An interaction between a leveraged ETF and futures in a crash investigated by an agent-based model
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December 12, 24
スライド概要
An interaction between a leveraged ETF and futures in a crash investigated by an agent-based model
Takanobu Mizuta
SPARX Asset Management Co. Ltd.
Note that the opinions contained herein are solely those of the authors and do not necessarily reflect those of SPARX Asset Management Co., Ltd.
In this study, the artificial market model was built by adding a market in which a L-ETF itself is traded and adding a erroneous orders leading turmoil implemented by the model of [Mizuta 2016] to the prior model of [Yagi 2020]. We investigated that sharply falling of the futures or the L-ETF affects the another market and effects of rebalancing trades of the L-ETF.
In the result, the erroneous orders to the L-ETF leads that market prices of the L-ETF fall, but the arbitrage-trades reduce the fall. Here, waiting buy orders to the futures, not to the L-ETF, are executed and reduced because a sell order of a normal agent executes a buy order of the AA and then the AA sells the futures. The buy orders of the AA to the L-ETF reduce the fall of the L-ETF. The reason why the AA can buy the L-ETF is that the AA can consume the waiting buy orders of the futures. In short, liquidity of the futures is consumed to prevent the fall of the L-ETF. This means that liquidity of the futures is used as hidden liquidity of the L-ETF.
In the case that the erroneous orders to the futures, the arbitrage-trades by the AA mix and combine liquidity of the L-ETF and futures, therefore, same falling happens even which to the L-ETF or the futures the erroneous orders come. The futures can gain only a less liquidity from the L-ETF that has a less liquidity, then this leads not to reduce falling so much. In real financial market, this means that investors of L-ETFs have great benefit by gaining much liquidity of futures, on the other hand, investors of futures do not have so benefit by gaining low liquidity of L-ETFs.
Larger the L-ETF, larger falling for both the L-ETF and the futures. When S0 > 100000, the simulation can not run due to too large falling of market prices. This means that markets are destroyed. When the futures fall, the L-ETF sells futures. Here, when the L-ETF is so big, the futures also fall, and then the number of futures the L-ETF should sell is larger, finally, the L-ETF should additionally sell the futures. Such process continues as loop and falling can not stop. This leads to destroy the markets.
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22-23 October 2024 2024 IEEE Computational Intelligence for Financial Engineering and Economics (IEEE CIFEr) https://www.stevens.edu/page-right-nav/cifer-conference-2024 https://www.stevens.edu/page-right-nav/cifer-conference-2024 An interaction between a leveraged ETF and futures in a crash investigated by an agent-based model Takanobu Mizuta SPARX Asset Management Co. Ltd. mizutata[at]gmail.com https://mizutatakanobu.com mizutata[at]gmail.com https://mizutatakanobu.com Isao Yagi Kogakuin University Note that the opinions contained herein are solely those of the authors and do not necessarily reflect those of SPARX Asset Management Co., Ltd. You can download this presentation material from https://mizutatakanobu.com/2024CIFEr.pdf https://mizutatakanobu.com/2024CIFEr.pdf 1
(1) Introduction (2) Our Model (3) Simulation Results (4) Conclusion You can download this presentation material from https://mizutatakanobu.com/2024CIFEr.pdf https://mizutatakanobu.com/2024CIFEr.pdf 2
(1) Introduction (2) Our Model (3) Simulation Results (4) Conclusion You can download this presentation material from https://mizutatakanobu.com/2024CIFEr.pdf https://mizutatakanobu.com/2024CIFEr.pdf 3
Leveraged ETF (L-ETF) ETF = Exchange Traded Funds is an investment fund traded on stock exchanges, much like stocks A Leveraged ETF is designed to deliver several times the return of its benchmark index (future), e.g. S&P 500, Euro Stoxx 50, FTSE 100, Nikkei 225 140 Index (Future) Leveraged ETF (x2) Value 120 100 80 60 0 2 4 6 8 10 12 14 days Actually, Leveraged ETF has to trade the Rebalance index on daily to maintain its leverage 4
Rebalance to maintain its leverage For example, when the index returns are +10% on day 1 and -10% on day 2 The Leverage is twice (x2) Index Days 0 1 2 (a) Return +10% -10% Leveraged ETF (x2) is Designed as (b) Return = 2x(a) +20% -20% Leveraged ETF (x2) Holding Index (future) Value (c) Value Will be by (b) +20% -20% $100 $120 $96 (d) Should have = 2x(c) x2 x2 x2 (e) Will be by (a) $200 +10% $200 $240 $220 $192 -10% $216 So, leveraged ETF Buy the futures when its price goes up Sell the futures when its price goes down This leads Momentum Trading This momentum trading Rebalance concern some people, This Rebalance makes Financial Market Unstable? (f) Rebalance = (d)-(e) +20 -24
