A Simple Model for Intra-day Trading

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1 A Simple Model for Intra-day Trading Anton Golub 1 1 Marie Curie Fellow, Manchester Business School April 15, 2011 Abstract Since currency market is an OTC market, there is no information about orders, open positions of traders. The objective of this project is to extract information about the positions, and generally about liquidity, from only the observed price curve. The goal of the model, is to generate realistic, sustainable and fractallike price curve with the simplest possible setting. Indeed, the only stylized feature implemented in this model is the separation of the market (see Section 2 for details). 1 Market Maker The model has two types of participants; the market maker and the traders. In this model, only the market maker can move the price curve, and this movement is based on his exposure E MM. The market maker s exposure is always equal to the net exposure of all the traders. For an example, assume that there are just two traders, trader A goes long (buys) 1000 units of a currency, while trader B goes short (sells) 500 units of the same currency. To satisfy both traders need, the market maker will have an exposure E MM = = 500. Let there be n traders and let e T i rader (t), i = 1,..., n be their exposure at time t R +, the exposure E MM (t) R of market maker, at time t equals E MM (t) = n i=1 e T i rader (t). The market maker s action, at time t R +, depends on the magnitude of his exposure E MM (t) with respect to T skew and T hedge, where T skew < T hedge, and T skew = T Hedge F H, where F H > 1 is the hedging factor. When certain conditions are met, the market maker adjusts E MM (t i ) (at time t i ) by skewing (changing) the price: The research leading to these results has received funding from the European Community s Seventh Framework Programme FP7-PEOPLE-ITN-2008 under grant agreement number PITN-GA The funding is gratefully acknowledged. 1

2 (i) When E MM (t i ) < T skew, market maker interacts with traders, but does not change the price. (ii) When T skew E MM (t i ) T hedge, the amount of time (in seconds) the market maker takes to change the price is proportional to the magnitude of E MM (t i ). For example, if E MM (t i ) = T skew, the market maker might take 100 seconds before he changes the price. If E MM (t i ) T hedge, the market maker will change the price in the next second. The time of response is a linear function of E MM (t i ) T skew and will range between t 1 (for E MM(t i ) = T hedge ) and t 2 seconds (for E MM(t i ) = T skew ), t 1 < t 2. Specifically, market maker keeps track when did he change the price the last time, that is, at (current) time t, maker maker knows the last time he changed the price t = sup{t [0, t : Market maker changed the price at time t}, if the following condition is satisfied t t t 2 > t 1 E MM ( t) T Skew + t T Hedge T 2, Skew the market maker will change the price. For example, if E MM (1) = 1000, the market maker might take 20 seconds before he changes the price curve in order to change his exposure. When the market make decides to changes the price, the amount of price change is fixed at 2 pips (1 pip=0.0001) to attract an order that will reduce his exposure. It is described later in the text how does the market maker change the price. (iii) When E MM (t i ) T hedge, the market maker will buy/sell all of it s exposure to get a neutral position (i.e. E MM (t i+1 ) = 0), but doing so he will move the market by 5 E MM (t i ) T Hedge pips. After the market maker has changed the price, E MM (t i+1 ) is set to zero. For E MM (t i ) < 0, i.e. market maker is in the short position, he will move the price upwards p(t i+1 ) ask = p(t i ) ask ( skew), and vice versa for E MM (t i ) > 0 p(t i+1 ) bid = p(t i ) bid ( skew), p(t i+1 ) ask = p(t i ) ask ( skew), p(t i+1 ) bid = p(t i ) bid ( skew). The variable skew is set such that skew equals 2 pips. The motivation for this behavior is the following: the model assumes that traders are more tolerant to having a negative P&L than a positive P&L, i.e. traders are more ready to take profit than to cut losses. This means that for E MM < 0, it is more likely for the market maker to force long traders (rather than the short traders) to exit the market by adjusting the price upwards. The long traders would close the trade (with a positive P&L), while the short traders would still keep their position open because of their greater tolerance to negative P&L. The market maker does not have any random behavior; all his actions are deterministic responses to the actions by the traders. 2

