Trading Financial Market s Fractal behaviour by Solon Saoulis CEO DelfiX ltd. (delfix.co.uk) Introduction In 1975, the noted mathematician Benoit Mandelbrot coined the term fractal (fragment) to define repetitively scaled, self-similar geometric formations in nature (as seen in ferns, trees, rivers, blood vessels etc). We have discovered that the prices of financial assets display persistent, stable and systematic dynamic characteristics within each specific timeframe. We have also identified that this price behaviour reappears in all timeframes as we zoom in, from very slow to very fast time periods, which is typical of fractal systems. In 2004, we devised a system, that follows price dynamic characteristics, by using an L shaped geometric pattern called the L-Wave. By adjusting the scaling of the L-Wave we capture the price dynamics in any timeframe. Using an optimized L-Wave as a potential dynamic footprint of the future expected price action, we can predict and trade any asset. The L-Wave s geometrical shape provides us with a comprehensive technical framework required for informed trading and risk control. Following the optimized L-Wave while tracking the market, we generate returns well above the average expected returns. Fractal Nature of the Financial Markets Mandelbrot used line multi-fractal techniques, to generate mimics of price data in order to calculate forecasting probabilities. In essense the methods published by Mandelbrot are great data fitting exercises, but give limited market behavioral insight. In contrast to Mandelbrot s method, we are using the L-Wave as a price fractal discovery process in a continous and systematic manner. Market volatility acts on the price as a diffusion process such that longer time horizons result in larger fluctuations. Trading in Financial markets is characterized by the investment holding period and risk profile. An investor looking to benefit from smaller moves such as a weekly or intraday fluctuation is aiming for smaller price returns with low risk. A pension investment fund that looks to earn returns in an equity or bond holding is hoping for large price returns over a longer time horizon. Financial charts present price information in bar form (see EUR/USD charts below). A single monthly bar will show the range (high to low) of the price action over a monthly period. Thus, the 24 monthly bars show a monthly price history summary over 2 years. Likewise 24 weekly chart bars show a history summary of approximately 6 months, 24 Daily bars about a month and 24 hourly bars 1 day etc. Keeping our chart window constant at 24 bars we can see that, as timeframe selection is zoomed into smaller bar periods, the average bar size is also reduced due to lower fluctuation (this is no surprise to option traders, as expected future average price, is estimated by multiplying initial 1 P a g e
price times the annual volatility times the square root of the normalized time of the holding period). Typical price bar fluctuation numbers (Average True Range) for the EUR/USD pair are shown below: Table 1: EUR/USD Average True Range Monthly Chart Weekly Chart Daily Chart Hourly Chart 540 pips 250 pips 133 pips 17 pips A fractal is a geometric figure that is divided into smaller versions of itself following a set of rules. Therefore, as the price bars become smaller as the time scale is reduced, the chart system lends itself to geometric fractal analysis. Dynamic Patterns Dynamics studies the way objects or dynamic systems respond to inputs, using calculus invented by Newton and Leibnitz. Although the maths behind Newtonian dynamics require continuous differentiability (i.e. smooth continous transitions from state to state) that prices do not exhibit, the observed price action takes forms similar to dynamic systems responses to inputs. A model of a dynamic system responds to a fixed continuous input like the market when surprise news comes out. Figure 1 below represents a lightly damped dynamic system response to a fixed persistent input. The top level it reaches is called the overshoot level and where the system settles is its steady state. The equivalent market behaviour would occur if there is low liquidity and price overreacts on the surprise news initially, only to correct back to its steady state after some price oscillation. When a system is highly damped, it exhibits the behaviour captured by Figure 2. Although this time we do not have overshoot, the system settles at the same level as the lightly damped system. The equivalent interpretation for this behaviour in the market is that there is a lot of liquidity and the market absorbs the news smoothly, getting to its steady state objective gradually. Fig. 1 Lightly damped dynamic system Fig. 2 Heavily damped dynamic system 2 P a g e
