Download Application for processing and option pricing by trinomial model

2nd International e-Conference on Optimization, Education and Data Mining in Science,
Engineering and Risk Management 2012 (OEDM SERM 2012)
Application for processing and option pricing by trinomial model
Říský, V.
University of Finance and Administration, Prague, Czech Republic
[email protected]
Abstract
The goal of this paper is to validate the hypotheses that it is possible to obtain an algorithm for the
trinomial model by extension of the algorithm for the binomial model, and that trinomial model
software can be obtained by a projection of the binomial model software using trinomial parameters.
Furthermore this algorithm can be used in calculations of empirical statistics, the Black-Scholes
model, the binomial model, the trinomial model compatible with Black-Scholes model, the
autonomous trinomial model, a valuation of options and premature assertions, and premature sales of
options.
Keywords: Black-Scholes model, binomial model, trinomial model, option pricing
I.
INTRODUCTION
This paper aims at claryfing multinomial models applied to financial option pricing. More
exactly, it aims at validation of hypothesis that it is possible to obtain an algorithm of the
trinomial model by extension of algorithm of the binomial model, and application of the
Black-Scholes model, the binomial model, the option pricing, the premature assertions, the
premature sales of options, and the autonomous trinomial model as well.
Initially, we have to learn and understand specific aspects of the Czech Republic – market
size, limited amount of subjects being authorized to use the financial options. These are
mostly large enterprises that ought to learn how to use various financial instruments and how
to handle risks. I think there is insufficient knowledge of these products in the Czech
Republic. The vast majority of enterprises do not have any complete information on these
products.
Employees of economic and foreign departments of large enterprises do not usually use these
instruments because they do not know them very well, and they are afraid of these processes.
They have not understood importance and significance of financial derivatives. Before 1989,
these instruments used to be applied to a limited extent. Therefore, the financial derivatives
were not largely taken into account, or included into ordinary working processes. Banks
mediating financial operations were not prepared enough to provide these services in the
1990’s. In many cases, they did not find their way to potential users of these services.
Nowadays, such banks are prepared enough to provide the financial derivatives to their
clients. To introduce and explain the instruments to other subjects – it is their most important
duty. Price is the most limiting factor of such financial instruments and processes. Concluded
contracts must be helpful and financially acceptable for enterprises. The question of price is to
play a significant role in the future.
This topic highly deserves our attention, as it belongs to up-to-date elements of the market
economy. Unfortunately, subjects concerned are not informed enough on them.
Higher option pricing models (it means higher than the binomial and the Black- Scholes ones)
are used to a very limited extent in the Czech Republic. Monitoring the market, we notice
traders use the Black- Scholes model to different purposes, than it has been designed and
projected to. The Black- Scholes model assumes an option price is lognormally distributed. If
it is not, the Black- Scholes model produces incorrect option prices. The lognormal
distribution is based on a hypothesis a share price runs a constant volatile process. Taking
© Publishing House Curriculum. ISBN 978-80-904948-4-8
70
2nd International e-Conference on Optimization, Education and Data Mining in Science,
Engineering and Risk Management 2012 (OEDM SERM 2012)
market option prices into account, we realise that this volatility (implied by the pricing
formula) is more determined by strike and time period, until the option expires, than constant
quantity.
Nowadays, Bloomberg terminal is the most famous product applying the trinomial model to
counting. Bloomberg terminal is a computer system allowing us to monitor and analyse data
produced by the financial market in real time. It also provides up-to-date information and
news concerning the financial market fluctuations, and various investment instruments trading
too. It provides detailed analyses. This is a commercial product, which is sold for the price of
$ 1,500 per month.[3] Book called Measuring and Controlling Interest Rate and Credit Risk
[1], and its page 188, demonstrate Bloomberg terminal using the trinomial model in its 2002
version.
MBRM - MB Risk Management is another interesting product. This is a commercial product
too, which was sold for the individual user’s price of £ 49,999 per computer in 2011.[4]
OptionMatrix: The advanced Derivates Calculator is another interesting application. It
supports more than 136 theoretical models projected to the financial derivatives.[5] The
application has been launched under GNU GPLv3 license; it is available in OS Windows
(DOS) version, as well as Linux/Unix version. Mac version is also to be developed in a short
time.
II.
VALIDATION OF THE HYPOTHESIS THAT IT IS POSSIBLE TO
OBTAIN AN ALGORITHM OF THE TRINOMIAL MODEL BY
EXTENSION OF ALGORITHM OF THE BINOMIAL MODEL
In From financial derivatives to option hedging [2], chapter 1.5. Multinomial model and its
options, the multinomial model, its definition, methods and steps are described in a pregnant
and readable way. From the very beginning of the chapter, the general multinomial financial
option pricing model is based on an algorithm of discreet s-fold multinomial theoretical
distribution. Authors of this book also state, this model can be approximated by
multidimensional Poisson distribution. In certain case, the model can be approximated by
multidimensional Hypergeometric distribution.
Comparison of the binomial and the trinomial model
S-fold multinomial distribution is a discreet theoretical distribution s-Multi(n,p1,….,ps-1)
having s-theoretical parameters of n,p1,…,ps-1 of a random vector X=[X1,...,Xs-1] (values of the
random X1,…,Xs-1 quantities the random vector comprises of are expressed as i1,…,is-1 =
0,1,…,n)." [2]
Comparison of the models:
• binomial model:
o it is based on s-fold multinomial distribution for s=2,
o the binomial distribution, expressed as Bi(n,p), is related to this model
during n period (∆t); there are two profit options available for each of
these periods (with p probability and 1-p probability) which are
characterized by index of decrease (d) and increase (u) of base share (S)
in t=0 time.
o probability function of the binomial distribution (for n ≥ j ≥ 0):
n j
n− j
Π j=
p (1− p )
j
where
Probability function for Bi(n,p)
()
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2nd International e-Conference on Optimization, Education and Data Mining in Science,
Engineering and Risk Management 2012 (OEDM SERM 2012)
•
trinomial model:
o it is based on s-fold multinomial distribution for s=3,
o the trinomial distribution, expressed as Tr(n,p1,p2), is related to this
model for n period (∆t); there are three profit options available for each
of these periods (with p1 probability, p2 probability and p3=1–p1–p2
probability) which are characterized by index of decrease and increase of
base share (S) in t=0 time;
these index options are expressed as I1, I2 and I3.
o probability function of the trinomial distribution:
Π i1 , i 2 =
n!

