liverpool complexity presentation

Alternative Models for Distributions of Returns of Stocks and Pricing Derivatives

José Augusto Carvalho FilhoMaster in Complexity and

its Interdisciplinary ApplicationsPavia, Italy

gugadrum@gmail.com

Prof. Giovani Lopes VasconcelosFederal University of PernambucoPhysics DepartmentRecife, Brazilgiovani@df.ufpe.br

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

http://www.unipv.it/complexity/