOptions")
## Examples from Chapter 2.11 in E.G. Haug's Option Guide (1997)
## Gap Option [2.11.1]:
xmpOptions("\nStart: Gap Option > ")
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]:
xmpOptions("\nNext: Cash or Nothing Option > ")
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]:
xmpOptions("\nNext: Two Asset Cash Or Nothing Option > ")
# 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]:
xmpOptions("\nNext: Asset Or Nothing Option > ")
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]:
xmpOptions("\nNext: Super Share Option > ")
SuperShareOption(S = 100, XL = 90, XH = 110, Time = 0.25, r = 0.10,
b = 0, sigma = 0.20)
## Binary Barrier Option [2.11.6]:
xmpOptions("\nNext: Binary Barrier Option > ")
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)
}
\keyword{math}