We have identified the above dynamic behaviour to be persistent in all financial price time series and in all timeframes (deviations from this behaviour occur only during periods of very low price activity). These are the signatures of the fractal behaviour of the price action which the L-Wave described below is designed to capture. The L-Wave Creation Having identified the fractal nature of the market and its displayed dynamic behaviour we construct the L-Wave to capture both these characteristics. Although we can demonstrate that the L-Wave can be deemed as a fractal itself, we will concenrate here with the inner workings of the L-Wave that will help us understand its construction. Using a similar process to that of creating fractals (i.e. like starting from the leaf to create the fern), the following steps are needed : 1. L-Wave Initiator 2. L-Wave Generator 3. L-Wave Recursion rules 1. L-Wave Initiator The rhombus shape below is the fractal unit used to generate the L-Wave step=fn(volatility,timeframe) slope= fn(momentum) The step of the initiator is based on the volatility of any specific timeframe. The slope can be calculated from the average momentum or any other method. 2.L-Wave Generator Using the fractal initiator as a building block, we can generate the two fundamental L-Waves needed to model the market. 3 P a g e
up L-Wave down L-Wave 3.L-Wave Recursion Rules Unlike Mandelbrot s fractal techniques which attempt to fit the entire market data series within broken line fractals, we use the L-Wave to continuously track market behaviour. Using a rule based on when the price breaks out of the L-Wave area, we determine the rule of recursion. For example, below we have an up L-Wave followed by a down L-Wave after breakout. Price Break out of up L-Wave 4 P a g e
Trading an Optimized L-Wave A specific parameterization of the L-Wave that is used for trading by DelfiX is the SolonWave TM. The SolonWave TM is an L-Wave designed to capture a market s dynamic behaviour in a unique and profitable way. In doing so, it renders all the trading information that is needed during a trading session. In the simulation of the response of a dynamic system to a sudden input below, the Overshoot point is captured by the level of the top left corner of the up pattern and the level below where the market settles is called the objective of the SolonWave TM. The SolonWave TM is automatically calibrating each asset in a specific timeframe in a way that captures its dynamic behaviour characteristics within that timeframe. The DelfiX calculations map market volatility characteristics to the unique shape of the SolonWave TM. This mechanism renders the active trading channel (width and slope), its direction and the target price levels of the move. The SolonWave TM construction allows for anchoring new extension waves so that move continuation and higher price volatility can be accommodated. In doing so, the SolonWave TM offers the framework within which successful trading decisions can be made in a variable volatility environment. The top corners of the upward wave are depicting the market critical levels of the underlying move. The first point of the upward move is called the Objective and signifies the first profit taking point. The next levels are the Overshoot and the Extreme (extreme overshoot point) where the market is expected to encounter strong resistance (or strong support for a downward wave). Breaking through the extreme signifies continuation of the move in the same direction. 5 P a g e
Trading the SolonWave TM Examples Below we show some charts that demonstrate the effectiveness of the DelfiX system. Each price bar on the chart represents the price activity within a unit of the timeframe (e.g. 1 hour for the hourly chart, 1 day for the Daily chart). The red patterns (related to red bars) are the selling waves and the green patterns (associated with green bars) are the buying waves. The pink solid line is the price's exponential average and the dotted blue line is the pattern's break reversal line. In order to minimize noise induced false breaks, the blue dotted line adds an extra level of optimized protection away from the boundary. When the market closes past the blue dotted line, the system changes direction. Green bars illustrate that the system is long (i.e. profiting from rising prices) and red bars that the system is short (i.e. profiting from falling prices). The last pattern is highlighted (red in this example) to illustrate the active pattern. The continuous switching between different up and down channels is executed automatically. Switching between up and down channels locks the position s profit (or loss) and provides the system with a systematic performance measure that characterizes its quality. In order to demonstrate our method and its fractal nature, we are zooming out from intraday hourly charts to the Daily and Weekly analysis focusing on the EUR/USD pair. A single timeframe EUR/USD hourly chart Note in the above chart, how the dynamic behaviour follows the objective and overshoot characteristics described above. Prices go to the objective most of the time and when they reach the overshoot they bounce back to the objective before the next action comes to continue or terminate the move. Additionally when the trading channel is broken (past the blue line) it is a confirmed change of direction which holds (usually with profit) until the next break. 6 P a g e
