Is Gaussian Volatility Necessary To
Price Equity Derivatives?
Master Thesis
Darius Kirevicius
University of Amsterdam, Amsterdam Business School
MSc Business Economics, Finance track
Table of Content
1 Introduction p. 03
2 Literture review
2.1 The Puzzle p. 06
2.2 Consequences of the Puzzle p. 15
2.3 Pricing of Derivative Securities p. 19
2.4 Pricing of American Type Derivative Securities p. 21
2.5 The Considered Method and Its Contribution p. 25
3 Hypothesis and Methodology
3.1 Hypothesis p. 26
3.2 Methodology p. 28
4 Data, Empirical Results and Discussion
4.1 The Data p. 35
4.2 Empirical Results p. 37
4.2.1 The Implied RND p. 37
4.2.2 Pricing of American Type Derivative Securities p. 39
4.2.3 Fragility of Considered Method p. 41
4.3 Discussion p. 43 6 Conclusion p. 46 References p. 49 Appendices Appendix 1 p. 51 Appendix 2 p. 54 Appendix 3 p. 58 Appendix 4 p. 75 Appendix 5 p. 81
1 Introduction
Modern financial asset pricing models often require that the underlying asset price changes are well approximated by the Gaussian (the so-called normal) distribution. In turn, financial return and risk are defined in terms of the distribution domain of attraction and scale parameters respectively. In particular, the Gaussian distribution mean and standard deviation (volatility) parameters. The Gaus-sian distribution mean and volatility parameters are essential to the risk-return trade-off framework and financial models that are derived from the framework. For instance, the volatility parameter is among key inputs in the Black-Scholes-Merton (BSM) contingent claim pricing model, etc. Further, the volatility parameter is used to express other risk measures: value-at-risk (VaR) and conditional VaR (CVaR). Both VaR and CVaR are widely used for financial risk management (e.g. fixed-income) and investment decision making (e.g. CVaR portfolio optimization) purposes. However, empirical evidence suggests that changes of financial asset prices and economic parameter values are not well approximated by the Gaussian distribution hypothesis (Cont 2001 and Jondeau, Poon & Rockinger 2007).
Empirical distributions of change of financial asset prices and economic parameter values exhibit higher peakedness around the domain of attraction and contain higher probability mass in the tails than expected under the Gaussian distribution hypothesis. In other words, empirical distribu-tions exhibit higher concentration around the mean and greater extremes events (larger deviadistribu-tions from the mean) than expected under the Gaussian distribution hypothesis. Majority of research focused on improving fits of modern financial models to empirical data add parameters (e.g. stochas-tic volatility, jumps, exponential or power law tails, etc.) that require additional estimation. Further, majority of the improvements are sensitive to distribution assumptions of the underlying process. Disagreement, however, exists regarding the distribution properties of the underlying process of changes of financial asset prices and economic parameter values.
The disagreement may be broadly characterized by two positions (Taleb 2009). The first position is that the underlying process distribution scale parameter is well defined. In other words, as the number of observations increases, the distribution scale parameter converges to some finite value. Convergence property is statistically convenient. It implies that a sample distribution scale parameter may be used to efficiently estimate the population scale parameter. Therefore, statistical inference of the sample may be generalized to describe the population in a non-trivial way. Further, the underlying process distribution scale parameter may be modelled using parametric distributions for which cumulative (CDF) and probability mass (PDF) distribution functions are available in closed-form expressions (e.g. Gaussian or exponential Pareto distributions for the tails). The availability of closed-form expressions (analytical solutions) of the distribution CDF and PDF functions of the underlying process enable efficient application to model uncertainty within the risk-return framework.
The second position is that the underlying distribution scale parameter is not well defined, hence, does not converge to some finite value as the number of observation increases. Although in a finite sample the distribution scale parameter is always well defined (finite), the distribution scale parameter will differ from sample to sample and increase with the number of observation (not converge). Non convergence property is not statistically convenient. In particular, a sample distribu-tion scale parameter may not be used to efficiently estimate the populadistribu-tion scale parameter. There-fore, statistical inference of the sample may not be generalized to describe the population in a non-trivial way. The non-convergence property poses significant challenges to methods using infinite scale distributions (e.g. Pareto-Levy) to fit empirical distributions of changes of financial assets and other economic parameters. Closed-form expressions of the distribution CDF and PDF functions are often not available, except in limit cases (e.g. Cauchy distribution is the limit case of the Pareto-Levy distribution). The available characteristic expressions of the CDF and PDF functions may be applied to model uncertainty within the risk-return framework with significantly less efficiency relative to the closed-form alternatives. Hence, theoretical and practical applications require specialist knowledge and high performance computing.
Is the disagreement regarding the distribution properties of the underlying process relevant when pricing financial assets? Taleb (2009) proposes that finiteness or in-finiteness of the distribu-tion scale parameter is irrelevant in the context of quantitative finance. Broadly speaking, Taleb’s (2009) suggestion is motivated by divergence between the theoretical financial modeling practice and the quantitative finance practice. The theoretical financial modeling practice is often concerned with distribution properties of the underlying process. For instance, fitting probability density func-tions to historical log-returns of the underlying asset. In other words, theory is often concerned with “raw” probabilities. The quantitative finance practice is concerned with the ultimate financial conse-quence - the payoff (profit or loss) - which is measured in terms of, for instance, Euro and Euro-cents instead of probabilities. In other words, the “raw” probability is irrelevant in practical financial decision making without considering the financial payoff (profit or loss) associated with the “raw” probability.
The “true” probability distribution of an underlying process (and its properties) is not observ-able. Only discrete instance of the stochastic processes are observable via market prices. The observable market prices may be considered as realizations of the supply and demand of financial instruments concerned with various states of underlying assets and economic parameter values. Further, one may argue that actions of agents (e.g. supply and demand) and not the existence of financial theories and models create markets. Therefore, application of theoretical financial model-ing is not particularly useful beyond attempts to describe option tradmodel-ing and quantitative finance practice within a sterilized world assumptions; especially in the context of risk management. Taleb (2009) suggest that practitioners of quantitative finance develop heuristic processes and numerical methods that circumvent theoretical shortcomings and allow practitioners focus on every-day busi-ness of making the markets.
Hence, how to efficiently price financial assets without engaging in the disagreement regard-ing the distribution properties of the underlyregard-ing process? In particular, the central thesis question is how to price equity derivatives without making assumptions about finiteness or in-finiteness of the distribution scale parameter values? To answer the central thesis questions, a numerical method (algorithm) is considered. The objective of the considered method is to minimizing errors between model and market prices of American type derivative securities. The performance efficiency of the considered numerical method is compared to the performance efficiency of the (conventional) binomial model with constant historic volatility estimated using daily log-returns of the underlying security. The performance efficiency is defined in terms of mean errors (ME), mean absolute devia-tion (MAD) of ME and MAD of MAD of ME. Finally, fragility of the considered method is examined by perturbing the lattice density of end-states and their respective implied RND probabilities. Fragility is measured in terms of non-linearity of mis-pricing errors (between model and market prices) with respect to the lattice density.