Difficulty of Empirical Study There are many empirical studies that produce both opposite results Difficulty of Empirical Study Empirical studies cannot be conducted to isolate the pure contributions of L-ETFs due to the many diverse factors affecting price formation in actual markets Artificial Market Model can agent-based model for a financial market 6
An artificial market model = an agent-based model for a financial market Virtual and Artificial financial Market built on Computers Models Include Agents (Artificial Investors) ＋ Price Mechanism (Artificial Exchange) Order Price Mechanism (Exchange) Agent (Investor) Each Agent determines an order by some rules, Price Mechanism gather agents orders and determines Market Price Determination market price Complete Computer Simulation needing NO Empirical Data ✓ can isolate the pure contributions of L-ETFs ✓ can be conducted to investigate situations that have never occurred in actual financial markets ✓ can effectively handling micro-macro feedback loops 7
Previous Studies by Artificial Market Models [Yagi 2016] When a L-ETF impacts market prices more than their volatility Market will be crushed [Yagi 2020] Using a continuous double auction Investigated rebalance-trades to less impact to market prices However, no previous study implemented a market in which a L-ETF itself is traded in an artificial market model and investigated effects of falling market prices of L-ETFs to those of futures. Therefore in this study [Yagi 2020] + L-ETF market Arbitrage Agent We investigated that sharply falling of the futures or the L-ETF affects the another market and effects of rebalancing trades of the L-ETF 8
(1) Introduction (2) Our Model (3) Simulation Results (4) Conclusion You can download this presentation material from https://mizutatakanobu.com/2024CIFEr.pdf https://mizutatakanobu.com/2024CIFEr.pdf 9
Overall view of our model Normal agents (NAs) 1000 agents trade only the futures trades place an order with100% High liquidity An exchange for the futures Arbitrage Agent (AA) Only one NAs 1000 agents trades only The L-ETF place an order with probability 20% Low liquidity trades An exchange for the L-ETF rebalance -trades holding the futures L-ETF agent Only one Holding the futures 10
Continuous Double Auction Waiting Orders Shares Sell 10 30 50 130 When sell order come here transaction immediately occurs Price 103 102 101 100 99 98 97 96 Shares Buy 150 When buy order come here transaction immediately occurs 70 Multiple buyers and sellers compete to buy and sell stocks in the market, and transactions can occur at any time whenever an offer to buy and an offer to sell match. 11
Normal Agents (NAs) This term is needed To replicate stylized facts 1000 agents j: agent number ordering in number order t: tick time Expected Return of each agent Technical 𝑡−1 𝑃 1 𝑃 𝑓 𝑡 𝑟𝑒,𝑗 = 𝑤1,𝑗 log 𝑡−1 + 𝑤2,𝑗 log 𝑡−𝜏𝑗 + 𝑤3,𝑗 𝜀𝑗𝑡 σ𝑖 𝑤𝑖,𝑗 𝑃 𝑃 noise wi , j and j Fundamental Pf Fundamental Price tj Random of Uniform Distribution P t Market Price at t Parameters for agents wi , j j i=1,3: 0～1 i=2: 0～100 0～10000 Random of Normal Distribution Average=0 σ=3% 10000 = constant This term is needed to prevent the prices go out far away To keep agents varied To keep simulation runs stably Expected Price 𝑡 𝑃 𝑒,𝑗 = 𝑃𝑡 exp( 𝑟 𝑡 𝑒,𝑗 ) of each agent 12
Fundamental Strategy Fundamental Price ＞ Market Price -> Expect + return Fundamental Price < Market Price -> Expect - return Technical Strategy (Historical Return) Historical Return ＞ 0 -> Expect + return Historical Return < 0 -> Expect - return Technical Strategy Fundamental Price Market Price Fundamental Strategy 13
Erroneous orders Erroneous orders period 30000-60000 Dropping depth time Within the period, with a constant probability 8% for the future, 40% for the L-ETF <- same total erroneous orders each order of the NAs is changed to sell with very low price Increasing such erroneous sell orders is what makes market prices fall 14
Arbitrage Agent (AA) Frequency of Trades AA t=1 t=2 t=3 t=4 t=5 time NAs The AA places an order and cancel an order any time 15
