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Alternative Models for Distributions of Returns of Stocks and Pricing Derivatives José Augusto Carvalho Filho Master in Co...
Outline <ul><ul><li>Phynance </li></ul></ul><ul><ul><li>Complex Systems </li></ul></ul><ul><ul><li>Derivatives </li></ul><...
Complex Systems A  complex system  is a  system  whose properties are not fully explained by an understanding of its compo...
Econophysics and Complexity <ul><ul><li>Stock exchanges, exchanges rates, derivatives markets, financial assets in general...
Brief History 1900 - Louis Bachelier and “Théorie de La Especulation”. 1905 - Albert Einstein’s brownian motion. 1908 – La...
1973 - Options contracts started to be traded in exchanges.  1973 - Fischer Black, Myron Scholes,  Robert Merton and the O...
Complexity in Financial World Inflation, unemployment Market movements EMERGENT BEHAVIOR Price adjustments Stock price mov...
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 p...
Option Market Sell Option (Put) On the maturity An option contract gives the right in buying or selling some asset  S  wit...
Option Market Therefore, the following question arises: How must an option contract be worth? An option represents the rig...
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) ...
Prices follow a Log-normal distribution. Returns follow a brownian motion  (Efficient Market Hypothesis). Price Dynamics S...
The solution of the Black and Scholes equation is known as Black and Scholes Formula. Risk Neutral Approach ( Merton 1973 ...
Data Ibovespa index is one of the most important stock market index in Latin America and is considered one of the thermome...
Ibovespa Time Series Year Closed Value
Return Starting from the 1 day return   time series we can generate returns time series for any time window t.
Return Time Series of Ibovespa Year Returns
Daily Returns Histograms Probability Density Function Returns
Histograms of Returns t=1 1 day returns Gaussian  σ = 0.028 Returns Probability Density Function
Histograms of Returns t=100 100 days returns Gaussian  σ = 0. 34 Returns Probability Density Function
(McCauley 2003) Let S(t) be the price of the financial asset in a time instant t. In an exponential model, the distributio...
In case the probability density function is exponential, then its cumulative distribution is exponential as well. Exponent...
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 Probability Density Function Exponential Fitting
Exponential Fitting Returns Probability Density Function
Cumulative Distributions of Daily Returns Returns Cumulative Distribution
Cumulative Distributions of Daily Returns Returns Cumulative Distribution
Variance of Returns Time window Variance
Collapse of Cumulative Distributions Normalized returns Cumulative Distribution
Normalized returns Cumulative Distribution Normalized returns Cumulative Distribution Collapse of Cumulative Distributions
Diffusive Process for t<30 Variance Time window Variance
Intraday Cumulative Distribution Cumulative Distribution Returns
Exponential Distribution for t >3h Power Law for t <3h? Diffusive Process Intraday Cumulative Collapse Cumulative Distribu...
Price of Ibovespa option  IBOVH on  the 16 th  of june, 2004. In that day the Ibovespa had its closed value R$ 22447. The ...
The central region of the empirical distributions of returns is well described by exponential functions for time windows u...
http://www.unipv.it/complexity/
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Liverpool Complexity Presentation

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Liverpool Complexity Conference 2005 - Presentation about Distribution Models for Ibovespa Index and Its Vanilla Options.

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Liverpool Complexity Presentation

  1. 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
  2. 3. Outline <ul><ul><li>Phynance </li></ul></ul><ul><ul><li>Complex Systems </li></ul></ul><ul><ul><li>Derivatives </li></ul></ul><ul><ul><li>Black and Scholes Theory </li></ul></ul><ul><ul><li>Statistical Analysis of Ibovespa Index </li></ul></ul><ul><ul><li>Price Dynamics </li></ul></ul><ul><ul><li>Exponential Model </li></ul></ul><ul><ul><li>Conclusions </li></ul></ul>
  3. 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 &quot;to plait or twine.&quot; (Gell-Mann).
  4. 5. Econophysics and Complexity <ul><ul><li>Stock exchanges, exchanges rates, derivatives markets, financial assets in general. </li></ul></ul>Some systems: <ul><ul><li>The word “econophysics” was used for the frst time in a Conference of Complex Systems in Calcuta in 1995. </li></ul></ul><ul><ul><li>Econophysics concerns in analyzing financial markets from a physics point of view, in order to describe complex systems in terms of simple models. </li></ul></ul><ul><ul><li>Application of methods of statistical mechanics, chaos theory, fractals and complex systems to economical and social systems. </li></ul></ul>
  5. 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
  6. 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
  7. 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
  8. 9. Complexity of Pricing
  9. 10. Complexity of Pricing
  10. 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.
  11. 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). `
  12. 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.
  13. 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
  14. 15. Prices follow a Log-normal distribution. Returns follow a brownian motion (Efficient Market Hypothesis). Price Dynamics Standard Model
  15. 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
  16. 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.
  17. 18. Ibovespa Time Series Year Closed Value
  18. 19. Return Starting from the 1 day return time series we can generate returns time series for any time window t.
  19. 20. Return Time Series of Ibovespa Year Returns
  20. 21. Daily Returns Histograms Probability Density Function Returns
  21. 22. Histograms of Returns t=1 1 day returns Gaussian σ = 0.028 Returns Probability Density Function
  22. 23. Histograms of Returns t=100 100 days returns Gaussian σ = 0. 34 Returns Probability Density Function
  23. 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
  24. 25. In case the probability density function is exponential, then its cumulative distribution is exponential as well. Exponential Distribution
  25. 26. For the exponential distribution the variance is given accordingly: 2 H=1 : diffusive (no memory) Variance
  26. 27. In a exponential model, the price of the option contracts must obey the following equations: Exponential Model for Options
  27. 28. Returns Probability Density Function Exponential Fitting
  28. 29. Exponential Fitting Returns Probability Density Function
  29. 30. Cumulative Distributions of Daily Returns Returns Cumulative Distribution
  30. 31. Cumulative Distributions of Daily Returns Returns Cumulative Distribution
  31. 32. Variance of Returns Time window Variance
  32. 33. Collapse of Cumulative Distributions Normalized returns Cumulative Distribution
  33. 34. Normalized returns Cumulative Distribution Normalized returns Cumulative Distribution Collapse of Cumulative Distributions
  34. 35. Diffusive Process for t<30 Variance Time window Variance
  35. 36. Intraday Cumulative Distribution Cumulative Distribution Returns
  36. 37. Exponential Distribution for t >3h Power Law for t <3h? Diffusive Process Intraday Cumulative Collapse Cumulative Distribution Normalized Returns
  37. 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
  38. 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
  39. 40. http://www.unipv.it/complexity/

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