1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
|
\name{BinomialTreeOptions}
\alias{BinomialTreeOptions}
\alias{CRRBinomialTreeOption}
\alias{JRBinomialTreeOption}
\alias{TIANBinomialTreeOption}
\alias{BinomialTreeOption}
\alias{BinomialTreePlot}
\title{Binomial Tree Option Model}
\description{
A collection and description of functions to valuate
options in the framework of the Binomial tree option
approach.
\cr
The functions are:
\tabular{ll}{
\code{CRRBinomialTreeOption} \tab CRR Binomial Tree Option, \cr
\code{JRBinomialTreeOption} \tab JR Binomial Tree Option, \cr
\code{TIANBinomialTreeOption} \tab TIAN Binomial Tree Option, \cr
\code{BinomialTreeOption} \tab Binomial Tree Option, \cr
\code{BinomialTreePlot} \tab Binomial Tree Plot. }
}
\usage{
CRRBinomialTreeOption(TypeFlag = c("ce", "pe", "ca", "pa"), S, X,
Time, r, b, sigma, n, title = NULL, description = NULL)
JRBinomialTreeOption(TypeFlag = c("ce", "pe", "ca", "pa"), S, X,
Time, r, b, sigma, n, title = NULL, description = NULL)
TIANBinomialTreeOption(TypeFlag = c("ce", "pe", "ca", "pa"), S, X,
Time, r, b, sigma, n, title = NULL, description = NULL)
BinomialTreeOption(TypeFlag = c("ce", "pe", "ca", "pa"), S, X,
Time, r, b, sigma, n, title = NULL, description = NULL)
BinomialTreePlot(BinomialTreeValues, dx = -0.025, dy = 0.4,
cex = 1, digits = 2, \dots)
}
\arguments{
\item{b}{
the annualized cost-of-carry rate, a numeric value;
e.g. 0.1 means 10\% pa.
}
\item{BinomialTreeValues}{
the return value from the \code{BinomialTreeOption} function.
}
\item{cex}{
a numerical value giving the amount by which the plotting text
and symbols should be scaled relative to the default.
}
\item{description}{
a character string which allows for a brief description.
}
\item{digits}{
an integer value, how many digits should be displayed in the
option tree?
}
\item{dx, dy}{
numerical values, an offset fine tuning for the placement of
the option values in the option tree.
}
\item{n}{
number of time steps; an integer value.
}
\item{r}{
the annualized rate of interest, a numeric value;
e.g. 0.25 means 25\% pa.
}
\item{S}{
the asset price, a numeric value.
}
\item{sigma}{
the annualized volatility of the underlying security,
a numeric value; e.g. 0.3 means 30\% volatility pa.
}
\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{"ce"}, \code{"ca"} for an
European or American call option or a \code{"pe"}, \code{"pa"}
for a put option, respectively.
}
\item{X}{
the exercise price, a numeric value.
}
\item{\dots}{
arguments to be passed.
}
}
\details{
\bold{CRR Binomial Tree Model:}
\cr\cr
Binomial models were first suggested by Cox, Ross and Rubinstein (1979),
CRR, and then became widely used because of its intuition and easy
implementation. Binomial trees are constructed on a discrete-time
lattice. With the time between two trading events shrinking to zero,
the evolution of the price converges weakly to a lognormal diffusion.
Within this mode the European options value converges to the value
given by the Black-Scholes formula.
\cr
\bold{JR Binomial Tree Model:}
\cr\cr
There exist many extensions of the CRR model. Jarrow and Rudd (1983),
JR, adjusted the CRR model to account for the local drift term. They
constructed a binomial model where the first two moments of the
discrete and continuous time return processes match. As a consequence
a probability measure equal to one half results. Therefore the CRR and
JR models are sometimes atrributed as equal jumps binomial trees and
equal probabilities binomial trees.
\cr
\bold{TIAN Binomial Tree Model:}
\cr\cr Tian (1993) suggested to match discrete and continuous local
moments up to third order.
Leisen and Reimer (1996) proved that the order of convergence in
pricing European options for all three methods is equal to one, and
thus the three models are equivalent.