The thesis research questions is motivated by the practical need to price derivative securi-ties consistently with market prices; while avoiding the disagreement regarding the distribution properties of the underlying process. Specifically, to minimize the mis-pricing error between model and market prices. Further, the thesis concern with pricing of (equity) derivative type securities is motivated by several factors. First, the price definition of equity derivative type security is an explicit function of the ultimate financial consequence - the payoff (profit or loss). The price of a derivative type security may be defined as the expected discounted payoff (Taleb, 2009). However, the price definition does not explicitly specify the distribution function of respective states of the underlying process. Consequentially, this allows some flexibility with respect to the probability distribution function. For instance, instead of specifying the distribution function explicitly, one may attempt to estimate the market implied RND probabilities as discrete instances. Therefore, examining pricing of derivative type securities from the implied RND approach may yield non-trivial insight into pricing of financial assets while avoiding the disagreement regarding properties of the underlying process. For instance, reduce mis-pricing between model and market prices relative to conventional models relying on the Gaussian scale (volatility) parameter.
Second, derivative type securities are concerned with payoffs under various states of the underlying asset and/or economic parameters. Thus, derivative type securities are instrumental in hedging against adverse financial payoffs associate with extreme (“tail”) events of the underlying process. Extreme (“tail”) events often have disproportionate financial consequences (positive or negative) relative to events near the expectation. Hence, efficient pricing of derivative type securities is relevant to anyone concerned with exposure and hedging against adverse financial payoffs due to various (extreme) states of the underlying process. Finally, examining fragility of the considered method with respect to the lattice density of end-states of the underlying process may reveal non-linearity of errors with respect to model parameters. Exposure of non-non-linearity of model errors with respect to parameter errors may enable market operators to further calibrate the model and reduce financial exposure to model parameter errors.
The ultimate objective of the thesis is to produce a process that estimates prices of Ameri-can type derivative securities accurately with respect to market prices. The author considers the objective to price to be different from the objective to value derivative type securities. The later implying intimate knowledge of the underlying process while the former implying minimization of error between model and market prices. Therefore, the method considered for the purpose of the thesis may lack theoretical elegance, nevertheless, retain numerical accuracy; which is the ultimate objective of the quantitative finance modeling practice. Hence, the research is striving to contribute to the existing literature of implied RND pricing of derivative type securities by producing a simple yet efficient implementation of the numerical method (algorithm) to efficiently price American type derivative securities. The considered method does not explicitly rely on the definition of the probabil-ity distribution function and the value of its scale parameter. Hence, the thesis attempts to contribute to the existing literature of pricing derivative type securities a method that accurately prices Ameri-can type derivative securities while avoiding the disagreement regarding the value of the distribution scale parameter of the underlying process.
The rest of the thesis is structured as follows. Section 2 reviews related literature. In particu-lar, Section 2 reviews issues relevant to the disagreement regarding distribution properties of the underlying process in the context of quantitative finance and pricing of derivative type securities. Further, implied RND estimation methods are reviewed. Their advantages and limitations are briefly discussed. Finally, literature relevant to the method considered for the purpose of the thesis is discussed. Section 3 discusses the thesis hypothesis and methodology, including the Mathematica code developed for the purpose of the thesis. Section 4 presents data and empirical results followed by the discussion. Section 5 discusses fragility of the considered method while Section 6 concludes the thesis.
Modern financial asset pricing models often require that the underlying asset price changes are well approximated by the Gaussian (the so-called normal) distribution. In turn, financial return and risk are defined in terms of the distribution domain of attraction and scale parameters respectively. In particular, the Gaussian distribution mean and standard deviation (volatility) parameters. The Gaus-sian distribution mean and volatility parameters are essential to the risk-return trade-off framework and financial models that are derived from the framework. For instance, the volatility parameter is among key inputs in the Black-Scholes-Merton (BSM) contingent claim pricing model, etc. Further, the volatility parameter is used to express other risk measures: value-at-risk (VaR) and conditional VaR (CVaR). Both VaR and CVaR are widely used for financial risk management (e.g. fixed-income) and investment decision making (e.g. CVaR portfolio optimization) purposes. However, empirical evidence suggests that changes of financial asset prices and economic parameter values are not well approximated by the Gaussian distribution hypothesis (Cont 2001 and Jondeau, Poon & Rockinger 2007).
Empirical distributions of change of financial asset prices and economic parameter values exhibit higher peakedness around the domain of attraction and contain higher probability mass in the tails than expected under the Gaussian distribution hypothesis. In other words, empirical distribu-tions exhibit higher concentration around the mean and greater extremes events (larger deviadistribu-tions from the mean) than expected under the Gaussian distribution hypothesis. Majority of research focused on improving fits of modern financial models to empirical data add parameters (e.g. stochas-tic volatility, jumps, exponential or power law tails, etc.) that require additional estimation. Further, majority of the improvements are sensitive to distribution assumptions of the underlying process. Disagreement, however, exists regarding the distribution properties of the underlying process of changes of financial asset prices and economic parameter values.
The disagreement may be broadly characterized by two positions (Taleb 2009). The first position is that the underlying process distribution scale parameter is well defined. In other words, as the number of observations increases, the distribution scale parameter converges to some finite value. Convergence property is statistically convenient. It implies that a sample distribution scale parameter may be used to efficiently estimate the population scale parameter. Therefore, statistical inference of the sample may be generalized to describe the population in a non-trivial way. Further, the underlying process distribution scale parameter may be modelled using parametric distributions for which cumulative (CDF) and probability mass (PDF) distribution functions are available in closed-form expressions (e.g. Gaussian or exponential Pareto distributions for the tails). The availability of closed-form expressions (analytical solutions) of the distribution CDF and PDF functions of the underlying process enable efficient application to model uncertainty within the risk-return framework.
The second position is that the underlying distribution scale parameter is not well defined, hence, does not converge to some finite value as the number of observation increases. Although in a finite sample the distribution scale parameter is always well defined (finite), the distribution scale parameter will differ from sample to sample and increase with the number of observation (not converge). Non convergence property is not statistically convenient. In particular, a sample distribu-tion scale parameter may not be used to efficiently estimate the populadistribu-tion scale parameter. There-fore, statistical inference of the sample may not be generalized to describe the population in a non-trivial way. The non-convergence property poses significant challenges to methods using infinite scale distributions (e.g. Pareto-Levy) to fit empirical distributions of changes of financial assets and other economic parameters. Closed-form expressions of the distribution CDF and PDF functions are often not available, except in limit cases (e.g. Cauchy distribution is the limit case of the Pareto-Levy distribution). The available characteristic expressions of the CDF and PDF functions may be applied to model uncertainty within the risk-return framework with significantly less efficiency relative to the closed-form alternatives. Hence, theoretical and practical applications require specialist knowledge and high performance computing.