3 2 Traders The key feature of the model is the separation of the market into three different geographical groups of participants (Europe, America and Asia) with trading hours (CET) EU: 7.00am-3.00pm, AM: 1.00pm-11.00pm, AS: 11.00pm-9.00am. Specifically, traders do not operate outside their local working hours. This is motivated by the seasonality pattern in the intraday volatility pattern in the real price curve that reflects intermittant trading activities separated by operation hours of different time zones. At the start of the simulation, a market sentiment is decided that lasts for a random time period and this market sentiment will influence will there be more traders with a long position or more traders with a short position. The time period of market sentiment is decided according to a uniform distribution [0, ˆt 1 = [0, ( U(0, 1)) marketcycle, where variable marketcycle is set to be equal 200,000 seconds, U(0, 1) is an uniform random variable on set [0, 1] R and the market sentiment can be either long or short. After the period [0, ˆt 1 has passed a new random period is decided, but this time with the opposite market sentiment. Specifically, if in the time period [0, ˆt 1 the market sentiment was short, in the next period [ˆt 1, ˆt 1 + ( U(0, 1)) marketcycle = [ˆt 1, ˆt 2 the market sentiment will be long, the opposite from the sentiment in the previous period. This process is iterated until the end of the simulation. When a trader is created in the simulation, it is first decided will he follow the market sentiment of not, that is will he be a long or a short trader. Specifically, lets assume that the time is t [ˆt i, ˆt i+1 and the market sentiment is short, the type of the position of the trader is decided by the following discrete random variable ( ) short trader long trader, p 1 p where p 0, 1. If it were in fact that the market sentiment is long the type of the position would be decided by the following discrete random variable ( ) short trader long trader. 1 p p The probability p 0, 1 is used if one wants to emphasize market trends, in which case p should be set larger than 0.5. Furthermore, next is decided if the trader, be it long or short trader, will be a trender or a counter trender. Similarly as before, this is decided by a discrete random variable ( trender counter trender q 1 q where q 0, 1. Finally, within the context of market separation, the trader is only allowed to enter (and exit) the market during his local working hours and then his entrance is subjected to a Bernoulli random variable, with a time changing probability parameter p(t). Specifically, let trader i (t) be i-th trader at time t 3 ),

4 [t wh start, t wh end ] where twh end are, respectively, the start and the end of his local working hours, probability of trader trader i (t) entering the market equals (( t wh )) start +twh end 2 ( ( t wh )) start t +twh end b 1 + p(t) = a b 1 + a, t [t wh start, t wh end ]. 0, t / [t wh start, t wh end ] start, t wh where b = 0.5 and a = 100, 000. Notice that the highest probability of trader entering the market, during his local working hours, equals b and it is when half of his local working hours have passed. Each trader will also keep track what was the minimum and the maximum price, be it from the time of his creation or from time he exited the market (it will be described later how the trader exits the market), that is, let t c i be the time of creation of trader trader i (t) who has not yet entered the market, he will keep track of the following prices p max i = max{p(t): t t c i}, p min i = min{p(t): t t c i}, let t e j be the time when a trader trader j(t) has exited the market (he has entered the market before time t e j ), he will keep track of the following prices p max j = max{p(t): t t e j}, p min j = min{p(t): t t e j}. Within the context of entering the market, there is no need for the traders that are currently in the market to keep track of the mentioned maximum and minimum price. Each trader is also assigned with a threshold parameter, that is, i-th trader has a parameter th i R + which is defined by th i = ( U(0, 1)) gr where U(0, 1) is an uniform random variable on set [0, 1] R and gr is a global variable equal to all traders. Let us now describe how do traders enter the market; first let us note that there can be four type of traders, a long trader can either be a trender or a counter trender, and a short trader can be either a trender or a counter trender. Lets assume that p(t) bid, p(t) ask is the bid/ask price at time t R + and that trader i (t) is the i-th trader and let he be a long trader who is a trender. He will enter the market, with a long position, if the difference between the current ask price p(t) ask and p min i is greater than th i, that is p(t) ask p min i th i. The interpretation is that a long trader, who is a trender is motivated to enter the market when he sees an upward price move. Similarly, a short trader, who is a trender will enter the market, with a short position, if the difference between the current bid price p(t) bid and p max i is smaller than th i, that is p(t) bid p max i th i. A long trader who is a counter trender will enter the market, with a long position, if the difference between p max i and the current ask price p(t) ask is less than th i, that is p(t) ask p max i th i. 4