Price Relationship between L-ETF and Futures Process of an arbitrage in real financial markets is very complex, e. g., a ratio that investors can equivalent-exchange the L-ETF for the futures are pass-depended of time evolution of their market prices. In this study, the process of an arbitrage was modelized very simply The Leverage is twice (x2) L-ETF Return = 2 × Futures Return Fundamental P Futures L-ETF This means 10000 5000 Market Prices Price Change Range 10000 about 5000 --- about same as futures Example of Price Change +500 +500 same L-ETF Price = Futures Price - 5000 Arbitrage Agent Example of Return 5% 10% Twice place orders to maintain the price relationship 16
Immediately executed arbitrage-trades L-ETF Sell Price 7 5030 10 5020 7 5010 12 5000 4990 Best Sell Price 8 Best Buy Price 4980 4970 4960 Buy AA 10 6 4 Futures Sell Price Buy 30 10400 44 10300 70 10200 134 10100 10000 120 9900 88 9800 52 9700 25 Usually best sell price > best buy price, rarely can be reversed when orders from two different exchanges are combined AA can earn to buy at the best sell price and sell at the best buy price If L-ETF best Sell + 5000 ＜ Future best Buy Buy L-ETF Sell Futures If Futures best Sell - 5000 ＜ L-ETF best Buy Buy Futures Sell L-ETF Taking all arbitrage opportunities at that moment 17
Arbitrage-trades by waiting orders Sell 7 10 L-ETF Price 5030 5020 5010 5000 4990 4980 4970 4960 Buy 1 10 6 4 Wait AA Sell 30 44 70 134 Immediately Sell After waiting order executed Futures 価格 Buy 10400 10300 10200 10100 10000 120 9900 88 9800 52 9700 25 AA places the waiting order to earn by immediately executing for another asset If L-ETF best Buy < Futures best Buy - 5000 ＜ L-ETF best Sell Placing buy waiting order at Futures best Buy – 5000 on L-ETF When the waiting order is executed, AA sells Futures L-ETF best Buy < Futures best Sell - 5000 ＜ L-ETF best Sell, vice versa 18
L-ETF Agent The L-ETFA decides every 10 if it rebalance-trades t=10 Holding Futures L-ETF Agent t=20 t=30 t=40 time NAs To maintain leverage twice, it should holds S0×(Future Price/10000) It trades when different more than 1% Buy the futures when its price goes up Sell the futures when its price goes down This Rebalance makes Financial Market Unstable? 19
(1) Introduction (2) Our Model (3) Simulation Results (4) Conclusion You can download this presentation material from https://mizutatakanobu.com/2024CIFEr.pdf https://mizutatakanobu.com/2024CIFEr.pdf 20
5200 Time evolution of L-ETF prices Liquidity of the futures is consumed to prevent the fall of the L-ETF 5000 Market prices AA reduces the fall 5100 4900 4800 4700 4600 4500 4400 4300 140000 130000 120000 110000 90000 100000 Time 80000 70000 60000 50000 40000 30000 20000 10000 0 Rebalance-trades make enlarge the fall of market prices 4200 Without either the erroneous orders nor the AA Without the AA Without the rebalance-trades With the rebalance-trades(S0=50000) 21
Arbitrage NAs Sell 7 10 7 12 8 L-ETF Price 5030 5020 5010 5000 4990 4980 4970 4960 Buy AA 10 6 4 Futures Sell Price Buy 30 10400 44 10300 70 10200 134 10100 10000 120 9900 88 9800 52 9700 25 NAs sell L-ETF -> AA buy L-ETF -> AA Sell Futures So, NAs sell futures indirectly Liquidity of the futures is consumed to prevent the fall of the L-ETF Liquidity of the futures is used as hidden liquidity of the L-ETF 22
Market prices of the futures 140000 130000 120000 9200 110000 4200 100000 9400 90000 4400 80000 9600 70000 4600 60000 9800 50000 4800 40000 10000 30000 5000 20000 10200 10000 5200 0 Market prices of the L-ETF Time evolution of L-ETF and Futures prices with AA Time The L-ETF (left) The futures (right) The two lines overlap each other when the market prices of the L-ETF change double with those of the futures 23
The amount of rebalance-sell orders 50000 20000 10000 5000 2000 1600 1400 1200 1000 800 600 400 200 0 0 Larger the L-ETF (larger S0), larger falling for both the L-ETF and the futures 800 700 600 500 400 300 200 100 0 without the AA Consuming liquidity of the futures reduces falling of the L-ETF, on the other hand, the futures also fall. Falling depth Case with the erroneous orders to the L-ETF for several S0 S0(the initial number of futures the L-ETFA holds) Falling depth of the L-ETF (left) Destroying markets with too big L-ETF (S0>100000) Falling depth of the futures (left) The amount of rebalance-sell orders of the AA (right) 24
1000 400 800 300 600 200 400 100 200 0 0 Investors of L-ETFs have great benefit by gaining much liquidity of futures. On the other hand, Investors of futures do not have The amount of rebalance-sell orders 500 50000 1200 20000 600 10000 1400 5000 700 2000 1600 0 But with AA the results are almost same as the previous case 800 without the AA Without AA, The falling is not so deep than the case of L-ETF. Erroneous orders to Futures are less than L-ETF Falling depth Case with the erroneous orders to the Futures for several S0 S0(the initial number of futures the L-ETFA holds) Falling depth of the L-ETF (left) Falling depth of the futures (left) The amount of rebalance-sell orders of the AA (right) 25