i1 ! i 2 !  n −


ij !
∑
j =1

2
p 1 i1 p 2 i 2

1 −


pj

2
∑
j =1
n−
2
∑ ij
j =1
.
Probability function for Tr(n,p1,p2)
Resemblance of these two relations (the probability function of the binomial
distribution and the probability function of the trinomial distribution) is obvious and not
random. Probability function of the quattronomial distribution follows in order to make the
probability functions complete:
Π i1 , i2 , i3 =
n!
3

i1 ! i2 ! i3 !  n − ∑ i j
j =1



p1i1 p 2 i2 p 3 i3  1 − ∑ p j 
j =1


3

!

n−
3
∑ ij
j =1
.
Probability function for Qu(n,p1,p2,p3)
Comparing the other relations, we develop a higher degree model by extension of parametres:
Base share prices for Bi and Tr, after n period expires:
Si1 ,i2 = I1i1 I 2i2 I 3
Theoretically correct price of purchase option for Bi and Tr:
1 n
〈C 〉 = n ∑ Π j C j
q j =0
Cj = max (0, Sj – X)
Theoretically correct price of
1 n
〈 P〉 = n ∑ Π j Pj
q j =0
Pj = max (0, X – Sj)
Internal value of purchase option
IVC kj = max 0, S kj − X
Internal value of sales option for Bi and Tr:
IVPjk = max 0, X − S kj
Relation between call and put
Tr in k time period:
(
)
(
)
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n−
2
∑ij
j =1
S
sales option for Bi and Tr:
for Bi and Tr:
American options for Bi and
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2nd International e-Conference on Optimization, Education and Data Mining in Science,
Engineering and Risk Management 2012 (OEDM SERM 2012)
Ci1k,i2 =
(
Pi1k,i2
(
Cik1 ,i2 ,i3 =
(
Pi1k,i2 ,i3
(
1
p1Cik1 ++1,1 i2 + p2Cik1 ,+i21+1 + (1 − p1 − p2 ) Ci1k,+i21
q
1
=
p1 Pi1k++1,1i2 + p2 Pi1k,i+21+1 + (1 − p1 − p2 ) Pi1k,i+21
q
)
)
1
p1Ci1k++1,1 i2 ,i3 + p2Ci1k,+i21+1,i3 + p3Ci1k,+i21,i3 +1 + (1 − p1 − p2 − p3 ) Ci1k,+i21,i3
q
1
=
p1 Pi1k++1,1i2 ,i3 + p2 Pi1k,i+21+1,i3 + p3 Pi1k,i+21,i3 +1 + (1 − p1 − p2 − p3 ) Pi1k,i+21,i3
q
)
)
Adding another parameters into the model (decrease and increase i – index, {whereas j in the
binomial model} and p - probability), we develop a higher degree multinomial model.
Number of parameters we have to add is determined by s-parameter (or s-1 parameter) of sfold multinomial distribution. Calculation of parameters is based on the secondary empiricstatistical analysis of empirical standard deviations and empirical parameters of obliquenesses
of decrease or increase index of the autonomous binomial model.
Previous text validates the hypothesis that it is possible to obtain an algorithm of the trinomial
model by extension of algorithm of the binomial model.
I came to conclusion that, extending (or distributing) parameters of share price increase or
decrease, converting increase or decrease probabilities, calculating increase or decrease index
and substituting the binomial relations by the trinomial relations, the trinomial model is
calculated in similar way to the binomial model (except for certain little differences).
Indexes of share price increase or decrease I1, I2, I3 of the autonomous trinomial model are
derived by the secondary empiric-statistical analysis of the empirical standard deviations and
the empirical parameters of obliquenesses of increase or decrease index of the autonomous
binomial model.[2]
There are two options of distribution of share price increase or decrease index resulting from
the analysis:
1. ub increase index is divided into two trinomial increase indexes (I1=u1t and
I2=u2t); db index is kept as the trinomial decrease index and expressed as I3=dt;
p-probability is divided into p1 and p2; p3=1-p probability stays unchanged.
2. db decrease index is divided into two trinomial decrease indexes (I2=d1t and
I3=d2t); ub index is kept as the trinomial increase index and expressed as I1=ut;
1-p probability is divided into p2 and p3; p1=p probability stays unchanged. [2]
Validation of the hypothesis in practice and brief description of the application
In practice, the hypothesis has been validated by developing the application calculating
elementary statistical measures and the financial derivatives. The option pricing according to
the Black-Scholes model, the binomial model and the trinomial model.
Stati application has been developed there; it is logically and intuitively controlled. We can
turn back in every single data processing step, revise and recount the results. Data may be
converted into a graph, if meaningful.
Financial derivatives tab has been developed to count the financial derivatives (as its name
indicates). The program is implemented in order to count the Black-Scholes model of
European and American no-dividend share option, the Black-Scholes model of European
continuous-dividend share option, the Black-Scholes model of American discreet-divident