5 The interpretation is that a long trader, who is a counter trender is motivated to enter the market when he sees an downward price move and hence, wants to counter the trend. Similarly, a short trader, who is a counter trender will enter the market, with a short position, if the difference between the current price p(t) bid and p min i is greater than th i, that is p(t) bid p min i th i. When a trader trader i (t) enters the market, his stop-loss and take-profit orders will immediately be set, let us assume that trader i (t) is a long trader, his stop-loss and take-profit will equal stoploss i = p(t) ask (1 + tl th i ), takep rofit i = p(t) ask (1 + pt th i ), similarly, if he is a short trader, his stop-loss and take-profit will equal stoploss i = p(t) bid (1 tl th i ), takep rofit i = p(t) bid (1 pt th i ), where tl = 3 pt. Long traders exit the market when the bid price is smaller than the stop-loss or greater than the take-profit and the short traders will exit the market when the ask price is smaller than the take profit or greater than the stop loss, or equivalently, if p&l(t) i is the profit and loss of i-th trader trader(t) i at time t, he will exit the market if p&l(t) i, tl th i ] [pt th i,. The size of the trade, when entering the market is uniformly distributed, that is if trader trader(t) i enters the market at time t the size of his position size i (t) equals size i (t) = maxgearing U(0, 1) where U(0, 1) is an uniform random variable and maxgearing R + and it is unchanged until he exits the market. When the trader s stop-loss or take-profit have been hit, his exit is subjected to the probability 1 p(t) where {p(t): t R + } is the probability of entering the market defined earlier. Once the trader exits the market, the maximum and minimum price he kept track off are reset to the current price, the size of his positions is set to equal zero and once again his type (long or short) and will he be a trender or a counter trender is decided in the way described earlier. The total number of traders currently in the market from each geographical location N Eu, N Am, N As, at every moment is not bounded. The number of free participants, n Eu, n Am and n As who can enter the market, are exogenously set reflecting the pressure that pushes each type of (geographically different) traders into the market - only the corresponding ratio is decided by the model (e.g. n Eu = n Am = 2n As ). 3 Simulating and Analyzing the Model The code is run over 84 trading days with 10 ticks per second or 72,576,000 ticks in total. The simulation (usually) takes a whole day in real time. The information produced are opening time, opening price, size of the position, stop-loss order, takeprofit order, and bid/ask price. 5

