A Simple Model for Intra-day Trading

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6 3.1 Defining directional change and overshoot Let {x(t): t R + } be the price curve and let x 0 = x(0) be the price at time t x dc 0 = t 0 = 0. To define a directional change one first has to chose a directional change threshold x dc ; it can be a chosen in a form of a percentage price change or a price change (for the sake of simplicity, we will consider the case when a price change is chosen), then a time t x dc 1 is found when the price changed by the threshold x dc. It can be the case that the price change of size x dc is an upward or a downward move, lets assume that the price change of size x dc was an upward move (the reasoning in case it was a downward move is analogous). The time when this price change of size x dc happened is noted, that is t x dc 1 = inf{t R + : x(t) = x 0 + x dc }. The time period [t x dc 0, t x dc 1 ] is called the directional change phase and the price change from x(t x dc 0 ) to x(t x dc 1 ) is called the directional change of size x dc. We now find the next downward price change of size x dc from the last extremum, namely the maxima. To be specific, we are searching for times t x dc 1, t x dc 2 defined in the following way (t x dc 2, t x dc 3 ) = {(s, t) t x dc 1, [s, : x(t) = x(s) x dc, x(s) = max{x(r): r t x dc 1, t }}. The time period [t x dc 1, t x dc 2 ] is called the overshoot phase and the price price change from x(t x dc 1 ) to x(t x dc 2 ) is called the overshoot associated with the price change from x(t x dc 0 ) to x(t x dc 1 ), which is in fact, as defined before, the directional change of size x dc. Again, the time period [t x dc 2, t x dc 3 ] is the directional change phase of size x dc, but now it was a downward price change of size x dc. We iterate this process further, to be specific, since the last directional change was an downward move, we are now searching for an upward move of size x dc. Now we seek for times t x dc 4, t x dc 5 defined in the following way (t x dc 4, t x dc 5 ) = {(s, t) t x dc 3, [s, : x(t) = x(s)+ x dc, x(s) = min{x(r): r t x dc 3, t }}. The time period [t x dc 3, t x dc 4 ] is the overshoot phase and the corresponding price change from x(t x dc 3 ) to x(t x dc 4 ) is the overshoot, similarly the time period [t x dc 4, t x dc 5 ] is the directional change phase and the price change from x(t x dc 4 ) to x(t x dc 5 ) is the directional change. Notice that the overshoot phase and the overshoot can be identified only when the next directional change (or equivalently directional change phase) is found. Intrinsic time for the threshold x dc is defined as set of times {t x dc k : k 2N + 1}, which correspond to the (physical) times when each of the directional change phases has ended. 3.2 The analysis To analyze the simulated price a set of thresholds {( x dc ) i : i = 1,..., n} is chosen and the price curve is dissected into directional change/overshoot phase. The smallest threshold ( x dc ) 1 is chosen to equal to the smallest percentage price change, that is { } x(t) x(s) ( x dc ) 1 = min x(s) : (s, t) R2 +, s < t. 6

9 where n N. When it is clear from the context, on which set counting is being done, we will drop the N( )[x] notation and write only N( ). Let f : R R be a function, we say that f has a scaling property if the following holds f(ax) f(x), x R where a R. A notable example of function with scaling law property is f(x) = c x α, x R where c, a R. Currently, there are 22 scaling laws found in the currency markets. An extensive list can be found in Dupuis, Glattfelder, Olsen. In this model we have concentrated only on five of the 22 mentioned. We list the them; average absolute price change sampled over time interval t: x p = ( ) t Ex(p), p = 1, 2 C x (p) average maximal price range over time interval t: x max p = ( ) t Emax(p), p = 1, 2 C max (p) number of directional changes to the directional threshold x dc : N( x dc ) = ( ) EN,dc xdc C N,dc These scaling laws are used for determining if the price curve is realistic or not. They are also used for normalizing data, such as position information, absolute price change, obtained from different (physical or intrinsic) time. At this stage of the research, there is no attempt being made to try to find the model parameters that would produce a price curve such that the parameters (namely the exponents) of the scaling laws would match the ones obtained from the real price curve. 5 Comment The challenges remain to find intuitive explanations for the patterns found, to relate this patterns to the structure of the model and to differentiate the findings between real dynamics and artifacts. References [1] Bianca R.D. and A. Dupuis (2010) Intraday Volatility Seasonality in a Separated Foreign Exchange Market Model, working paper, Olsen Ltd. [2] J.B. Glattfelder, A.Dupuis and R.B.Olsen (2010) Patterns in High-Frequency FX data: Discovery of 12 empirical scaling laws, working paper, Olsen Ltd. 9

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