Is the disagreement regarding the distribution properties of the underlying process relevant when pricing financial assets? Taleb (2009) proposes that finiteness or in-finiteness of the distribu-tion scale parameter is irrelevant in the context of quantitative finance. Broadly speaking, Taleb’s (2009) suggestion is motivated by divergence between the theoretical financial modeling practice and the quantitative finance practice. The theoretical financial modeling practice is often concerned with distribution properties of the underlying process. For instance, fitting probability density func-tions to historical log-returns of the underlying asset. In other words, theory is often concerned with “raw” probabilities. The quantitative finance practice is concerned with the ultimate financial conse-quence - the payoff (profit or loss) - which is measured in terms of, for instance, Euro and Euro-cents instead of probabilities. In other words, the “raw” probability is irrelevant in practical financial decision making without considering the financial payoff (profit or loss) associated with the “raw” probability.
The “true” probability distribution of an underlying process (and its properties) is not observ-able. Only discrete instance of the stochastic processes are observable via market prices. The observable market prices may be considered as realizations of the supply and demand of financial instruments concerned with various states of underlying assets and economic parameter values. Further, one may argue that actions of agents (e.g. supply and demand) and not the existence of financial theories and models create markets. Therefore, application of theoretical financial model-ing is not particularly useful beyond attempts to describe option tradmodel-ing and quantitative finance practice within a sterilized world assumptions; especially in the context of risk management. Taleb (2009) suggest that practitioners of quantitative finance develop heuristic processes and numerical methods that circumvent theoretical shortcomings and allow practitioners focus on every-day busi-ness of making the markets.
Hence, how to efficiently price financial assets without engaging in the disagreement regard-ing the distribution properties of the underlyregard-ing process? In particular, the central thesis question is how to price equity derivatives without making assumptions about finiteness or in-finiteness of the distribution scale parameter values? To answer the central thesis questions, a numerical method (algorithm) is considered. The objective of the considered method is to minimizing errors between model and market prices of American type derivative securities. The performance efficiency of the considered numerical method is compared to the performance efficiency of the (conventional) binomial model with constant historic volatility estimated using daily log-returns of the underlying security. The performance efficiency is defined in terms of mean errors (ME), mean absolute devia-tion (MAD) of ME and MAD of MAD of ME. Finally, fragility of the considered method is examined by perturbing the lattice density of end-states and their respective implied RND probabilities. Fragility is measured in terms of non-linearity of mis-pricing errors (between model and market prices) with respect to the lattice density.
The thesis research questions is motivated by the practical need to price derivative securi-ties consistently with market prices; while avoiding the disagreement regarding the distribution properties of the underlying process. Specifically, to minimize the mis-pricing error between model and market prices. Further, the thesis concern with pricing of (equity) derivative type securities is motivated by several factors. First, the price definition of equity derivative type security is an explicit function of the ultimate financial consequence - the payoff (profit or loss). The price of a derivative type security may be defined as the expected discounted payoff (Taleb, 2009). However, the price definition does not explicitly specify the distribution function of respective states of the underlying process. Consequentially, this allows some flexibility with respect to the probability distribution function. For instance, instead of specifying the distribution function explicitly, one may attempt to estimate the market implied RND probabilities as discrete instances. Therefore, examining pricing of derivative type securities from the implied RND approach may yield non-trivial insight into pricing of financial assets while avoiding the disagreement regarding properties of the underlying process. For instance, reduce mis-pricing between model and market prices relative to conventional models relying on the Gaussian scale (volatility) parameter.
Second, derivative type securities are concerned with payoffs under various states of the underlying asset and/or economic parameters. Thus, derivative type securities are instrumental in hedging against adverse financial payoffs associate with extreme (“tail”) events of the underlying process. Extreme (“tail”) events often have disproportionate financial consequences (positive or negative) relative to events near the expectation. Hence, efficient pricing of derivative type securities is relevant to anyone concerned with exposure and hedging against adverse financial payoffs due to various (extreme) states of the underlying process. Finally, examining fragility of the considered method with respect to the lattice density of end-states of the underlying process may reveal non-linearity of errors with respect to model parameters. Exposure of non-non-linearity of model errors with respect to parameter errors may enable market operators to further calibrate the model and reduce financial exposure to model parameter errors.
The ultimate objective of the thesis is to produce a process that estimates prices of Ameri-can type derivative securities accurately with respect to market prices. The author considers the objective to price to be different from the objective to value derivative type securities. The later implying intimate knowledge of the underlying process while the former implying minimization of error between model and market prices. Therefore, the method considered for the purpose of the thesis may lack theoretical elegance, nevertheless, retain numerical accuracy; which is the ultimate objective of the quantitative finance modeling practice. Hence, the research is striving to contribute to the existing literature of implied RND pricing of derivative type securities by producing a simple yet efficient implementation of the numerical method (algorithm) to efficiently price American type derivative securities. The considered method does not explicitly rely on the definition of the probabil-ity distribution function and the value of its scale parameter. Hence, the thesis attempts to contribute to the existing literature of pricing derivative type securities a method that accurately prices Ameri-can type derivative securities while avoiding the disagreement regarding the value of the distribution scale parameter of the underlying process.
The rest of the thesis is structured as follows. Section 2 reviews related literature. In particu-lar, Section 2 reviews issues relevant to the disagreement regarding distribution properties of the underlying process in the context of quantitative finance and pricing of derivative type securities. Further, implied RND estimation methods are reviewed. Their advantages and limitations are briefly discussed. Finally, literature relevant to the method considered for the purpose of the thesis is discussed. Section 3 discusses the thesis hypothesis and methodology, including the Mathematica code developed for the purpose of the thesis. Section 4 presents data and empirical results followed by the discussion. Section 5 discusses fragility of the considered method while Section 6 concludes the thesis.
Modern financial asset pricing models often require that the underlying asset price changes are well approximated by the Gaussian (the so-called normal) distribution. In turn, financial return and risk are defined in terms of the distribution domain of attraction and scale parameters respectively. In particular, the Gaussian distribution mean and standard deviation (volatility) parameters. The Gaus-sian distribution mean and volatility parameters are essential to the risk-return trade-off framework and financial models that are derived from the framework. For instance, the volatility parameter is among key inputs in the Black-Scholes-Merton (BSM) contingent claim pricing model, etc. Further, the volatility parameter is used to express other risk measures: value-at-risk (VaR) and conditional VaR (CVaR). Both VaR and CVaR are widely used for financial risk management (e.g. fixed-income) and investment decision making (e.g. CVaR portfolio optimization) purposes. However, empirical evidence suggests that changes of financial asset prices and economic parameter values are not well approximated by the Gaussian distribution hypothesis (Cont 2001 and Jondeau, Poon & Rockinger 2007).
Empirical distributions of change of financial asset prices and economic parameter values exhibit higher peakedness around the domain of attraction and contain higher probability mass in the tails than expected under the Gaussian distribution hypothesis. In other words, empirical distribu-tions exhibit higher concentration around the mean and greater extremes events (larger deviadistribu-tions from the mean) than expected under the Gaussian distribution hypothesis. Majority of research focused on improving fits of modern financial models to empirical data add parameters (e.g. stochas-tic volatility, jumps, exponential or power law tails, etc.) that require additional estimation. Further, majority of the improvements are sensitive to distribution assumptions of the underlying process. Disagreement, however, exists regarding the distribution properties of the underlying process of changes of financial asset prices and economic parameter values.