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

7 The other thresholds are chosen to be linear in log scale, specifically log( x dc ) i+1 log( x dc ) i = c, i 1,..., n 1, where c > 0 is a constant and the last threshold is chosen to be smaller than the largest percentage price move { } x(t) x(s) ( x dc ) n max x(s) : (t, s) R2 +, s < t. The constant c is used to get statistical significance, when analyzing the price curve with different thresholds and the number of thresholds, that is n will depend on the constant c we choose, i.e. a smaller c will yield a larger number of thresholds, larger c will yield smaller number of thresholds. This helps to reduce the level of complexity of real-world time series and to define an activity-based time-scale (intrinsic time). For each ( x dc ) i, the size of the positions and P&L or traders is examined. For the whole spectrum of thresholds { x dc }, (scaled) net variation in volume and percentage of the total volume in each regime is examined. The trades are analyzed in different ways; all trades, trades that ended in a loss, pairs of directional-changes/overshoots where overshoot x dc and the trades ended in a loss, pairs of directional-changes/overshoots where overshoot x dc and the trades that ended in a loss. The model generates tick-by-tick prices based on specific set of parameter values. Market makers parameters will be removed once the real price curve is fed into the model. Due to the complexity of the model and trading behavior, it is not likely that one could estimate these parameters from the real price curves. In all of the cases examined, regardless of the threshold x dc or phase chosen, the size of trade initiated by the traders is uniformly distributed, while the number of traders is jointly determined by N and n. The only parameter that is estimated from the real data and implemented is the market separation. All of the other parameters in the model are set arbitrarily and it is not likely that they can be statistically estimated. The only thing one could do is to vary the parameter values and check their impact on the simulated price curve. 4 Real Price Curve In this section we will list some of the stylized facts found in the currency markets. While the list will not be exhaustive, it will be a guideline in determining if the price curve generated by the model is realistic. The stylized facts will include: first orders negative autocorrelation in returns, increasing fat-tailness in price returns distribution as data frequency increases, scaling laws, seasonal heteroskedasticity in form of distinct daily clusters of volatility (the cornerstone of our model). It is highly unlikely that our model will be able to capture all of the stylized facts found in the FX markets, due to its simple characteristics. We will briefly address which of the stylized, if any, are found in the model. 4.1 First order negative autocorrelation in returns Goodhard (1989) and Goodhard and Figliouli (1991) were the first to report the existence of negative first-order autocorrelation of returns at highest frequencies, 7

8 which disappears once the price formation process is over. The mentioned negative autocorrelation is found up to a lag of 4 minutes, as for longer lags the autocorrelation mainly lies within the 95% confidence intervals of an independent and identical Gaussian random distribution. A first explanation of this fact is that traders have diverging opinions about the impact of news of the direction of the price - contrary to the conventional assumption that the foreign exchange market is composed of homogeneous traders who would share the same view about the effect of news so that no negative autocorrelation of the returns would be observed. Secondary explanation of this negative autocorrelation is the tendency of the market maker to skew the spread in a particular direction when they have order imbalance. A third explanation is that even without order imbalance or diverging opinions of the price, certain banks systematically publish higher bid-ask spreads. A model for the bid/ask bounce, in modeling transaction data in the stock market was proposed by Roll (1984). The idea is that the two prices, bid and ask, can be hit randomly according to the number of buyers and sellers in the market. If the number of buyers is equal to the number of the sellers it will produce a negative autocorrelation of transaction returns at high-frequency. Our model does not exhibit such a characteristic as the market makers always keeps the bid-ask spread equal to 2 pips (1 pip = ). 4.2 Distributional properties of returns The model presented provides a unique opportunity to explore the dynamics of mentioned information, as the price curve, generated by the model, evolve. Therefore, it is analyzed how absolute percentage price changes and closed trades relate. Specifically, at time horizon is chosen, be it in physical t or intrinsic time t, the corresponding absolute percentage price changes are obtained, it is then observed how many closed trades happened. This dynamics is observed for different time horizon. In order to make this dynamics comparable across different time horizons, appropriate rescaling had to be done for absolute percentage price changes and for number of closed trades. It is well know that the absolute percentage price changes have a scaling behavior and it holds in the model, but as the information about closed trades is not available, no claims can be made that the scaling law holds. Furthermore, it is disputable if the mentioned scaling law holds in the model, further analysis is needed in that area. At this stage it is not clear if this dynamics provides incremental information about the model. 4.3 Scaling laws First, let us define notation. Averaging operator is defined as p : R n R x p = ( 1 n n k=1 x p k ) 1 p, x = (x1,..., x n ) R n where n N and p is taken to be either 1 or 2. Furthermore, for x = (x 1,..., x n ) R n we define a counting operator N( )[x]: R N as N(x )[x] = card{i {1,..., n}: x i = x } 8

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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