}
\note{
Note, the \code{BinomialTree} and \code{BinomialTreePlot} are preliminary
implementations.
}
\value{
The option price, a numeric value.
}
\references{
Broadie M., Detemple J. (1994);
\emph{American Option Evaluation: New Bounds, Approximations,
and a Comparison of Existing Methods},
Working Paper, Columbia University, New York.
Cox J., Ross S.A., Rubinstein M. (1979);
\emph{Option Pricing: A Simplified Approach},
Journal of Financial Economics 7, 229--263.
Haug E.G. (1997);
\emph{The complete Guide to Option Pricing Formulas},
McGraw-Hill, New York.
Hull J.C. (1998);
\emph{Introduction to Futures and Options Markets},
Prentice Hall, London.
Jarrow R., Rudd A. (1983);
\emph{Option Pricing},
Homewood, Illinois, 183--188.
Leisen D.P., Reimer M., (1996);
\emph{Binomial Models for Option Valuation -- Examining and
Improving Convergence},
Applied Mathematical Finanace 3, 319--346.
Tian Y. (1993);
\emph{A Modified Lattice Approach to Option Pricing},
Journal of Futures Markets 13, 563--577.
}
\author{
Diethelm Wuertz for the Rmetrics \R-port.
}
\examples{
## Cox-Ross-Rubinstein Binomial Tree Option Model:
# Example 14.1 from Hull's Book:
CRRBinomialTreeOption(TypeFlag = "pa", S = 50, X = 50,
Time = 5/12, r = 0.1, b = 0.1, sigma = 0.4, n = 5)
# Example 3.1.1 from Haug's Book:
CRRBinomialTreeOption(TypeFlag = "pa", S = 100, X = 95,
Time = 0.5, r = 0.08, b = 0.08, sigma = 0.3, n = 5)
# A European Call - Compare with Black Scholes:
CRRBinomialTreeOption(TypeFlag = "ce", S = 100, X = 100,
Time = 1, r = 0.1, b = 0.1, sigma = 0.25, n = 50)
GBSOption(TypeFlag = "c", S = 100, X = 100,
Time = 1, r = 0.1, b = 0.1, sigma = 0.25)@price
## CRR - JR - TIAN Model Comparison:
# Hull's Example as Function of "n":
par(mfrow = c(2, 1), cex = 0.7)
steps = 50
CRROptionValue = JROptionValue = TIANOptionValue =
rep(NA, times = steps)
for (n in 3:steps) {
CRROptionValue[n] = CRRBinomialTreeOption(TypeFlag = "pa", S = 50,
X = 50, Time = 0.4167, r = 0.1, b = 0.1, sigma = 0.4, n = n)@price
JROptionValue[n] = JRBinomialTreeOption(TypeFlag = "pa", S = 50,
X = 50, Time = 0.4167, r = 0.1, b = 0.1, sigma = 0.4, n = n)@price
TIANOptionValue[n] = TIANBinomialTreeOption(TypeFlag = "pa", S = 50,
X = 50, Time = 0.4167, r = 0.1, b = 0.1, sigma = 0.4, n = n)@price
}
plot(CRROptionValue[3:steps], type = "l", col = "red", ylab = "Option Value")
lines(JROptionValue[3:steps], col = "green")
lines(TIANOptionValue[3:steps], col = "blue")
# Add Result from BAW Approximation:
BAWValue = BAWAmericanApproxOption(TypeFlag = "p", S = 50, X = 50,
Time = 0.4167, r = 0.1, b = 0.1, sigma = 0.4)@price
abline(h = BAWValue, lty = 3)
title(main = "Convergence")
data.frame(CRROptionValue, JROptionValue, TIANOptionValue)
## Plot CRR Option Tree:
# Again Hull's Example:
CRRTree = BinomialTreeOption(TypeFlag = "pa", S = 50, X = 50,
Time = 0.4167, r = 0.1, b = 0.1, sigma = 0.4, n = 5)
BinomialTreePlot(CRRTree, dy = 1, cex = 0.8, ylim = c(-6, 7),
xlab = "n", ylab = "Option Value")
title(main = "Option Tree")
}
\keyword{math}
|
