STATS 242: Final Project High-Frequency Trading and Algorithmic Trading in Dynamic Limit Order

Transcription

1 STATS 242: Final Project High-Frequency Trading and Algorithmic Trading in Dynamic Limit Order Note : R Code and data files have been submitted to the Drop Box folder on Coursework Yifan Wang wangyf@stanford.edu 1

2 Yifan Wang STATS 242 Final Project 1 Introduction 1 Introduction In the high-frequency trading, the market maker faces an inventory risk due to the di usive nature of the stocks mid-price and a transactions risk due to a Poisson arrival of market buy and sell order. Studies of the mechanisms involved in trading financial assets have traditionally focused on quote-driven markets, where a market maker or dealer centralizes buy and sell orders and provides liquidity by setting bid and ask quotes, and an alternative order-driven trading system. The dynamics of a limit order book resembles in many aspects that of a queuing system. Limit orders wait in a queue to be executed against market orders (or canceled). An important motivation for modelling high-frequency dynamics of order books is to use the information on the current state of the order book to predict its short-term behavior. Therefore, focus on conditional probabilities of events, given the state of the order book. In this project, I replicate the simulation of the high-frequency trading in a limit order book, (Avellaneda and Stoikov 26) as understanding process for the mechanism of market making inventory strategy, and further consider the stochastic model for order book dynamics (Cont, Stoikov and Talreja 29). Then, focus on the e ective of the mid-price trend prediction and the liquidity parameter of Poisson process. In practical, calculate out the signals by moving average, realized volatility for order risk management, and applicate statistical methods confirm the dynamic spreads and the cancellations. 2 Inventory Strategy Simplicity the limit order book for understanding the inventory control mechanism. This part of objective is to replicate the Avellaneda and Stoikov (29). The solution in a two step procedure: First, the dealer computes a personal indi erence valuation given specific personal inventory. Second, calculate the bid and ask quotes to the market limit order book. 2.1 The optimizing with finite and infinite horizons For simplicity, assume the mid-price of stock follow ds u = dw u (1) with initial value S t = s, and W t is a standard one-dimensional Brownian Motion and is constant. For optimizing with finite horizon, the objective is to maximize the expected exponential utility of profile. The indi erence prices given by personal inventory, based on the frozen inventory strategy. 2 q 2 2 (T t) v(x, s, q, t) =E t [ exp( (x + qs T ))] = exp( x)exp( qs)exp 2 Then, define the indi erence bid and ask prices, which e ected by current portfolio and plus one stock. The indi erence bid price r b and the indi erence ask price r a are implicitly by the relation v(x + r a (s, q, t),s,q 1,t)=v(x, s, q, t) and v(x r b (s, q, t),s,q+1,t)=v(x, s, q, t) r a (s, q, t) =s +(1 2q) 2 (T t) 2 and r b (s, q, t) =s +( 1 2q) 2 (T t) 2 2 Inventory Strategy continued on next page... Page 2 of 11

3 Yifan Wang STATS 242 Final Project 2 Inventory Strategy (continued) If the market maker is long stock (q >), the indi erence price is below the mid-price, indicating a desire to liquidate the inventory by selling stock. On the other hand, if the market maker is short stock (q <), the di erence price is above the mid-price, since the market maker is willing to buy stock at a higher price. r(s, q, t) =s q 2 (T t) Further, considering the dynamic mid-price on time-varity liquidity, the value function of an infinite horizon objective Z 1 v(x, s, q) =E exp(!t)exp( (x + qs t ))dt 2.2 The limit order and trading intensity The limit order of ask quotes q a and of bid quotes q b can be continuously updated at no cost. The distance a = p a s and b = s p b (2) the current shape of the limit order book determine the priority of execution when market orders get executed. Use the zero-intelligence model (Smith, Farmer, Gillemot and Krishnamurthy) to assume the market buy (or sell) orders will hit the limit ask (or bid) orders at Poisson rate b( b ), a decreasing function of b (or Poisson rate a( a ), a increasing function of a). The wealth and inventory are now stochastic and depend on the arrival of market orders. Indeed, the wealth in cash jump dx t = p a dn a t p b dn b t (3) where N a t and N b t are the amount of stocks, the number of stocks at time t is q t = N b t N a t. For the trading intensity, consider the statistics on (i) the overall frequency of market order (ii) the distribution of their size and (iii) the temporary impact of a large market order. In pervious researches, the density of market order size is f Q (x) / x 1 for large x, with =1.53 in Gopikrishnan et al. for U.S. stocks, =1.4 in Maslow and Mills for shares on the NASDAQ and =1.5 in Gabaix et al. for the Paries Bourse. The market maker controls the bid and ask prices and therefore indirectly influences the order flow. So, further considering the realistic functional forms for the intensities a( a ) and b ( b ) Poisson rate. In Cont, Stoikov and Talreja (29), they assume orders to be unit size, compute the average sizes of market order S m, limit order S l, and canceled order S c and choose the size unit to be the average size of a limit order S l, as zero-intelligence model. The limit order arrival rate function for 1 apple i apple 5 can be estimated by ˆ(i) = N l (i) T (4) where N l (i) is the total number of limit orders that arrived at a distance i from the opposite best quote, and T is the total trading time in sample. In Avellaneda and Stoikov (26), the order arrival terms are highly non-linear. Therefor, they suggest 2 Inventory Strategy continued on next page... Page 3 of 11

