# Where is the Value in High Frequency Trading?

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14 which, if we drop the superscripts and the functional dependence of asset holding on prices, is ) W 3 = W + θ P 3 (θ + i) P ( θ + i. Substituting for wealth in the utility function, we obtain: ( ( ) ) E [U (W 3 ) F ] = exp a θ [µ P ] ip W ( θ + i + ) a σ (θ ). Trader LT s final asset demand, θ LT, is such that it maximizes this expression. Hence, θ LT (P ) = µ P aσ. () Similarly, LT s and MMs asset demands at t = can be computed and we obtain θ LT (P ) = θ MM (P ) = µ P aσ. () Market clearing requires that changes in asset holdings equal to zero: ( ) ( i + θ LT + θ LT θ LT ) ( + M θ MM θ MM ) = 0. Using the date market clearing condition ( i θ LT ) Mθ MM = 0 implies that θ LT + θ LT + Mθ MM = 0, so that the equilibrium price is P = µ, and the optimal asset positions at the end of period are θ LT = θ LT = θ MM = 0. The HFT, on the other hand, will extract trading surplus from the assets transacted equal to 6 θ LT ( P + LT ) + i LT + θ LT ( P LT ) θ LT LT +M θ MM ( P MM ) θ MM 6 We are assuming that the market makers and LT are net sellers of the asset. This is confirmed as in equilibrium MMs will be net buyers in the first period and have zero net final holdings (and θ LT θ LT has the same sign as MMs changes in asset holdings at date two). MM. 4

15 3.3 The general model: Trading at t = Traders at date t = anticipate what will happen at t =. They also know that there will be public information revealed prior to date two trading so that P is normally distributed with mean µ and variance σ (we keep a constant variance solely to reduce notational clutter). Traders future wealth can be written as W3 LT = W0 LT + θ LT P 3 + ( θ LT θ LT W MM 3 = W MM 0 + θ MM P 3 + ( θ MM θ MM ) P (θ LT ) P (θ MM ) LT θ θ MM LT ( i ) LT θ LT, θ LT P + ) MM θ MM MM P θ MM.(3) Using E [P F ] = E [E [P 3 F ] F ] = µ, simplifies the expression for traders wealth, and it is straightforward to derive optimal demands: θ LT = θ MM = ( µ P aσ LT ), (4) ( µ P aσ MM ). (5) Notice that traders anticipate that their current asset demand (and hence positions at the end of trading at t = ) will affect future trading and hence the haircuts they will have to pay. The market clearing condition is now i + θ LT + Mθ MM = 0. From this we obtain the market clearing price: and traders asset demand: P = µ LT + M MM + iaσ, (6) M + θ LT = θ MM = i M + + aσ M ( MM M + ( MM i M + aσ M + LT ), (7) LT ). (8) 5

16 Again, the HFT is able to extract trading surplus and generate profits of i θ LT ( P LT ) LT + M θ MM ( P + MM ) MM, where i θ LT ( P LT ) M = M + i M M + MM LT aσ + LT aσ. In summary, from the above analysis we extract the following conclusions: Theorem 3.. For a given order imbalance of magnitude i:. Market clearing prices are. Asset trading is: P = µ LT + M MM + iaσ, P = µ. M + Price LT LT MM (total) P LT (i M MM LT M+ P LT aσ P + MM aσ ) + LT aσ P i LT aσ P - LT P + LT M+ (i + M MM LT aσ aσ ) LT + LT aσ aσ P + MM M M+ The HFT acts as counterparty to all these trades. M (i MM M+ (i MM M MM aσ ) aσ LT M MM aσ M MM aσ ) LT +M MM aσ aσ Thus, the HFT affects the market clearing price. Traders, anticipating the haircuts imposed by the HFT, amplify the selling pressure of the liquidity trader, driving the market clearing price in the first period down further than it would have been without HF trading. Also, the presence of the HFT introduces additional prices at which transaction takes place. These price movements around the market clearing price can be interpreted as additional microstructure volatility. Furthermore, it can be seen from the table that most of the volume takes place around the market clearing price, and very little at the market clearing price, which 6

