Preferred gradients : evidence of emergent symmetry in financial markets

**Authors:**Beukes, Gideon Jacobus**Date:**2012-06-19**Subjects:**Preferred Gradient Hypothesis , Random Walk Hypothesis , Symmetry (Mathematics) , Gradients , Financial instruments , Technical analysis (Investment analysis)**Type:**Mini-Dissertation**Identifier:**uj:8769 , http://hdl.handle.net/10210/5120**Description:**M.Comm. , The dependence claim of the Random Walk Hypothesis, formulated by Louis Bachelier in 1900, states that asset price movement is based on a stochastic process (Shafer & Vovk, 2001) in which subsequent prices in a series are random in relation to previous ones (Malkiel, 2003:3). Malkiel (1973) identifies a major implication of the Random Walk Hypothesis, namely that future directions cannot be extrapolated from past actions. In the context of financial markets, the implication is a rejection of the notion that fundamental and technical analyses have value as instruments of investment analysis. The Preferred Gradient Hypothesis, developed by Daan Joubert, an independent technical analyst, rejects the dependence claim of the Random Walk Hypothesis by suggesting the existence of a form of regularity in asset price movement. The regularity, based on the notion that trend reversals tend to occur along preferred gradients, is claimed to manifest as an emergent phenomenon with its origin in the complex, self-adaptive nature of the financial markets. The phenomenon is claimed to manifest across time scales and financial instruments. Three objectives are formulated for the purposes of the study, each relating to a specific claim of the Preferred Gradient Hypothesis. The primary objective is to verify objectively whether the presence of preferred gradients on price charts can be shown to be statistically significant. The secondary objectives are to verify objectively whether the presence of preferred gradients across financial instruments and time scales can be shown to be statistically significant. The samples employed in the study are drawn from two populations and consist of two components, namely the exchange rates of a set of ten currencies against the United States Dollar and the prices of five commodities – Brent crude oil, ethanol, gold, silver and soybeans. The sample data, which consist of daily commodity prices and exchange rates, are converted to datasets spanning daily, weekly and monthly timescales by means of a custom-developed software application entitled the Preferred Gradient Hypothesis Workbench. A subset of each dataset is selected for analysis based on the criterion that the particular subset exhibits enough price variation to enable sensible analysis. Two charts are constructed for each dataset, namely a control (R) chart and a treated (P) chart by using the Preferred Gradient Hypothesis-prescribed method. Each of the charts is analysed by the Preferred Gradient Hypothesis Workbench and the total number of intersections between trend lines and trend reversals, recorded. A trend line is deemed to have intersected a trend reversal if it passes the reversal within a distance of one percent of the total range of the chart’s Y-axis while intersecting the chart’s X-axis at the same point as the reversal. The number of intersections on each pair of control and treated charts are stored as datasets to which the t-Test for Paired Samples is applied. Statistical analysis based on the t-Test for Paired Samples with an alpha value of 0.05 results in the research hypothesis being rejected for the daily and weekly exchange rate datasets as well as for the daily commodity price dataset, but not for the monthly exchange rate dataset. Evaluation of these results, in terms of the tests formulated for each research objective, indicates that the notion that the presence of preferred gradients on price charts is statistically significant cannot be rejected. The notions that its presence is statistically significant over different time scales and financial instruments are, however, rejected. The statistical analysis is followed by a discussion of the research results in which the results are interpreted within the context of the research objectives. The section also features a discussion regarding problems that were encountered during the research and software development processes. The study is concluded with suggestions regarding methodological improvements as well as the identification of topics relating to the research subject which may be investigated in future research projects.**Full Text:**