4 Yifan Wang STATS 242 Final Project 2 Inventory Strategy (continued) an asymptotic expansion of in the inventory variable q, and a linear approximation of the order arrival terms. The exponential arrival rates a( )= b ( )=Ae k After proving process with functions of indi erence price and distance above, obtain the indi erence price as for the frozen inventory problem. r(s, t) =s q 2 (T t) (5) then set a ask/bid spread by a + b = 2 (T t)+ 2 ln(1 + k ) (6) around this indi erence price. Note that if taken a quadratic approximation of the order arrival term, the sensitivity would solve a non-linear PDE. 2.3 Numerical simulations The simulation is obtained through the following procedure: At time t, compute the market maker quotes a and b. At time t + dt, update the variables. With the probability a ( a )dt, the inventory variable decreases by one and the wealth increases by s + a. With the probability b ( b )dt, the inventory increases by one and the wealth decreases by s b The mid-price is updated by a random increment ± p dt. In simulation, I set the same parameters on both inventory strategy and symmetric strategy. The symmetric strategy uses the same bid/ask spread as the inventory strategy, but send bid/ask orders by the bid/ask price around the mid-price. Then, run this simulation process 5 times with specific to compare the two strategy. The results of 5 simulations show that the inventory strategy position is considered less risky. Inventory Strategy Symmetric Strategy Price Ask Price Mid price Indifference Bid Price Price Ask Price Mid price Predict Price Bid Price Figure 1: The mid-price, indi erence price and optimal bid and ask price, with s = 1,T =1, =2,dt =.5,q =, =.1,k = 1.5,A= Inventory Strategy continued on next page... Page 4 of 11

5 Yifan Wang STATS 242 Final Project 2 Inventory Strategy (continued) Inventory Symmetric PnL PnL Figure 2: The PnL of the inventory strategy and symmetric one, with s = 1,T =1, =2,dt=.5,q =, =.1,k = 1.5,A= 14. =.1 Spread PnL std.pnl FinalQ std.finalq Inventory Symmetric =.5 Spread PnL std.pnl FinalQ std.finalq Inventory Symmetric =.1 Spread PnL std.pnl FinalQ std.finalq Inventory Symmetric Table 1: 5 simulations with specific. Frequency Inventory Strategy Symmetric Strategy PnL Figure 3: The PnL distributions of 5 simulation with = Discussion The numerical simulation helps to understand the mechanism of inventory control strategy for market making, on specific liquidity and volatility. But, in the real market, the liquidity and volatility are time variety, so the situations are complex to consider the risk managements, such as the prediction of mid-price and the 2 Inventory Strategy continued on next page... Page 5 of 11

6 Yifan Wang STATS 242 Final Project 2 Inventory Strategy (continued) prediction of spreads to decide the bid/ask price and the bid/ask quotes for inventory control. For example, in an up-trend, the probability of the limit bid order hit the market sell order is less than the probability of the limit ask order hit the market buy order. If the limit ask and bid orders sent in a same time, the intensity and distance should adjust time-variety. I try the realized volatility to adjust the in original simulating model RV n = p n i=1 log(p t+1/p t ), but the issue about how to adjust the intensity, which followed by Poisson distribution, can t figure out. In addition, the proper trading times in the real market should be considered, it doesn t like the simulation model which may trade per 5ms. Inventory Figure 4: The inventory strategy in 215/7/28 IF.CFE high-frequency data, 3242 observations per day. 3 Dynamic Algo HFT Generally, if analysis the U.S. stock markets TAQ and aggregate the data by unique time stamp, the durations show the U-Shape, which means that the transactions are more crazy at very beginning and ending of the intraday. The orders arrival rates followed by Poisson distributions is so-called liquidity time variety. Unfortunately, the TAQ data is not available in the financial market of China. The only high-frequency data we can get is the 5ms-tick levels with bid and ask price. Obviously, the dataset miss some information like durations in TAQ obstructs the analysis of arrival rate. The liquidity information include the limit order arrival rate which show in limit order book, and the market order arrival rate which show in volume. Hence, I attempt to combine algorithmic trading methods and statistical methods in high-frequency trading by three procedures. First, convert the inventory risk due to the di usive of price to the short-behaviour trend prediction by moving average methods for improving the probability of pairs order deal in proper sequence. This algorithmic method can decrease the risk of failing to pairs ask/bid price. Second, consider the statistical methods to analysis the stylize of spread and volume, and modeling the proper spread and the cancellation of limit orders. Third, the risk management for some fake signals risk by VaR adjusted RV. 3 Dynamic Algo HFT continued on next page... Page 6 of 11