The disagreement may be broadly characterized by two positions (Taleb 2009). The first position is that the underlying process distribution scale parameter is well defined. In other words, as the number of observations increases, the distribution scale parameter converges to some finite value. Convergence property is statistically convenient. It implies that a sample distribution scale parameter may be used to efficiently estimate the population scale parameter. Therefore, statistical inference of the sample may be generalized to describe the population in a non-trivial way. Further, the underlying process distribution scale parameter may be modelled using parametric distributions for which cumulative (CDF) and probability mass (PDF) distribution functions are available in closed-form expressions (e.g. Gaussian or exponential Pareto distributions for the tails). The availability of closed-form expressions (analytical solutions) of the distribution CDF and PDF functions of the underlying process enable efficient application to model uncertainty within the risk-return framework.
The second position is that the underlying distribution scale parameter is not well defined, hence, does not converge to some finite value as the number of observation increases. Although in a finite sample the distribution scale parameter is always well defined (finite), the distribution scale parameter will differ from sample to sample and increase with the number of observation (not converge). Non convergence property is not statistically convenient. In particular, a sample distribu-tion scale parameter may not be used to efficiently estimate the populadistribu-tion scale parameter. There-fore, statistical inference of the sample may not be generalized to describe the population in a non-trivial way. The non-convergence property poses significant challenges to methods using infinite scale distributions (e.g. Pareto-Levy) to fit empirical distributions of changes of financial assets and other economic parameters. Closed-form expressions of the distribution CDF and PDF functions are often not available, except in limit cases (e.g. Cauchy distribution is the limit case of the Pareto-Levy distribution). The available characteristic expressions of the CDF and PDF functions may be applied to model uncertainty within the risk-return framework with significantly less efficiency relative to the closed-form alternatives. Hence, theoretical and practical applications require specialist knowledge and high performance computing.
Is the disagreement regarding the distribution properties of the underlying process relevant when pricing financial assets? Taleb (2009) proposes that finiteness or in-finiteness of the distribu-tion scale parameter is irrelevant in the context of quantitative finance. Broadly speaking, Taleb’s (2009) suggestion is motivated by divergence between the theoretical financial modeling practice and the quantitative finance practice. The theoretical financial modeling practice is often concerned with distribution properties of the underlying process. For instance, fitting probability density func-tions to historical log-returns of the underlying asset. In other words, theory is often concerned with “raw” probabilities. The quantitative finance practice is concerned with the ultimate financial conse-quence - the payoff (profit or loss) - which is measured in terms of, for instance, Euro and Euro-cents instead of probabilities. In other words, the “raw” probability is irrelevant in practical financial decision making without considering the financial payoff (profit or loss) associated with the “raw” probability.
The “true” probability distribution of an underlying process (and its properties) is not observ-able. Only discrete instance of the stochastic processes are observable via market prices. The observable market prices may be considered as realizations of the supply and demand of financial instruments concerned with various states of underlying assets and economic parameter values. Further, one may argue that actions of agents (e.g. supply and demand) and not the existence of financial theories and models create markets. Therefore, application of theoretical financial model-ing is not particularly useful beyond attempts to describe option tradmodel-ing and quantitative finance practice within a sterilized world assumptions; especially in the context of risk management. Taleb (2009) suggest that practitioners of quantitative finance develop heuristic processes and numerical methods that circumvent theoretical shortcomings and allow practitioners focus on every-day busi-ness of making the markets.
Hence, how to efficiently price financial assets without engaging in the disagreement regard-ing the distribution properties of the underlyregard-ing process? In particular, the central thesis question is how to price equity derivatives without making assumptions about finiteness or in-finiteness of the distribution scale parameter values? To answer the central thesis questions, a numerical method (algorithm) is considered. The objective of the considered method is to minimizing errors between model and market prices of American type derivative securities. The performance efficiency of the considered numerical method is compared to the performance efficiency of the (conventional) binomial model with constant historic volatility estimated using daily log-returns of the underlying security. The performance efficiency is defined in terms of mean errors (ME), mean absolute devia-tion (MAD) of ME and MAD of MAD of ME. Finally, fragility of the considered method is examined by perturbing the lattice density of end-states and their respective implied RND probabilities. Fragility is measured in terms of non-linearity of mis-pricing errors (between model and market prices) with respect to the lattice density.
The thesis research questions is motivated by the practical need to price derivative securi-ties consistently with market prices; while avoiding the disagreement regarding the distribution properties of the underlying process. Specifically, to minimize the mis-pricing error between model and market prices. Further, the thesis concern with pricing of (equity) derivative type securities is motivated by several factors. First, the price definition of equity derivative type security is an explicit function of the ultimate financial consequence - the payoff (profit or loss). The price of a derivative type security may be defined as the expected discounted payoff (Taleb, 2009). However, the price definition does not explicitly specify the distribution function of respective states of the underlying process. Consequentially, this allows some flexibility with respect to the probability distribution function. For instance, instead of specifying the distribution function explicitly, one may attempt to estimate the market implied RND probabilities as discrete instances. Therefore, examining pricing of derivative type securities from the implied RND approach may yield non-trivial insight into pricing of financial assets while avoiding the disagreement regarding properties of the underlying process. For instance, reduce mis-pricing between model and market prices relative to conventional models relying on the Gaussian scale (volatility) parameter.
Second, derivative type securities are concerned with payoffs under various states of the underlying asset and/or economic parameters. Thus, derivative type securities are instrumental in hedging against adverse financial payoffs associate with extreme (“tail”) events of the underlying process. Extreme (“tail”) events often have disproportionate financial consequences (positive or negative) relative to events near the expectation. Hence, efficient pricing of derivative type securities is relevant to anyone concerned with exposure and hedging against adverse financial payoffs due to various (extreme) states of the underlying process. Finally, examining fragility of the considered method with respect to the lattice density of end-states of the underlying process may reveal non-linearity of errors with respect to model parameters. Exposure of non-non-linearity of model errors with respect to parameter errors may enable market operators to further calibrate the model and reduce financial exposure to model parameter errors.
The ultimate objective of the thesis is to produce a process that estimates prices of Ameri-can type derivative securities accurately with respect to market prices. The author considers the objective to price to be different from the objective to value derivative type securities. The later implying intimate knowledge of the underlying process while the former implying minimization of error between model and market prices. Therefore, the method considered for the purpose of the thesis may lack theoretical elegance, nevertheless, retain numerical accuracy; which is the ultimate objective of the quantitative finance modeling practice. Hence, the research is striving to contribute to the existing literature of implied RND pricing of derivative type securities by producing a simple yet efficient implementation of the numerical method (algorithm) to efficiently price American type derivative securities. The considered method does not explicitly rely on the definition of the probabil-ity distribution function and the value of its scale parameter. Hence, the thesis attempts to contribute to the existing literature of pricing derivative type securities a method that accurately prices Ameri-can type derivative securities while avoiding the disagreement regarding the value of the distribution scale parameter of the underlying process.
