An extended Speculation Game for the recovery of Hurst exponent of financial time series
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New Mathematics and Natural Computationc (cid:13)
World Scientific Publishing Company
AN EXTENDED SPECULATION GAME FOR THE RECOVERY OFHURST EXPONENT OF FINANCIAL TIME SERIES
KEI KATAHIRA ∗ Graduate School of Frontier Sciences, The University of Tokyo,5-1-5 Kashiwanoha, Kashiwa-shi, Chiba-ken 277-8563, [email protected]
YU CHEN
Graduate School of Frontier Sciences, The University of Tokyo,5-1-5 Kashiwanoha, Kashiwa-shi, Chiba-ken 277-8563, [email protected]
Received 8 November 2018Revised Day Month YearThe speculation game is an agent-based toy model to investigate the dynamics of thefinancial market. Our model has achieved the reproduction of 10 of the well-knownstylized facts for financial time series. However, there is also a divergence from thebehavior of real market. The market price of the model tends to be anti-persistent tothe initial price, resulting in the quite small value of Hurst exponent of price change. Toovercome this problem, we extend the speculation game by introducing a perturbativepart to the price change with the consideration of other effects besides pure speculativebehaviors.
Keywords : Cognitive agent-based model; Round-trip trading; Financial stylized facts.
1. Introduction
The financial time series of asset returns have several qualitative properties collec-tively called stylized facts. For example, one of the well known stylized facts is calledheavy tails, which describes that the probability distribution of price returns hasfatter tails than those of Gaussian distribution 1 ,
2. Cont summarized 11 currentlywell known stylized facts including heavy tails 3. They are quite nontrivial featureswhich can be observed in different markets and instruments.To investigate the emerging mechanism of stylized facts, we build a novel simpleagent-based model named Speculation Game 4, which has two distinct featurescomparing with preceding agent-based market models. First, it takes account of round-trip trades by extending the decision-making structure of Minority Game ∗ Research Fellow of Japan Society for the Promotion of Science1 eptember 9, 2019 1:12 WSPC/INSTRUCTION FILE ws-nmnc K. Katahira and Y. Chen
5. Second, a mutual projection between players’ realistic and cognitive worlds isimplemented explicitly. As a result of these novelties, Speculation Game succeedsin reproducing 10 out of the 11 well known stylized facts (see Table 1).
Table 1. The reproducibility of stylized facts in Speculation Game. Symbol “+” shows that themodel successfully recovers the property, and symbol “ − ” means that it does not. Stylized fact ReproducibilityVolatility clustering +Intermittency +Heavy tails +Absence of autocorrelation in returns +Slow decay of autocorrelation in volatilities +Volume/volatility correlation +Aggregational Gaussianity +Conditional heavy tails +Asymmetry in time scales +Leverage effect +Gain/loss asymmetry − However, there is still a divergence from the behavior of real market. The marketprice p ( t ) of Speculation Game tends to stick around the initial price as Fig. 1displays (the parameters are fixed as N = 1000, M = 5, S = 2, B = 9, C = 3in this study), which causes an anti-persistent price change with very small Hurstexponents. As a power exponent of regression line of standard deviation σ ( τ ) (orseveral other forms) on time scale τ , Hurst exponent H describes the long-termtrend in the price time series. Standard deviations of price changes on differenttime scales can be calculated as follows, σ ( τ ) = p h ( p ( t + τ ) − p ( t )) i − h p ( t + τ ) − p ( t ) i . (1.1)When H = 0 .
5, a price time series is a random walk. When
H > .
5, it shows apersistent trend while it has a mean-reverting feature when
H < .
5. Hurst exponentin Speculation Game is found as H ≃ . R ≃ .
86 in Fig. 2.Objective of this study is to overcome the price sticking problem and recoverHurst exponent to the normal level like ones in the real financial markets. Partic-ularly, some perturbation is added to price change as the consideration of othereffects besides pure speculative behaviors.eptember 9, 2019 1:12 WSPC/INSTRUCTION FILE ws-nmnc
The Recovery of Hurst Exponent in Speculation Game p ( t ) to stick around theinitial price (= 100) for whole time periods. Fig. 2. A poor regression of 100-trial averaged σ ( τ ) on τ for 50,000 time steps.