Mechanism of destroying markets with too big L-ETF (S0>100000) The L-ETF rebalancesells futures The number of futures the L-ETF should sell is larger The futures fall The futures fall more The L-ETF rebalancesells futures When the L-ETF is so big, the sells largely impact to market prices of the futures Sells largely impact to market prices, the futures fall more, the L-ETF should additionally sell the futures Such process continues as loop and falling can not stop This result is consistent with [Yagi 2016] 26
(1) Introduction (2) Our Model (3) Simulation Results (4) Conclusion You can download this presentation material from https://mizutatakanobu.com/2024CIFEr.pdf https://mizutatakanobu.com/2024CIFEr.pdf 27
Summary (1/2) ✓ In this study, the artificial market model was built by adding a market in which a L-ETF itself is traded and adding a erroneous orders leading turmoil implemented by the model of [Mizuta 2016] to the prior model of [Yagi 2020]. We investigated that sharply falling of the futures or the L-ETF affects the another market and effects of rebalancing trades of the L-ETF. ✓ In the result, the erroneous orders to the L-ETF leads that market prices of the L-ETF fall, but the arbitrage-trades reduce the fall. Here, waiting buy orders to the futures, not to the L-ETF, are executed and reduced because a sell order of a normal agent executes a buy order of the AA and then the AA sells the futures. The buy orders of the AA to the LETF reduce the fall of the L-ETF. The reason why the AA can buy the L-ETF is that the AA can consume the waiting buy orders of the futures. In short, liquidity of the futures is consumed to prevent the fall of the L-ETF. This means that liquidity of the futures is used as hidden liquidity of the L-ETF. You can download this presentation material from https://mizutatakanobu.com/2024CIFEr.pdf https://mizutatakanobu.com/2024CIFEr.pdf 28
Summary (2/2) ✓ In the case that the erroneous orders to the futures, the arbitrage-trades by the AA mix and combine liquidity of the L-ETF and futures, therefore, same falling happens even which to the L-ETF or the futures the erroneous orders come. The futures can gain only a less liquidity from the L-ETF that has a less liquidity, then this leads not to reduce falling so much. In real financial market, this means that investors of L-ETFs have great benefit by gaining much liquidity of futures, on the other hand, investors of futures do not have so benefit by gaining low liquidity of L-ETFs. ✓ Larger the L-ETF, larger falling for both the L-ETF and the futures. When S0 > 100000, the simulation can not run due to too large falling of market prices. This means that markets are destroyed. When the futures fall, the L-ETF sells futures. Here, when the LETF is so big, the futures also fall, and then the number of futures the L-ETF should sell is larger, finally, the L-ETF should additionally sell the futures. Such process continues as loop and falling can not stop. This leads to destroy the markets. You can download this presentation material from https://mizutatakanobu.com/2024CIFEr.pdf https://mizutatakanobu.com/2024CIFEr.pdf 29
Future Works ✓ The difference in results in the case of erroneous orders to the futures or L-ETF may depend only on the lower liquidity. This is a future work. ✓ Note that an arbitrage in real financial markets is extremely more complex than our model. In the actual, market prices of L-ETFs do not track preciously like as our model. Some process existing in real financial market but not included in our model could be important. All of them are future works. You can download this presentation material from https://mizutatakanobu.com/2024CIFEr.pdf https://mizutatakanobu.com/2024CIFEr.pdf 30
References [Yagi 2016] Yagi, I., Mizuta, T., “Analysis of the Impact of Leveraged ETF Rebalancing Trades on the Underlying Asset Market Using Artificial Market Simulation”, 12th Artificial Economics Conference, September 20-21, 2016 http://ae2016.it/public/ae2016/files/ssc2016_Mizuta.pdf ⚫ [Yagi 2016] Yagi, I., Mizuta, T., “Analysis of the Impact of Leveraged ETF Rebalancing Trades on the Underlying Asset Market Using Artificial Market Simulation”, 12th Artificial Economics Conference, September 20-21, 2016 http://ae2016.it/public/ae2016/files/ssc2016_Mizuta.pdf [Yagi 2020] Yagi, I., Maruyama, S., Mizuta, T., “Trading strategies of a leveraged ETF in a continuous double auction market using an agent-based simulation”, Complexity, 2020 https://doi.org/10.1155/2020/3497689 ⚫ [Yagi 2020] Yagi, I., Maruyama, S., Mizuta, T., “Trading strategies of a leveraged ETF in a continuous double auction market using an agent-based simulation”, Complexity, 2020 https://doi.org/10.1155/2020/3497689 You can download this presentation material from https://mizutatakanobu.com/2024CIFEr.pdf https://mizutatakanobu.com/2024CIFEr.pdf 31