The rest of the thesis is structured as follows. Section 2 reviews related literature. In particu-lar, Section 2 reviews issues relevant to the disagreement regarding distribution properties of the underlying process in the context of quantitative finance and pricing of derivative type securities. Further, implied RND estimation methods are reviewed. Their advantages and limitations are briefly discussed. Finally, literature relevant to the method considered for the purpose of the thesis is discussed. Section 3 discusses the thesis hypothesis and methodology, including the Mathematica code developed for the purpose of the thesis. Section 4 presents data and empirical results followed by the discussion. Section 5 discusses fragility of the considered method while Section 6 concludes the thesis.
Modern financial asset pricing models often require that the underlying asset price changes are well approximated by the Gaussian (the so-called normal) distribution. In turn, financial return and risk are defined in terms of the distribution domain of attraction and scale parameters respectively. In particular, the Gaussian distribution mean and standard deviation (volatility) parameters. The Gaus-sian distribution mean and volatility parameters are essential to the risk-return trade-off framework and financial models that are derived from the framework. For instance, the volatility parameter is among key inputs in the Black-Scholes-Merton (BSM) contingent claim pricing model, etc. Further, the volatility parameter is used to express other risk measures: value-at-risk (VaR) and conditional VaR (CVaR). Both VaR and CVaR are widely used for financial risk management (e.g. fixed-income) and investment decision making (e.g. CVaR portfolio optimization) purposes. However, empirical evidence suggests that changes of financial asset prices and economic parameter values are not well approximated by the Gaussian distribution hypothesis (Cont 2001 and Jondeau, Poon & Rockinger 2007).
Empirical distributions of change of financial asset prices and economic parameter values exhibit higher peakedness around the domain of attraction and contain higher probability mass in the tails than expected under the Gaussian distribution hypothesis. In other words, empirical distribu-tions exhibit higher concentration around the mean and greater extremes events (larger deviadistribu-tions from the mean) than expected under the Gaussian distribution hypothesis. Majority of research focused on improving fits of modern financial models to empirical data add parameters (e.g. stochas-tic volatility, jumps, exponential or power law tails, etc.) that require additional estimation. Further, majority of the improvements are sensitive to distribution assumptions of the underlying process. Disagreement, however, exists regarding the distribution properties of the underlying process of changes of financial asset prices and economic parameter values.
The disagreement may be broadly characterized by two positions (Taleb 2009). The first position is that the underlying process distribution scale parameter is well defined. In other words, as the number of observations increases, the distribution scale parameter converges to some finite value. Convergence property is statistically convenient. It implies that a sample distribution scale parameter may be used to efficiently estimate the population scale parameter. Therefore, statistical inference of the sample may be generalized to describe the population in a non-trivial way. Further, the underlying process distribution scale parameter may be modelled using parametric distributions for which cumulative (CDF) and probability mass (PDF) distribution functions are available in closed-form expressions (e.g. Gaussian or exponential Pareto distributions for the tails). The availability of closed-form expressions (analytical solutions) of the distribution CDF and PDF functions of the underlying process enable efficient application to model uncertainty within the risk-return framework.
The second position is that the underlying distribution scale parameter is not well defined, hence, does not converge to some finite value as the number of observation increases. Although in a finite sample the distribution scale parameter is always well defined (finite), the distribution scale parameter will differ from sample to sample and increase with the number of observation (not converge). Non convergence property is not statistically convenient. In particular, a sample distribu-tion scale parameter may not be used to efficiently estimate the populadistribu-tion scale parameter. There-fore, statistical inference of the sample may not be generalized to describe the population in a non-trivial way. The non-convergence property poses significant challenges to methods using infinite scale distributions (e.g. Pareto-Levy) to fit empirical distributions of changes of financial assets and other economic parameters. Closed-form expressions of the distribution CDF and PDF functions are often not available, except in limit cases (e.g. Cauchy distribution is the limit case of the Pareto-Levy distribution). The available characteristic expressions of the CDF and PDF functions may be applied to model uncertainty within the risk-return framework with significantly less efficiency relative to the closed-form alternatives. Hence, theoretical and practical applications require specialist knowledge and high performance computing.
Is the disagreement regarding the distribution properties of the underlying process relevant when pricing financial assets? Taleb (2009) proposes that finiteness or in-finiteness of the distribu-tion scale parameter is irrelevant in the context of quantitative finance. Broadly speaking, Taleb’s (2009) suggestion is motivated by divergence between the theoretical financial modeling practice and the quantitative finance practice. The theoretical financial modeling practice is often concerned with distribution properties of the underlying process. For instance, fitting probability density func-tions to historical log-returns of the underlying asset. In other words, theory is often concerned with “raw” probabilities. The quantitative finance practice is concerned with the ultimate financial conse-quence - the payoff (profit or loss) - which is measured in terms of, for instance, Euro and Euro-cents instead of probabilities. In other words, the “raw” probability is irrelevant in practical financial decision making without considering the financial payoff (profit or loss) associated with the “raw” probability.
The “true” probability distribution of an underlying process (and its properties) is not observ-able. Only discrete instance of the stochastic processes are observable via market prices. The observable market prices may be considered as realizations of the supply and demand of financial instruments concerned with various states of underlying assets and economic parameter values. Further, one may argue that actions of agents (e.g. supply and demand) and not the existence of financial theories and models create markets. Therefore, application of theoretical financial model-ing is not particularly useful beyond attempts to describe option tradmodel-ing and quantitative finance practice within a sterilized world assumptions; especially in the context of risk management. Taleb (2009) suggest that practitioners of quantitative finance develop heuristic processes and numerical methods that circumvent theoretical shortcomings and allow practitioners focus on every-day busi-ness of making the markets.
Hence, how to efficiently price financial assets without engaging in the disagreement regard-ing the distribution properties of the underlyregard-ing process? In particular, the central thesis question is how to price equity derivatives without making assumptions about finiteness or in-finiteness of the distribution scale parameter values? To answer the central thesis questions, a numerical method (algorithm) is considered. The objective of the considered method is to minimizing errors between model and market prices of American type derivative securities. The performance efficiency of the considered numerical method is compared to the performance efficiency of the (conventional) binomial model with constant historic volatility estimated using daily log-returns of the underlying security. The performance efficiency is defined in terms of mean errors (ME), mean absolute devia-tion (MAD) of ME and MAD of MAD of ME. Finally, fragility of the considered method is examined by perturbing the lattice density of end-states and their respective implied RND probabilities. Fragility is measured in terms of non-linearity of mis-pricing errors (between model and market prices) with respect to the lattice density.
The thesis research questions is motivated by the practical need to price derivative securi-ties consistently with market prices; while avoiding the disagreement regarding the distribution properties of the underlying process. Specifically, to minimize the mis-pricing error between model and market prices. Further, the thesis concern with pricing of (equity) derivative type securities is motivated by several factors. First, the price definition of equity derivative type security is an explicit function of the ultimate financial consequence - the payoff (profit or loss). The price of a derivative type security may be defined as the expected discounted payoff (Taleb, 2009). However, the price definition does not explicitly specify the distribution function of respective states of the underlying process. Consequentially, this allows some flexibility with respect to the probability distribution function. For instance, instead of specifying the distribution function explicitly, one may attempt to estimate the market implied RND probabilities as discrete instances. Therefore, examining pricing of derivative type securities from the implied RND approach may yield non-trivial insight into pricing of financial assets while avoiding the disagreement regarding properties of the underlying process. For instance, reduce mis-pricing between model and market prices relative to conventional models relying on the Gaussian scale (volatility) parameter.
