liverpool complexity presentation
TRANSCRIPT
Alternative Models for Distributions of Returns of Stocks and Pricing Derivatives
José Augusto Carvalho FilhoMaster in Complexity and
its Interdisciplinary ApplicationsPavia, Italy
Prof. Giovani Lopes VasconcelosFederal University of PernambucoPhysics DepartmentRecife, [email protected]
11-14 September 2005
Phynance and Complexity:
OutlineOutline
PhynancePhynanceDerivativesDerivatives
Black and Scholes TheoryBlack and Scholes Theory
Statistical Analysis of Statistical Analysis of Ibovespa IndexIbovespa Index
Price Dynamics Price Dynamics
Exponential ModelExponential Model
ConclusionsConclusions
Complex SystemsComplex Systems
Complex SystemsComplex SystemsA A complex systemcomplex system is a is a system whose properties are not fully whose properties are not fully
explained by an understanding of its component parts. Complex explained by an understanding of its component parts. Complex systems consist of a large number of mutually interacting and systems consist of a large number of mutually interacting and
interwoven parts, entities or agents. They are woven out of many interwoven parts, entities or agents. They are woven out of many parts, the Latin parts, the Latin complexuscomplexus comes from the Greek comes from the Greek plekopleko or or plektosplektos, ,
meaning "to plait or twine." meaning "to plait or twine."
(Gell-Mann). (Gell-Mann).
Econophysics and ComplexityEconophysics and Complexity
Stock exchanges, exchanges rates, derivatives markets, financial assets in
general.
Some systems:
The word “econophysics” was used for the frst time in a Conference of Complex Systems in Calcuta in 1995.
Econophysics concerns in analyzing financial markets from a physics point of view, in order to describe complex systems in terms of simple models.
Application of methods of statistical mechanics, chaos theory, fractals and complex systems to economical and social systems.
Brief History
1900 - Louis Bachelier and “Théorie de La Especulation”.1905 - Albert Einstein’s brownian motion.1908 – Langevin’s equation.1908 – Perrin confirms Einstein’s work.1963 – Benoit Mandelbrot and Levy Distributions.1964 - Paul Sammuelson – Modern Theory of pricing.
Louis Bachelier
1973 - Options contracts started to be traded in exchanges. 1973 - Fischer Black, Myron Scholes, Robert Merton and the Option Pricing Theory.90’s - Phynance. 1997 - Merton and Scholes and the Nobel Prize.
Brief History
Louis Bachelier
Complexity in Financial WorldFIELD Finance Economics
AGENT Investors ConsumersHETEROGENEITY Risk preferences,
InformationTastes, incomes
ORGANIZATION Mutual Funds, market makers
Families, firms
ADAPTATION Learning Affect of advertising, education
FEEDBACK Success or failure Buying, selling tradingDYNAMICS Stock price
movementsPrice adjustments
EMERGENT BEHAVIOR Market movements Inflation, unemployment
Complexity of Pricing
Complexity of Pricing
Option Market
An option contract gives the right in buying or selling some asset S with a predermined price K (strike price) in a future
date T (maturity).
Buy Option (Call)
On the Maturity
if S(T) > K, the holder exercise the option. Buy the asset for K, sell on the market for S making a profit of (S-K).
`
if S(T) <K, the holder does not exercise the option. The option is worthless.
Sell Option (Put)On the maturity
if S(T) >K, the holder does not exercise the option. The option is worthless.
if S(T) < K, the holder exercise the option. Sell the asset for K, buy on market for S making a profit of (S-K).
`
An option contract gives the right in buying or selling some asset S with a predermined price K (strike price) in a future
date T (maturity).
Option Market
Therefore, the following question arises:
How must an option contract be worth?How must an option contract be worth?
An option represents the right. In this sense, one should pay for that.
Option Market
Price DynamicsLet S(t) be the price of the financial asset
in a time t.
μ : average rate of returnσ : volatilidyX(t) : brownian motion
Prices follow a Log-normal distribution.
Returns follow a brownian motion (Efficient Market Hypothesis).
Price DynamicsStandard Model
The solution of the Black and Scholes equation is known as Black and Scholes Formula.
Risk Neutral Approach (Merton 1973)
With μ= r
Black and Scholes Formula
Data
Ibovespa index is one of the most important stock market index in Latin America and is considered one of the
thermometer of the brazilian economy as well.
Intraday values of Ibovespa for every 15 minutes, from 1998 to 2001. Total of 19959 quotations.
Closed values of the Ibovespa, from january 1968 to february 2004. Total of 8889 traded days.
Ibovespa Time Series
Year
Clos
ed V
alue
Starting from the 1 day return time series we can generate returns time series for any time
window t.
Return
Return Time Series of Ibovespa
Year
Retu
rns
Daily Returns HistogramsPr
obab
ility
Den
sity
Fu
ncti
on
Returns
Histograms of Returns t=1
1 day returnsGaussian σ=0.028
Returns
Prob
abili
ty D
ensi
ty
Func
tion
100 days returnsGaussian σ=0.34
Returns
Prob
abili
ty D
ensi
ty F
unct
ion
Histograms of Returns t=100
(McCauley 2003)
Let S(t) be the price of the financial asset in a time instant t. In an exponential model, the distribution of the returns
f(x,t) is given as following:
Exponential Distribution
In case the probability density function is exponential, then its cumulative distribution is exponential as well.
Exponential Distribution
For the exponential distribution the variance is given accordingly:
2 H=1 : diffusive (no memory)
Variance
In a exponential model, the price of the option contracts must obey the following equations:
Exponential Model for Options
Returns
Prob
abili
ty D
ensi
ty
Func
tion
Exponential Fitting
Exponential Fitting
Returns
Prob
abili
ty D
ensi
ty
Func
tion
Cumulative Distributions of Daily Returns
Returns
Cum
ulat
ive
Dis
trib
utio
n
Cum
ulat
ive
Dis
trib
utio
nCumulative Distributions of Daily Returns
Returns
Variance of Returns
Time window
Varia
nce
Collapse of Cumulative Distributions
Normalized returns
Cum
ulat
ive
Dis
trib
utio
n
Normalized returns
Cum
ulat
ive
Dis
trib
utio
n
Normalized returns
Cum
ulat
ive
Dis
trib
utio
n
Collapse of Cumulative Distributions
Diffusive Process for t<30
Time window
Varia
nce
Variance
Intraday Cumulative DistributionCu
mul
ativ
e D
istr
ibut
ion
Returns
Exponential Distribution for t >3h
Power Law for t <3h?
Diffusive Process
Intraday Cumulative CollapseCu
mul
ativ
e D
istr
ibut
ion
Normalized Returns
Price of Ibovespa option IBOVH on
the 16th of june, 2004. In that day the
Ibovespa had its closed value R$
22447. The option maturity is 18th of
august 2004.
Pricing Analysis for Ibovespa Options
The central region of the empirical distributions of returns is well described by exponential functions for time windows up to t=30 days. For bigger time windows the distribution is gaussian, as one can expect.
In both regimes (daily and intraday) the variance increases in a linear fashion with respect to time, indicating that the underlying stochastic process is diffusive.
The exponential behavior has been found within the intraday regime from t>3h on.
The exponential model seems to correctly describe the price of the Ibovespa options.
Conclusions