2. Speculation Game and its Extension
Speculation Game is a repeated game in which N players in a gamified market,with an initial market wealth and randomly given S strategies, compete with eachother to increase wealth through round-trip trades. The game proceeds with updatesalternating between players realistic and cognitive worlds.At discrete time t , player i using her best strategy j ∗ ( ∈ S ) takes an action a j ∗ i ( t ) from the set: buy (= 1), sell (= − j ∗ , the quantity of order q i ( t ) is decided both with hermarket wealth w i ( t ) and the board lot amount B , the latter of which describes theease of placing orders with multiple quantities, as follows, q i ( t ) = ⌊ w i ( t ) B ⌋ , (2.2)where ⌊· · · ⌋ stands for a flooring operator. Note that the closing quantity q i ( t ) of around-trip trade is required to be equal to the opening one q i ( t ). Also, the player’sinitial market wealth w i (0) is decided with a uniformly distributed random number U [0 , w i (0) = ⌊ B + U [0 , ⌋ . (2.3)If w i ( t ) < B as a result of round-trip trade, the player will be forced to leave themarket and substituted by a new player.Following the order imbalance equation defined by Cont and Bouchaud 6, themarket price change ∆ p is calculated as follows by letting the initial market priceas p (0) = 100, ∆ p = p ( t ) − p ( t −
1) = 1 N N X i =1 a j ∗ i ( t ) q i ( t ) . (2.4)Then, the quantized price movement h ( t ) (the last digit in the history H ( t )) isdecided by the magnitude correlation between ∆ p and the cognitive threshold C ,eptember 9, 2019 1:12 WSPC/INSTRUCTION FILE ws-nmnc K. Katahira and Y. Chen the latter of which can be explained as a threshold value for the players to recognizea big price move: h ( t ) = p > C, C ≥ ∆ p > , p = 0 , − − C ≤ ∆ p < , − p < − C. (2.5)To find an action of the best strategy a j ∗ i ( t ), the player with memory M at firsttakes the reference of the last M digits of history H ( t ). Next, she looks up strategy j ∗ to obtain a recommended action corresponding to the historical pattern (seeTable 2). However, when the recommendation is the same as the opening order a j ∗ i ( t ), the player will hold the position. Table 2. Example of a strategy for M = 3. Each strategy table holds recommended actionscorresponding to whole historical patterns. History Recommended action − − − − − − − − − − − − − − − − − j ∗ , all strategies are evaluated through virtualround-trip trades in the background similarly. Hence, performance of the strategiesare assessed in the cognitive world, in terms of the accumulated strategy gains G ji ( t )( j ∈ S ) calculated with cognitive price P ( t ) corresponding to the rough informationof ∆ p in H ( t ). Letting P (0) = 0, P ( t ) is updated as P ( t ) = P ( t −
1) + h ( t ) . (2.6)The gain of strategy j in a round-trip trade ∆ G ji ( t ) reads as∆ G ji ( t ) = a ji ( t )( P ( t ) − P ( t )) . (2.7)Thus, the accumulated strategy gain G ji ( t ) is calculated by: G ji ( t ) = G ji ( t ) + ∆ G ji ( t ) . (2.8)eptember 9, 2019 1:12 WSPC/INSTRUCTION FILE ws-nmnc The Recovery of Hurst Exponent in Speculation Game Whenever the accumulated gain of the strategy in use G j ∗ i ( t ) is updated, all G ji ( t )will be reviewed to renew the strategy with the best performance. If the best strategyhappens to be one of the unused strategies with a virtual trade ongoing, the virtualposition will be closed immediately before the player switches the strategy at thenext time step. Note that the evaluating system is developed by considering that theinvesting strategies should be evaluated through by round-trip trades, as Katahiraand Akiyama pointed out 7.Since the self-financing assumption is not required in Speculation Game, when around-trip trade is closed in the realistic world, the player’s market wealth w i ( t ) isupdated with an investment adjustment ∆ w i ( t ), which is the conversion of strategygain in the cognitive world with q i ( t ) taken into consideration, w i ( t ) = w i ( t ) + ∆ w i ( t ) = w i ( t ) + f (∆ G j ∗ i ( t ) q i ( t )) , (2.9)where f can be an arbitrary function. In the game, ∆ w i ( t ) = ∆ G j ∗ i ( t ) q i ( t ) is usedfor the simplicity.As an extension to the original model, a random perturbation U [ − P b, P b ) isfurther added to ∆ p as∆ p = 1 N N X i =1 a j ∗ i ( t ) q i ( t ) + U [ − P b, P b ) , (2.10)where the first term of Eq. (2.10) could be considered as the effect given by the purespeculative players while the second one as other effects by other types of traderssuch as long term value traders, hedgers, and so on. Due to this extension, ∆ p nevertakes zero so that H ( t ) become a quaternary time series instead of a quinary one.
3. Result and Discussion
By adding the random perturbation, the dynamics of market price in SpeculationGame becomes more realistic. As Fig. 3 displays, p ( t ) can escape from the initialprice range and shows long-term fluctuating structures when P b = 0 .
25. Accord-ingly, Hurst exponent also increases to the normal level ( H ≈ . ∼ .
7) as H ≃ . R ≃ .
99) as the regression line shown in Fig. 4. It can alsobe inferred from these results that the long-term fluctuations and trends originatedfrom those non-speculative behaviors.On the other hand, the further increment of perturbation would weaken thestylized properties as well. For instance, the probability distributions of price re-turns have less heavier tails as more perturbation is added, which is displayed inFig. 5. Since the similar propensity is confirmed for many of other reproduced styl-ized features in Speculation Game, speculative behaviors can be considered as themain source for the emergence of stylized facts. Note that adding moderate priceperturbation (
P b ≃ . ∼ .
3) is favorable in the viewpoint of efficiency to enhanceHurst exponent because excess
P b contribute less to the increment of H as Fig. 6eptember 9, 2019 1:12 WSPC/INSTRUCTION FILE ws-nmnc K. Katahira and Y. Chen shows. Also, it is reasonable that the perturbation can not make
H > . Fig. 3. The more realistic movement of p ( t ) withlong-term fluctuations when P b = 0 .
25. Fig. 4. The recovery of Hurst exponent and thebetter regression when
P b = 0 . P b as 0.25, 0.50, and 0.75. Fig. 6. The decay in the increment of Hurstexponent as
P b increases.
4. Conclusion and Future Work
To conclude, adding moderate perturbation to ∆ p can solve the price sticking prob-lem and recover Hurst exponent for our Speculation Game. Simulation results alsoindicate that the non-speculative behaviors generate the long-term fluctuations andtrends while speculative ones contribute to the emergence of well-known financialstylized facts.For future research, decision-making structures for fundamentalists and newstraders need to be developed for the generation of a long-term trend. Adding per-eptember 9, 2019 1:12 WSPC/INSTRUCTION FILE ws-nmnc The Recovery of Hurst Exponent in Speculation Game turbation is the simplest way to include the effects brought by those non-pure spec-ulative behaviors, though these effects should be further verified in a more concreteway as well. Acknowledgments
This work was supported by JSPS KAKENHI grant number JP17J09156.
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