2. Alternative Models for Distributions of Returns of Stocks and Pricing Derivatives José Augusto Carvalho Filho Master in Complexity and its Interdisciplinary Applications Pavia, Italy [email_address] Prof. Giovani Lopes Vasconcelos Federal University of Pernambuco Physics Department Recife, Brazil [email_address] Phynance and Complexity: 11-14 September 2005
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4. Complex Systems A complex system is a system whose properties are not fully explained by an understanding of its component parts. Complex systems consist of a large number of mutually interacting and interwoven parts, entities or agents. They are woven out of many parts, the Latin complexus comes from the Greek pleko or plektos , meaning "to plait or twine." (Gell-Mann).
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6. 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
7. 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
8. Complexity in Financial World Inflation, unemployment Market movements EMERGENT BEHAVIOR Price adjustments Stock price movements DYNAMICS Buying, selling trading Success or failure FEEDBACK Affect of advertising, education Learning ADAPTATION Families, firms Mutual Funds, market makers ORGANIZATION Tastes, incomes Risk preferences, Information HETEROGENEITY Consumers Investors AGENT Economics Finance FIELD
11. 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.
12. Option Market Sell Option (Put) On the maturity 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). 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). `
13. Option Market Therefore, the following question arises: How must an option contract be worth? An option represents the right. In this sense, one should pay for that.
14. Price Dynamics Let S(t) be the price of the financial asset in a time t. μ : average rate of return σ : volatilidy X(t) : brownian motion
15. Prices follow a Log-normal distribution. Returns follow a brownian motion (Efficient Market Hypothesis). Price Dynamics Standard Model
16. 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
17. 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.
22. Histograms of Returns t=1 1 day returns Gaussian σ = 0.028 Returns Probability Density Function
23. Histograms of Returns t=100 100 days returns Gaussian σ = 0. 34 Returns Probability Density Function
24. (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
25. In case the probability density function is exponential, then its cumulative distribution is exponential as well. Exponential Distribution
26. For the exponential distribution the variance is given accordingly: 2 H=1 : diffusive (no memory) Variance
27. In a exponential model, the price of the option contracts must obey the following equations: Exponential Model for Options
37. Exponential Distribution for t >3h Power Law for t <3h? Diffusive Process Intraday Cumulative Collapse Cumulative Distribution Normalized Returns
38. Price of Ibovespa option IBOVH on the 16 th of june, 2004. In that day the Ibovespa had its closed value R$ 22447. The option maturity is 18 th of august 2004. Pricing Analysis for Ibovespa Options
39. 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
![Alternative Models for Distributions of Returns of Stocks and Pricing Derivatives José Augusto Carvalho Filho Master in Complexity and its Interdisciplinary Applications Pavia, Italy [email_address] Prof. Giovani Lopes Vasconcelos Federal University of Pernambuco Physics Department Recife, Brazil [email_address] Phynance and Complexity: 11-14 September 2005](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)