Second, derivative type securities are concerned with payoffs under various states of the underlying asset and/or economic parameters. Thus, derivative type securities are instrumental in hedging against adverse financial payoffs associate with extreme (“tail”) events of the underlying process. Extreme (“tail”) events often have disproportionate financial consequences (positive or negative) relative to events near the expectation. Hence, efficient pricing of derivative type securities is relevant to anyone concerned with exposure and hedging against adverse financial payoffs due to various (extreme) states of the underlying process. Finally, examining fragility of the considered method with respect to the lattice density of end-states of the underlying process may reveal non-linearity of errors with respect to model parameters. Exposure of non-non-linearity of model errors with respect to parameter errors may enable market operators to further calibrate the model and reduce financial exposure to model parameter errors.
The ultimate objective of the thesis is to produce a process that estimates prices of Ameri-can type derivative securities accurately with respect to market prices. The author considers the objective to price to be different from the objective to value derivative type securities. The later implying intimate knowledge of the underlying process while the former implying minimization of error between model and market prices. Therefore, the method considered for the purpose of the thesis may lack theoretical elegance, nevertheless, retain numerical accuracy; which is the ultimate objective of the quantitative finance modeling practice. Hence, the research is striving to contribute to the existing literature of implied RND pricing of derivative type securities by producing a simple yet efficient implementation of the numerical method (algorithm) to efficiently price American type derivative securities. The considered method does not explicitly rely on the definition of the probabil-ity distribution function and the value of its scale parameter. Hence, the thesis attempts to contribute to the existing literature of pricing derivative type securities a method that accurately prices Ameri-can type derivative securities while avoiding the disagreement regarding the value of the distribution scale parameter of the underlying process.
The rest of the thesis is structured as follows. Section 2 reviews related literature. In particu-lar, Section 2 reviews issues relevant to the disagreement regarding distribution properties of the underlying process in the context of quantitative finance and pricing of derivative type securities. Further, implied RND estimation methods are reviewed. Their advantages and limitations are briefly discussed. Finally, literature relevant to the method considered for the purpose of the thesis is discussed. Section 3 discusses the thesis hypothesis and methodology, including the Mathematica code developed for the purpose of the thesis. Section 4 presents data and empirical results followed by the discussion. Section 5 discusses fragility of the considered method while Section 6 concludes the thesis.
2 Literature Review
2.1 The Puzzle: Underlying Process Distribution Scale Parameter Value
Application of the Gaussian distribution to describe changes of financial asset prices dates back to the doctoral thesis of Luis Bachelier (1900). Bachelier (1900) applied the stochastic Brownian motion - the so-called Brownian random walk - to describe changes of speculative prices of the Paris stock exchange. After translation of Bachelier’s work into English in 1964, use of the stochas-tic Brownian motion and the Gaussian distribution has become synonymous with modern quantita-tive finance theories and models. Theories that are closely related to the Bachelier’s work include the efficient market hypothesis (EMH), the capital asset pricing model (CAPM), the Black-Scholes-Merton dynamic hedging framework, etc. Common feature shared among the mentioned and other models is the use of the Gaussian distribution scale parameter - the standard deviation (volatility) - to define financial risk.
The Gaussian distribution scale parameter - the volatility - is used to describe variability of financial asset price changes around the expectation (mean). The Gaussian volatility parameter is among key input parameters in modern quantitative finance theories and models. For instance, the volatility parameter is used in attempts to measure financial risk in mean-variance portfolio optimiza-tion techniques (e.g. CAPM) and dynamically hedge financial risk (e.g. Black-Scholes-Merton opoptimiza-tion pricing model). Also, the volatility parameter is among key inputs in computing other measures that attempt to improve financial risk metrics, for instance, value-at-risk (VaR) and conditional VaR (CVaR). Both VaR and CVaR are widely used for risk-management purposes and investment decision making. Hence, the underlying process distribution scale parameter (e.g. the Gaussian volatility) is among essential parameters in quantitative finance theories and models based on the risk-return framework. Further, the Gaussian volatility parameter is synonymous with financial risk measure.
Properties of the stochastic Brownian motion and the Gaussian distribution are convenient. The Gaussian distribution moments are well defined. Hence, given a sufficient number of observa-tions one may generalize the sample statistics to describe the population properties. In other words, this enables to apply statistics outside of the sample in a non-trivial way. Further, well defined second moment of the Gaussian distribution (variance for mean-centered distribution), enables the use of the linear and non-linear optimization in the Euclidean (L2) norm. Therefore, econometric techniques based on the variance minimization (e.g. generalized least squares, GARCH and its derivatives, etc.) are well defined. Further, since sample statistics may be used to generalize out-side of the sample, the variance minimization econometric techniques may be applied also in non-trivial way. Nevertheless, empirical evidence suggests that the Gaussian distribution and the stochas-tic Brownian motion do not accurately approximate distribution of changes of financial asset prices and economic parameter values.
Research of Mitchell (1915), Olivier (1927), Mills (1927), Larson (1960), and Osbourne (1959 and 1962) indicated that the Gaussian distribution does not accurately approximate changes of financial asset prices and economic parameter values as hypothesized in Bachelier’s models. Mandelbrot (1963) rejected the Gaussian distribution hypothesis in favor of the Pareto-Levy distribu-tion hypothesis for changes of certain speculative prices (e.g. cotton, etc.). Consequentially, the stochastic Brownian motion is rejected in favor of the Levy flight. The Levy flight is a random walk process under the Pareto-Levy distribution. Previous and Mandelbrot’s (1963) research suggests that empirical distributions exhibit leptokurtosis. In other words, empirical distributions contain higher probability mass in the tails (more extreme deviation from the mean) than expected under the Gaussian distribution hypothesis.
The Pareto-Levy distribution may be characterized by its characteristic function and four parameters. The univariate characteristic function (cf) of i.i.d. stable random variables, x œ X, satisfies
Application of the Gaussian distribution to describe changes of financial asset prices dates back to the doctoral thesis of Luis Bachelier (1900). Bachelier (1900) applied the stochastic Brownian motion - the so-called Brownian random walk - to describe changes of speculative prices of the Paris stock exchange. After translation of Bachelier’s work into English in 1964, use of the stochas-tic Brownian motion and the Gaussian distribution has become synonymous with modern quantita-tive finance theories and models. Theories that are closely related to the Bachelier’s work include the efficient market hypothesis (EMH), the capital asset pricing model (CAPM), the Black-Scholes-Merton dynamic hedging framework, etc. Common feature shared among the mentioned and other models is the use of the Gaussian distribution scale parameter - the standard deviation (volatility) - to define financial risk.