7 Yifan Wang STATS 242 Final Project 3 Dynamic Algo HFT (continued) 3.1 Moving Average and Volatility For controlling the risk of trend, consider the probability of execution order. In the model, I calculate the moving average in 3 ticks (MA3) and the moving average in 6 ticks (M6), then compute the volatility of di erence between MA3 and M6 to sort the 4 groups signals, including the sell order signals or the buy order signals for into the market or out the market, and use di erent thresholds to adjuste the probability of the signals. Then, generate out 4 signals like (Figure 5) IF price Signal: sell in Signal: sell out Signal: buy in Signal: buy out Figure 5: The intraday trading signals at 215/7/28 IF.CFE futures series. Note that the thresholds can be set by personal risky reversal or by the policy limited in specific market. In another words, we can adjust the transactions times by internal personal reasons or by external financial markets regulatory. 3.2 Spread and Volume Use the statistical methods to check the stylize of the spread by multiple intervals. Based on the test, the spread and volume show autocorrelations and the delta spread and delta volume show the negative autocorrelations at 1-lag. Further, to check the best bid/ask price and the spread in 5/1/15 ticks intervals. Intervals Min. 1st Qu. Median Mean 3rd Qu. Max. 1-tick tick tick tick Table 2: The spread in 1/5/1/15 ticks intervals. For modeling the spread, dynamic fitting the ARMA model of delta spread S t update as time and forecast 1 ahead step S t+1. The VAR model with S and V is not significant and not suitable to use. Depends on the statistics of spread in multiple intervals, I simply set the 5-tick interval which be the upper limit waiting time before to cancel the order, and send the order with the prediction of mid-price. 3 Dynamic Algo HFT continued on next page... Page 7 of 11

8 Yifan Wang STATS 242 Final Project 3 Dynamic Algo HFT (continued) Series spread Series diff(spread) ACF ACF Series spread Series diff(spread) Partial ACF Partial ACF Series volume Series diff(volume) ACF ACF Series volume Series diff(volume) Partial ACF Partial ACF Figure 6: The ACF and PACF of spread (S and S) and volume (V and V ). 3.3 Risk Managements Generally, the value at risk (VaR) is a useful index for risk management. In this case, considering the fat tail may generate out some fake signals, I use the value at risk (VaR) by the dynamic adjusted RV to set the loss accounts at time t. Alternatively, simply setting the limit account in trading system. 3.4 Strategy Performance By implemented the trading strategy above, the performance formate below the Table 3. Depends on the backtesting, the signals could be optimized may include the volatility and multiple interval moving average. Because in this case, I only use the MA3, MA6 and MA9. Some longer moving average (like 5-min level) could be the trend signs for omitting fake signals which short-behaviour can t beat the longer-trend. 6/12 7/28 8/3 7/3 7/3(No RM) PnL (points) Sharp Ratio PnL (per Pairs) Trading Cost Table 3: The performance reports with risk management. Page 8 of 11

9 Yifan Wang STATS 242 Final Project 3 Dynamic Algo HFT price IF pnl Figure 7: The intraday trading signals at 215/6/12 IF.CFE futures series. price IF pnl Figure 8: The intraday trading signals at 215/7/28 IF.CFE futures series. Page 9 of 11

10 Yifan Wang STATS 242 Final Project 3 Dynamic Algo HFT price IF pnl Figure 9: The intraday trading signals at 215/8/3 IF.CFE futures series. price IF pnl Figure 1: The intraday trading signals at 215/7/3 IF.CFE futures series. Page 1 of 11

11 Yifan Wang STATS 242 Final Project 4 Reference 4 Reference Ruey S. Tsay (23), An Introduction to Analysis of Financial Data with R T.Z Lai and H. Xing (28), Statistical Models and Methods for Financial Markets Marco Avellaneda and Sasha Stoikov (26), High-Frequency Trading in a Limit Order Book Rama Cont, Sasha Stoikov and Rishi Talreja (29), A Stochastic Model for Order Book Dynamics Rama Cont, Statistical Modeling of High-Frequency Financial Data: Facts, Models and Challenges Page 11 of 11