**Authors:**Beukes, Gideon Jacobus**Date:**2012-06-19**Subjects:**Preferred Gradient Hypothesis , Random Walk Hypothesis , Symmetry (Mathematics) , Gradients , Financial instruments , Technical analysis (Investment analysis)**Type:**Mini-Dissertation**Identifier:**uj:8769 , http://hdl.handle.net/10210/5120**Description:**M.Comm. , The dependence claim of the Random Walk Hypothesis, formulated by Louis Bachelier in 1900, states that asset price movement is based on a stochastic process (Shafer & Vovk, 2001) in which subsequent prices in a series are random in relation to previous ones (Malkiel, 2003:3). Malkiel (1973) identifies a major implication of the Random Walk Hypothesis, namely that future directions cannot be extrapolated from past actions. In the context of financial markets, the implication is a rejection of the notion that fundamental and technical analyses have value as instruments of investment analysis. The Preferred Gradient Hypothesis, developed by Daan Joubert, an independent technical analyst, rejects the dependence claim of the Random Walk Hypothesis by suggesting the existence of a form of regularity in asset price movement. The regularity, based on the notion that trend reversals tend to occur along preferred gradients, is claimed to manifest as an emergent phenomenon with its origin in the complex, self-adaptive nature of the financial markets. The phenomenon is claimed to manifest across time scales and financial instruments. Three objectives are formulated for the purposes of the study, each relating to a specific claim of the Preferred Gradient Hypothesis. The primary objective is to verify objectively whether the presence of preferred gradients on price charts can be shown to be statistically significant. The secondary objectives are to verify objectively whether the presence of preferred gradients across financial instruments and time scales can be shown to be statistically significant. The samples employed in the study are drawn from two populations and consist of two components, namely the exchange rates of a set of ten currencies against the United States Dollar and the prices of five commodities – Brent crude oil, ethanol, gold, silver and soybeans. The sample data, which consist of daily commodity prices and exchange rates, are converted to datasets spanning daily, weekly and monthly timescales by means of a custom-developed software application entitled the Preferred Gradient Hypothesis Workbench. A subset of each dataset is selected for analysis based on the criterion that the particular subset exhibits enough price variation to enable sensible analysis. Two charts are constructed for each dataset, namely a control (R) chart and a treated (P) chart by using the Preferred Gradient Hypothesis-prescribed method. Each of the charts is analysed by the Preferred Gradient Hypothesis Workbench and the total number of intersections between trend lines and trend reversals, recorded. A trend line is deemed to have intersected a trend reversal if it passes the reversal within a distance of one percent of the total range of the chart’s Y-axis while intersecting the chart’s X-axis at the same point as the reversal. The number of intersections on each pair of control and treated charts are stored as datasets to which the t-Test for Paired Samples is applied. Statistical analysis based on the t-Test for Paired Samples with an alpha value of 0.05 results in the research hypothesis being rejected for the daily and weekly exchange rate datasets as well as for the daily commodity price dataset, but not for the monthly exchange rate dataset. Evaluation of these results, in terms of the tests formulated for each research objective, indicates that the notion that the presence of preferred gradients on price charts is statistically significant cannot be rejected. The notions that its presence is statistically significant over different time scales and financial instruments are, however, rejected. The statistical analysis is followed by a discussion of the research results in which the results are interpreted within the context of the research objectives. The section also features a discussion regarding problems that were encountered during the research and software development processes. The study is concluded with suggestions regarding methodological improvements as well as the identification of topics relating to the research subject which may be investigated in future research projects.**Full Text:**

Symmetry methods and conservation laws applied to the Black-Scholes partial differential equation

**Authors:**McDonald, Ruth Leigh**Date:**2012-07-03**Subjects:**Applied mathematics , Symmetry (Mathematics) , Conservation laws (Mathematics) , Differential equations , Differential equations, Partial**Type:**Thesis**Identifier:**uj:8783 , http://hdl.handle.net/10210/5141**Description:**M.Sc. , The innovative work of Black and Scholes [1, 2] extended the mathematical understanding of the options pricing model, beginning the deliberate study of the theory of option pricing. Its impact on the nancial markets was immediate and unprecedented and is arguably one of the most important discoveries within nance theory to date. By just inserting a few variables, which include the stock price, risk-free rate of return, option's strike price, expiration date, and an estimate of the volatility of the stock's price, the option-pricing formula is easily used by nancial investors. It allows them to price various derivatives ( nancial instrument whose price and value are derived from the value of assets underlying them), including options on commodities, nancial assets and even pricing of employee stock options. Hence, European1 and American2 call or put options on a non-dividend-paying stock can be valued using the Black-Scholes model. All further advances in option pricing since the Black-Scholes analysis have been re nements, generalisations and expansions of the original idea presented by them.**Full Text:**

**Authors:**McDonald, Ruth Leigh**Date:**2012-07-03**Subjects:**Applied mathematics , Symmetry (Mathematics) , Conservation laws (Mathematics) , Differential equations , Differential equations, Partial**Type:**Thesis**Identifier:**uj:8783 , http://hdl.handle.net/10210/5141**Description:**M.Sc. , The innovative work of Black and Scholes [1, 2] extended the mathematical understanding of the options pricing model, beginning the deliberate study of the theory of option pricing. Its impact on the nancial markets was immediate and unprecedented and is arguably one of the most important discoveries within nance theory to date. By just inserting a few variables, which include the stock price, risk-free rate of return, option's strike price, expiration date, and an estimate of the volatility of the stock's price, the option-pricing formula is easily used by nancial investors. It allows them to price various derivatives ( nancial instrument whose price and value are derived from the value of assets underlying them), including options on commodities, nancial assets and even pricing of employee stock options. Hence, European1 and American2 call or put options on a non-dividend-paying stock can be valued using the Black-Scholes model. All further advances in option pricing since the Black-Scholes analysis have been re nements, generalisations and expansions of the original idea presented by them.**Full Text:**

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