The Gaussian distribution scale parameter - the volatility - is used to describe variability of financial asset price changes around the expectation (mean). The Gaussian volatility parameter is among key input parameters in modern quantitative finance theories and models. For instance, the volatility parameter is used in attempts to measure financial risk in mean-variance portfolio optimiza-tion techniques (e.g. CAPM) and dynamically hedge financial risk (e.g. Black-Scholes-Merton opoptimiza-tion pricing model). Also, the volatility parameter is among key inputs in computing other measures that attempt to improve financial risk metrics, for instance, value-at-risk (VaR) and conditional VaR (CVaR). Both VaR and CVaR are widely used for risk-management purposes and investment decision making. Hence, the underlying process distribution scale parameter (e.g. the Gaussian volatility) is among essential parameters in quantitative finance theories and models based on the risk-return framework. Further, the Gaussian volatility parameter is synonymous with financial risk measure.
Properties of the stochastic Brownian motion and the Gaussian distribution are convenient. The Gaussian distribution moments are well defined. Hence, given a sufficient number of observa-tions one may generalize the sample statistics to describe the population properties. In other words, this enables to apply statistics outside of the sample in a non-trivial way. Further, well defined second moment of the Gaussian distribution (variance for mean-centered distribution), enables the use of the linear and non-linear optimization in the Euclidean (L2) norm. Therefore, econometric techniques based on the variance minimization (e.g. generalized least squares, GARCH and its derivatives, etc.) are well defined. Further, since sample statistics may be used to generalize out-side of the sample, the variance minimization econometric techniques may be applied also in non-trivial way. Nevertheless, empirical evidence suggests that the Gaussian distribution and the stochas-tic Brownian motion do not accurately approximate distribution of changes of financial asset prices and economic parameter values.
Research of Mitchell (1915), Olivier (1927), Mills (1927), Larson (1960), and Osbourne (1959 and 1962) indicated that the Gaussian distribution does not accurately approximate changes of financial asset prices and economic parameter values as hypothesized in Bachelier’s models. Mandelbrot (1963) rejected the Gaussian distribution hypothesis in favor of the Pareto-Levy distribu-tion hypothesis for changes of certain speculative prices (e.g. cotton, etc.). Consequentially, the stochastic Brownian motion is rejected in favor of the Levy flight. The Levy flight is a random walk process under the Pareto-Levy distribution. Previous and Mandelbrot’s (1963) research suggests that empirical distributions exhibit leptokurtosis. In other words, empirical distributions contain higher probability mass in the tails (more extreme deviation from the mean) than expected under the Gaussian distribution hypothesis.
The Pareto-Levy distribution may be characterized by its characteristic function and four parameters. The univariate characteristic function (cf) of i.i.d. stable random variables, x œ X, satisfies
Application of the Gaussian distribution to describe changes of financial asset prices dates back to the doctoral thesis of Luis Bachelier (1900). Bachelier (1900) applied the stochastic Brownian motion - the so-called Brownian random walk - to describe changes of speculative prices of the Paris stock exchange. After translation of Bachelier’s work into English in 1964, use of the stochas-tic Brownian motion and the Gaussian distribution has become synonymous with modern quantita-tive finance theories and models. Theories that are closely related to the Bachelier’s work include the efficient market hypothesis (EMH), the capital asset pricing model (CAPM), the Black-Scholes-Merton dynamic hedging framework, etc. Common feature shared among the mentioned and other models is the use of the Gaussian distribution scale parameter - the standard deviation (volatility) - to define financial risk.
The Gaussian distribution scale parameter - the volatility - is used to describe variability of financial asset price changes around the expectation (mean). The Gaussian volatility parameter is among key input parameters in modern quantitative finance theories and models. For instance, the volatility parameter is used in attempts to measure financial risk in mean-variance portfolio optimiza-tion techniques (e.g. CAPM) and dynamically hedge financial risk (e.g. Black-Scholes-Merton opoptimiza-tion pricing model). Also, the volatility parameter is among key inputs in computing other measures that attempt to improve financial risk metrics, for instance, value-at-risk (VaR) and conditional VaR (CVaR). Both VaR and CVaR are widely used for risk-management purposes and investment decision making. Hence, the underlying process distribution scale parameter (e.g. the Gaussian volatility) is among essential parameters in quantitative finance theories and models based on the risk-return framework. Further, the Gaussian volatility parameter is synonymous with financial risk measure.
Properties of the stochastic Brownian motion and the Gaussian distribution are convenient. The Gaussian distribution moments are well defined. Hence, given a sufficient number of observa-tions one may generalize the sample statistics to describe the population properties. In other words, this enables to apply statistics outside of the sample in a non-trivial way. Further, well defined second moment of the Gaussian distribution (variance for mean-centered distribution), enables the use of the linear and non-linear optimization in the Euclidean (L2) norm. Therefore, econometric techniques based on the variance minimization (e.g. generalized least squares, GARCH and its derivatives, etc.) are well defined. Further, since sample statistics may be used to generalize out-side of the sample, the variance minimization econometric techniques may be applied also in non-trivial way. Nevertheless, empirical evidence suggests that the Gaussian distribution and the stochas-tic Brownian motion do not accurately approximate distribution of changes of financial asset prices and economic parameter values.
Research of Mitchell (1915), Olivier (1927), Mills (1927), Larson (1960), and Osbourne (1959 and 1962) indicated that the Gaussian distribution does not accurately approximate changes of financial asset prices and economic parameter values as hypothesized in Bachelier’s models. Mandelbrot (1963) rejected the Gaussian distribution hypothesis in favor of the Pareto-Levy distribu-tion hypothesis for changes of certain speculative prices (e.g. cotton, etc.). Consequentially, the stochastic Brownian motion is rejected in favor of the Levy flight. The Levy flight is a random walk process under the Pareto-Levy distribution. Previous and Mandelbrot’s (1963) research suggests that empirical distributions exhibit leptokurtosis. In other words, empirical distributions contain higher probability mass in the tails (more extreme deviation from the mean) than expected under the Gaussian distribution hypothesis.
The Pareto-Levy distribution may be characterized by its characteristic function and four parameters. The univariate characteristic function (cf) of i.i.d. stable random variables, x œ X, satisfies (1) fxHwL = EA‰ÂwxE = ExpBÂwd - †gw§aJ1 - Âb w †w§tanI pa 2MNF if a ¹≠ 1 ExpBÂwd - †gw§ J1 + Âb w †w§ p 2LnH†w§LNF if a = 1
where a, b, g, and d are the characteristic exponent, the index of skewness, the scale parameter, and the domain of attraction (Mandelbrot 1997, p. 446). Often the Pareto-Levy distribution is referred to as the stable distribution. This reference is closely linked to the property of stability (Mandlbrot 1997). In short, the stability property suggests that the sum of two (or more) Pareto-Levy distributed random variables is also Pareto-Levy distributed. Note that stability property also applies to the Gaussian distribution. Hence, both the Pareto-Levy and the Gaussian distributions are from the stable distribution family. In fact, the Pareto-Levy distribution generalizes the Gaussian distribu-tion.