18 In the Appendix we discuss another case (where the HFTs make no distinctions and apply the same haircut to all trades). Given these assumptions on the HFTs informational advantages, the HFTs problem is to maximize profits, which from Theorem 3. are: ( M L M + i ) ( L M aσ + S M + i M ) ( S aσ + L i ) ( L aσ + S i (M + ) ) S aσ. Lemma 3.4. The optimal haircuts are S = (M + ) iaσ, L = 4 M + M + iaσ = S ( M + ). This result backs the intuition that the haircut imposed on large trades is bigger than that applied to small trades, and the increase in haircut for large trades is proportional to the (average) number of other, non-liquidity seeking, market participants. Moreover, we see that with more MMs present in the market, the HFT increases the haircut imposed on the LT in date and on LT in date ( L / and S 0 as M ). Figure below shows that the haircut for the large (small) trades is increasing (decreasing) in M. Interestingly, the initial holdings of LT are the same as without a HFT: θ LT = i M +. Figure shows the optimal asset holdings, θ t and θ t, for the liquidity traders and MMs. 8

19 Figure : Optimal Haircuts for large and small trades 0.5 T = T M = M = T Demands assuming i = a = σ = Number of Market Makers Figure : Optimal asset holdings when the HFT sets a haircut for large trades and another haircut for small trades. θ T = θ M θ T θ M = θ T θ M T = θ = θ M θ T = θ M = θ M θ M = θ M T = θ θ T Demands assuming i = a = σ = Number of Market Makers 9

21 HFT intermediates every trade in order to ensure it does not end up with a non-zero stock of assets. Third, although the MMs also pay a haircut in each transaction, the average price they pay to purchase shares at date is lower than the price paid in the absence of the HFT, and the average price at which MMs sell at date is also lower than that received if there were no HFTs extracting trading surplus. The overall effect of lower buy prices and lower sell prices for MMs is that expected returns from holding inventories from date to is higher when a HFT extracts surplus on both rounds of trades. However, the higher return does not imply greater expected profits for MMs. Fourth, HF trading increases the volatility of prices. 4. Number of Market Makers Our model allows us to derive the impact of HFT s surplus extraction activities on the number of MMs, and hence on the true supply of liquidity in the market. We proceed as in GM and assume that the cost of entry for the MM is c. This cost is sunk before knowing the liquidity shock i which we assume to be normally distributed and independent of the other shocks affecting prices. At time t = 0 the expected utility of an individual MM is E [ U ( W MM 3 c ) F 0 ], where W MM 3 is given by (3). Free entry of MM will occur until E [ U ( W3 MM c ) ] [ ( ) ] F0 = E U W MM F0, (9) and recall that θ MM = 0. The expected value and variance of wealth in period three are 0 E [W 3 F ] = W0 MM c + aσ ( θ MM ) θ M MM + ( ( MM aσ ) ( ) + MM ) and Var [ W M 3 ] F = σ ( θ MM ). (0) Hence the entry condition (9) becomes

23 so that most studies focus on measuring liquidity in two ways. One measure of liquidity is how much of the quantity i the price sensitive liquidity trader (LT) is willing to sell in period at the market clearing price, in reaction to the price impact of his trade. According to this quantity metric liquid markets would be those where LT carries the smallest holdings over to the next period (that is θ T is smallest recall that LT sells ( i θ T ) in period ). In the particular case analyzed here, where the HFT sets a haircut for small trades and a haircut for large trades, the LT carries over an amount θ T = i/(m + ) which is the same as what LT carries over if there were no HFT, so that the presence of the HFT would not alter liquidity at all. The other measure of liquidity is to focus on price impact. As we have seen above, our analysis indicates that price impact might be a better metric. This is because although LT sells the same amount in period with and without an HFT, the amount of cash he obtains (and hence the liquidity he attains from the sale) is much lower due to the presence of the HFT both because of the direct loss from the haircuts, but also the additional losses from lower market clearing prices. Thus, it is liquidity traders, and not MMs (whose expected utility and numbers stay the same), the ones who take the brunt of the presence of HFTs. Using our model, we can calculate the effect of HFTs on the prices and returns faced by all traders, and compare them to those obtained in the absence of a HFT. When there is no HFT the market clearing price (for buyers and sellers) are: P =0 = µ iaσ M +, P =0 = µ. iaσ This implies that MMs receive a liquidity premium at date t = of M+, and obtain an expected return of E [ r =0 F ] = iaσ M + P =0. () 3