Multi-timeframe Daily overlay on EUR/USD hourly chart Above we can see the bigger picture. The boundary of the Daily Overlay is the area that guides the Hourly market pointing at the Daily objective level further down. Multi-timeframe Weekly overlay on EUR/USD daily chart Zooming out of the Daily system into the Weekly system we discover why the market has stopped moving further up (after hitting the boundary of the Weekly up wave) and that the Daily is pointing below the boundary of the weekly, threatening a reversal on the weekly chart. 7 P a g e
SolonWave TM Performance Results Table 2 below shows the consistency and effectiveness of the SolonWave TM across many important timeframes. The Wins% ratio indicates the number of successful trades out of 100 and is a useful indicator for confidence in the decision making of the system. The 3yr returns illustrate the returns on capital over 3 consecutive years and the Sharpe ratio is a measure of the returns given their volatility. The Hits demonstrate the effectiveness of the SolonWave TM in capturing the critical market levels. This gives us a measure of success and confidence in the profit taking decisions during trading. We can conclude that based on these results the Hourly and Daily are more of trading timeframes, as the rapidly declining probabilities of the Overshoot and Extreme levels, help the profit taking decisions. The Weekly and Monthly timeframes are more akin to long term investment decision timeframes as there is more directional behaviour based on the way the hits are achieved. Table 2: EUR/USD Performance Statistics How Investors can use the SolonWave TM Which asset? The SolonWave TM is a directional tool that works on all assets like FX, Equities, Commodities and Bonds. In our view it should work on any tradable asset whether this is a spread or an equity ratio. Different markets display different sensitivities and the automatic scaling of the SolonWave TM captures this behaviour better on some assets than others. Using SolonWave TM s effectiveness we can select a portfolio of assets which display better dynamic behaviour. We have categorized assets based on their Sharpe ratio results. This together with the wins% gives us a good indication of which assets display good dynamic behaviour for trading purposes and for constructing our portfolio. Which Timeframe? Choosing the appropriate timeframe should include consideration of the average fluctuation numbers given in table 1. The bigger the fluctuation the bigger the risk when deciding to switch a 8 P a g e
position around. For example, the monthly 540 pips is a very large risk to take on in order to follow such a timeframe. Additionally, in order to minimize slippage we require liquid assets that display continuous behaviour within the selected timeframe. For example FX is a continuous market and therefore usually no gaps appear in the data. However, the equity markets do not trade continuously and therefore gaps affect their performance. To manage gaps better, one may need to look at investment horizons like the weekly analysis and adjust the position size to accommodate higher fluctuations. Portfolio Management Due to its stability characteristics the SolonWave TM renders itself for managing portfolios of assets. Below we display the results obtained by running an equally weighted portfolio of five FX pairs. The low drawdowns associated with this portfolio make it perfect for trading. Trading Options The SolonWave TM is an ideal tool for managing options trading. The low frequencies such as the Daily and Weekly can help investors to decide on the direction of their trade. Whereas the critical levels (objective, overshoot and extreme) and their probabilities, can be used as reference to select the strike of the option (especially for binaries). The higher frequencies, such as the hourly or faster, can be used for managing the residual hedged risk of the option (delta-gamma). 9 P a g e
Conclusions Human nature behaves naturally in dynamic ways and over time we have developed an implicit sense for interpreting financial dynamic information in a way that is reflected into the price action. This behaviour can be captured by an assimilated dynamic mechanism like the L-Wave. We have demonstrated that using fractal dynamics concepts incorporated into the L-Wave, traders and investors can benefit across all timeframes. This new technology adds a geometrical and strongly visual solution to the toolbox of financial engineering. The fractal nature of the L-Wave makes it appropriate for trading multiple timeframes and for helping to run a well-managed portfolio of multiple assets. References The (mis)behaviour of Markets, Benoit B. Mandelbrot (2004) The Complete Guide to Option Pricing Formulas, Espen Gaarder Haug (1998) 10 P a g e