When the characteristic parameter is exactly equal to two, a = 2 and b œ [-1, 1], the character-istic function of the Pareto-Levy distribution become the Fourier transform of the Gaussian probabil-ity distribution function (Mandelbrot 1997, p. 246). The characteristic exponent, a œ (0, 2], deter-mines the total probability mass contained in the tails of the distribution (Mandelbrot 1997, p. 446). In other words, it determines the rate at which the tails of the distribution tapper off (McCulloch 1986). The skewness parameter, b œ [-1, 1] determines the degree of asymmetry of the distribution with respect to its domain of attraction, d œ R. When b = 0, the distribution is symmetric around the domain of attraction, d. The distribution scale parameter, g œ (0, ¶), determines the dispersion of the distribution with respect to the domain of attraction, d. In other words, the scale parameter, g, determines how compressed or stretched the distribution is with respect to its domain, d. The domain of attraction may be interpreted as the center of the distribution - the mean. Figure 1 below illustrates shapes of the symmetric Pareto-Levy probability mass distribution function (PDF) under different characteristic exponent, a, parameter values.
Notice that as the characteristic parameter, a, increases (e.g. from 0.5 to 2.0 with incre-ments of 0.5), the distribution becomes less peaked and contains a smaller probability mass in the tails. When a œ [1, 2), the distribution moments beyond the first (e.g. mean) are not well defined; do not converge to finite values. Moreover, when a œ (0, 1), moments beyond the zero moment (e.g. random variable instance) are not well defined. This implies that the population mean does not converge to some finite value. Therefore, the population distribution is approaching the domain of attraction asymptotically yet never converging. Significant disadvantage of the Pareto-Levy distribu-tion is that analytic expressions of CDF and PDF funcdistribu-tions are available only for the Gaussian and Cauchy distributions (Koutrouvelis 1981, p. 918). For a detailed review of properties of the Pareto-Levy distribution and its application to financial modeling refer to Rachev & Mittnik (2000).
where a, b, g, and d are the characteristic exponent, the index of skewness, the scale parameter, and the domain of attraction (Mandelbrot 1997, p. 446). Often the Pareto-Levy distribution is referred to as the stable distribution. This reference is closely linked to the property of stability (Mandlbrot 1997). In short, the stability property suggests that the sum of two (or more) Pareto-Levy distributed random variables is also Pareto-Levy distributed. Note that stability property also applies to the Gaussian distribution. Hence, both the Pareto-Levy and the Gaussian distributions are from the stable distribution family. In fact, the Pareto-Levy distribution generalizes the Gaussian distribu-tion.
When the characteristic parameter is exactly equal to two, a = 2 and b œ [-1, 1], the character-istic function of the Pareto-Levy distribution become the Fourier transform of the Gaussian probabil-ity distribution function (Mandelbrot 1997, p. 246). The characteristic exponent, a œ (0, 2], deter-mines the total probability mass contained in the tails of the distribution (Mandelbrot 1997, p. 446). In other words, it determines the rate at which the tails of the distribution tapper off (McCulloch 1986). The skewness parameter, b œ [-1, 1] determines the degree of asymmetry of the distribution with respect to its domain of attraction, d œ R. When b = 0, the distribution is symmetric around the domain of attraction, d. The distribution scale parameter, g œ (0, ¶), determines the dispersion of the distribution with respect to the domain of attraction, d. In other words, the scale parameter, g, determines how compressed or stretched the distribution is with respect to its domain, d. The domain of attraction may be interpreted as the center of the distribution - the mean. Figure 1 below illustrates shapes of the symmetric Pareto-Levy probability mass distribution function (PDF) under different characteristic exponent, a, parameter values.
Notice that as the characteristic parameter, a, increases (e.g. from 0.5 to 2.0 with incre-ments of 0.5), the distribution becomes less peaked and contains a smaller probability mass in the tails. When a œ [1, 2), the distribution moments beyond the first (e.g. mean) are not well defined; do not converge to finite values. Moreover, when a œ (0, 1), moments beyond the zero moment (e.g. random variable instance) are not well defined. This implies that the population mean does not converge to some finite value. Therefore, the population distribution is approaching the domain of attraction asymptotically yet never converging. Significant disadvantage of the Pareto-Levy distribu-tion is that analytic expressions of CDF and PDF funcdistribu-tions are available only for the Gaussian and Cauchy distributions (Koutrouvelis 1981, p. 918). For a detailed review of properties of the Pareto-Levy distribution and its application to financial modeling refer to Rachev & Mittnik (2000).
Figure 1. Symmetric PDF of the Pareto-Levy Distribution
a = 0.5 a = 1.0 a = 1.5 a = 2.0 -10 -5 0 5 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Random Variable P roba bi li ty D ens it y
The Pareto-Levy distribution better captures properties exhibited by empirical distributions of changes of financial asset prices and other economic parameter values. In particular, higher concen-tration around the mean and larger extreme deviations from the mean. Mandelbrot (1963 and 1997) proposed that distributions of changes of financial asset prices and economic parameter values are well approximated by the Levy distribution with infinite moments beyond the first (the Pareto-Levy-Mandelbrot hypothesis, a œ (1, 2)). In other words, the Pareto-Pareto-Levy-Mandelbrot hypothesis implies that empirical distributions have well defined domains of attraction (mean) and not well defined variance, hence standard deviation (volatility). Therefore, standard statistic measures and techniques based on the Gaussian distribution standard deviation (e.g., covariance, generalized least squares) are not well defined. Figure 2 below illustrates non-convergence property of the distribution second moment (variance) of the S&P500 index historical daily log-returns and the expected send moment performance under the Pareto-Levy distribution hypothesis, including the Gaussian hypothesis limit case.
The top graph summarizes the cumulative second moment of daily log-returns of the S&P500 index between 3 January 1950 and 24 May 2014. Horizontal axis denotes the number of observations (total of 16201 observations) while the vertical axis denotes the cumulative non-centered second moment of the sample scaled by the size of the sample. In other words, the sam-ple Gaussian variance. Notice that as the number of observations increases, the empirical distribu-tion cumulative second moment also increases. The daily Gaussian volatility of the S&P500 index between 3 January 1950 and 24 May 2014 was 0.9744% (to 4 d.p.); See Appendix 1 for summary statistics of the S&P500 index price changes. In other words, as the sample size increases the Gaussian variance, hence, standard deviation, also increases. The bottom left graph summarizes the expected cumulative second moment performance under the Gaussian distribution fit. In con-trast to empirical returns (top graph), the cumulative second moment under the Gaussian distribu-tion hypothesis is well defined. As the number of observadistribu-tions increases the second moment con-verges (quickly) to a well behaved value. The bottom right graph summarizes the expected second moment performance under the Levy distribution fit. The second moment under the Pareto-Levy distribution hypothesis does not exhibit as well defined behavior as under the Gaussian distribu-tion hypothesis. As the number of observadistribu-tions increases, the cumulative second moment also increases. Even if convergence to some well defined value exists, it is (very) slow and may take infinite number of observations to eventually converge.