purchase share option, the binomial model of pricing of European and American no-dividend
© Publishing House Curriculum. ISBN 978-80-904948-4-8
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2nd International e-Conference on Optimization, Education and Data Mining in Science,
Engineering and Risk Management 2012 (OEDM SERM 2012)
share option, and the trinomial autonomous model of pricing of no-dividend share option as
well.
Black-Scholes model
Hedge ratio calculation, probability of application in time period, and call option and put
option values.
Binomial model
Individual sections are calculated during the expiry period. In table u-column, there are
increase indexes; in d-column, there are decrease indexes; in p-column, there are probabilities
of increase; in 1-p-collumn, there are probabilities of decrease; in S-column, there are base
share acceptable prices; in C-column, there are purchase share acceptable prices; in P-column,
there are sales share acceptable prices; in the last Π-column, there are probabilities of
purchase option acceptable price. Call option, put option and Put-call parita values are also
calculated there; Put-call parita means a relation between option bonuses of the
corresponding call options and put options (the call options and the put options having the
same basic instrument, the same price of implementation and the same expiry date). Using
Put-call parita method, we calculate e.g. option bonus, or we deduce put option features from
the already calculated option bonus, or from derived call option features.
Trinomial model
Individual sections are calculated during the expiry period. In table Πij-column, there are
probabilities of purchase option acceptable prices; in Sij-column, there are base share
acceptable prices; in Cij-column, there are purchase option acceptable prices; in Pij-column,
there are sales option acceptable prices. Call option, put option and Put-call parita values are
also calculated there; Put-call parita means a relation between option bonuses of the
corresponding call options and put options (the call options and the put options having the
same basic instrument, the same price of implementation and the same expiry date). Using
Put-call parita method, we calculate e.g. option bonus, or we deduce put option features from
the already calculated option bonus, or from derived call option features. Increase indexes or
decrease indexes are put down there (I1, I2 and I3).
The application also includes Tree of development of B\base share prompt price, Tree of
development of put/call option internal price, and Tree of development of purchase/sales
option price. In a textfield, the applied model puts down value, date and time, when the
calculation has been done, particular calculation parameters and the calculation result.
III.
CONCLUSION
In conclusion, we have to be aware of a little attention paid to the option pricing by the
trinomial and the quattronomial models in the Czech Republic. Nowadays, there is a lack of
specialized literary sources dealing with the option pricing by the trinomial and the
quattronomial models. Therefore, I highly appreciate the title From financial derivatives to
option hedging by Pavlát and Záškodný; it deals with the above-mentioned issue in a
theoretical way in its second part.
There are no obstacles hindering us from launching the multinomial models of a higher
degree in practice. No matter, these option pricing models are more precious and more exact.
Czech institutions operating the financial derivatives ought to focus on the application of the
multinomial models in practice.
The application has significantly contributed to calculations of the financial derivatives, but
© Publishing House Curriculum. ISBN 978-80-904948-4-8
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2nd International e-Conference on Optimization, Education and Data Mining in Science,
Engineering and Risk Management 2012 (OEDM SERM 2012)
also to implementation of analytic and synthetic models of the empiric statistics; such models
have a significant cognitive dimension. It shows how the analysis, the synthesis and the
abstraction are used and applied. The analysis, the synthesis and the abstraction are crucial for
us to reach a correct result.
REFERENCES
[1] FABOZZI, F. J. MANN, S. V. CHOUDHRY, M. 2003. Measuring and Controlling Interest Rate and Credit
Risk. England: John Wiley & Sons, 2003. ISBN 0-471-26806-2.
[2] PAVLÁT, V. ZÁŠKODNÝ, P. 2012. Od finančních derivátů k opčnímu hedgingu. Praha: Curriculum, 2012.
ISBN 978-80-904948-3-1.
[3] http://www.bloomberg.com/professional/software_support/
[4] www.mbrm.com/index.shtml
[5] http://opensourcefinancialmodels.com/
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