References (Reviews of agent-based models for a financial market) Review (2020) Mizuta https://arxiv.org/abs/1906.06000 of an agent-based An agent-based model model for designing for designing a financial a financial market market that works well, CIFEr 2020 arXiv Review of an agent-based model for designing a financial market Mizuta (2020) An agent-based model for designing a financial market that works well, CIFEr 2020 arXiv https://arxiv.org/abs/1906.06000 Slide: https://mizutatakanobu.com/2021kyushu.pdf YouTube: https://youtu.be/rmlb72ykmlE Slide: https://mizutatakanobu.com/2021kyushu.pdf YouTube: https://youtu.be/rmlb72ykmlE Mizuta (2022) Artificial Intelligence (AI) for Financial Markets: A Good AI for Designing Better Financial Markets and a Bad AI for Manipulating Markets https://doi.org/10.1007/978-981-19-0937-5_13 Mizuta (2022) Artificial Intelligence (AI) for Financial Markets: A Good AI for Designing Better Financial Markets and a Bad AI for Manipulating Markets https://doi.org/10.1007/978-981-19-0937-5_13 Citing many Mizuta https://ssrn.com/abstract=2710495 (2016) previous A Brief studies Review of Recent Artificial Market Simulation Studies for Financial Market Regulations And/Or Rules, SSRN Working Paper Series Citing many previous studies Mizuta (2016) A Brief Review of Recent Artificial Market Simulation Studies for Financial Market Regulations And/Or Rules, SSRN Working Paper Series https://ssrn.com/abstract=2710495 You can download this presentation material from https://mizutatakanobu.com/2024CIFEr.pdf https://mizutatakanobu.com/2024CIFEr.pdf 32
Appendix You can download this presentation material from https://mizutatakanobu.com/2024CIFEr.pdf https://mizutatakanobu.com/2024CIFEr.pdf 33
Order Price and Buy or Sell Price Order Price of each NA P t o, j Sell(one unit) Random (Uniformly Distribution) Expected Price of each NA P e , j = P exp( r e , j ) t t t Buy(one unit) To replicate many waiting limit orders, order price is scattered around expected price NA places one buy order when order price > expected price NA places one sell order when order price < expected price 34
Verification: Stylized Facts The purpose of simulation is understanding the reasons and mechanism, not replicating ALL Stylized Facts The simplicity of the model is very important because unnecessary replication of macro phenomena leads to models that are overfitted and too complex. Such models prevent understanding and discovery of mechanisms affecting price formation because of the increase in related factors. Many empirical studies, e.g., Sewell 2006 have shown that both stylized facts (fat-tail and volatilityclustering) exist statistically in almost all financial markets. Conversely, they also have shown that only the fat-tail and volatility-clustering are stable for any asset and in any period because financial markets are generally unstable. Fat-tail 1 to 100 kurtosis of price returns is positive Volatility-clustering 0 to 0.2 square returns have a positive auto-correlation The magnitudes of these values are unstable and vary greatly depending on the asset and/or period. For the above reasons, an artificial market model should replicate these values as significantly positive and within a reasonable range as I mentioned. It is not essential for the model to replicate specific values of stylized facts because the values of these facts are unstable in actual financial markets. 35
Stylized Facts The model of Chiarella (2002) is very simple but replicates long-term statistical characteristics observed in actual financial markets: a fat tail and volatility clustering. In contrast, Mizuta (2013) replicates high-frequency micro structures, such as execution rates, cancel rates, and one-tick volatility, that cannot be replicated with the model of Chiarella (2002). The simplicity of the model is very important for this study, because unnecessary replication of macro phenomena leads to models that are overfitted and too complex. Such models prevent understanding and discovery of mechanisms affecting price formation because of the increase in related factors. 36