24 On the other hand, when there is an HFT, buyers and sellers face different prices. The MM pays a haircut to the HFT in date (when he buys θ Mat P + MM per share, and then θ MM θ MM at P per share) so that the average price is P M = P + θ M θ M M = µ iaσ M + iaσ 4 M + = P =0 iaσ 4 M +. (3) From which we see that he obtains a 5% higher liquidity premium (after including the haircut paid to the HFT). Similarly, at date P M = µ M + θ M θ M M = P =0 iaσ 4 M +, (4) so that he ends up selling back at a lower price (relative to the case without HFT), and thereby losing the initial extra liquidity premium garnered from the first transaction. Note that if we compute the expected excess return obtained by the MM: r = ( P M / P M ) we have E [r] = iaσ M + P M > E [ r =0 ]. That is, the expected returns in the presence of HFT are higher than in the absence of HFT. As mentioned above, this higher return is purely an artifact of the lowering of both buy and sell prices a similar (though reversed) effect would appear if the initial liquidity imbalance had a different sign (that is, if i < 0). i As for liquidity traders, the HFT extracts surplus from LT in date (by first buying θ LT at P LT and then date (Figure 3) is θ LT θ LT at P ). The the average price received by T at P T = P i θ T i θ T T. (5) 4

25 Figure 3: Average price for LT in period.5 With HFT No HFT Average price for LT in period, i = a = and µ = Number of Market Makers At date, after the haircut the average price he sells his remaining assets is: P T = µ T + θ T θ T T = P =0 iaσ 4 M +. (6) From Equation (6) we can see that the liquidity premium increases in proportion to the size of the trade. This implies that LT receives less money for liquidating his position at both dates (relative to the case without HFTs) as can be seen in Figure 3. Similarly, the average price paid by LT in period is P T = µ T + θ T i T. (7) So that LT also suffers as she pays more to acquire her position than when there is no HFT. Note that LT s asset demand at date is distorted by the presence of the HFT, as he anticipates that he will pay a (lower) haircut when selling at date. 5

26 Figure 4: Average price for LT (sells) and LT (buys) in period.5.4 LT (sells) with HFT No HFT LT (buys) with HFT Average price for LT and LT in period, i = a = and µ = Number of Market Makers We see that the liquidity traders, LT and LT, bear the brunt of the haircut imposed by the monopolist HFT. Although MMs suffer the loss of consumer (date ) and supplier (date ) surpluses, the liquidity premium received by the MMs for shares at date is greater (the equilibrium price is lower) than in the absence of the HFT, and the net effect on his expected wealth is zero. 6

27 With HFT No HFT.9 Average price for MM in period, i = a = and µ = Number of Market Makers. With HFT No HFT.05 Average price for MM in period, i = a = and µ = Number of Market Makers 7

28 4.4 Price volatility When there is no HFT intermediating transactions between LTs and MMs there is one transaction price at date, P =0, and another transaction price at date, µ. However, when a monopolistic HFT intermediates all transactions the tape will record four prices in date, { P T, P, P + M, P }, and four (five or six if MM LT ) in date, { µ MM, µ, µ + T, µ}. Therefore, it is inevitable to observe price volatility within dates and which is solely caused by the HFT s presence. Figure 5 shows the volatility of prices in dates and that result from the HFT intermediating all trades. 8 Furthermore, Figure 6 depict the standard deviation of prices at both dates. We see that price volatility is much higher when an HFT operates in the markets. Note however that in our model this increased volatility does not generate additional risk for traders. Traders, in their evaluation of price risk evaluation in their objective functions, recognize that the microstructure noise generated by the HFT translates into a deterministic effect on their execution costs while leaving unaffected their final holdings. 9 5 Conclusions We have studied the impact of a monopolistic HFT that extracts trading surplus in transactions between liquidity traders and market makers. Clearly, the HFT comes out making profits. As a monopolist, she would enter the market only if these profits exceed the necessary entry costs. Introducing competition between HFTs would reduce these profits until HFT profits just cover investment costs for marginal HFTs. Thus the social value of HFT profits revolves around the 8 To calculate the mean and variance of prices we also use the price at time t = 0 and assume it is the fundamental value µ, thus the mean and variance of the price in date are P = µ + 4P + ( ) M T and V [P ] = { (µ ) ( P + P T 5 5 P ) ( + P P ) + (P + M P ) }. Similarly, to calculate the mean and variance of prices with date we assume that the first observation is µ. 9 This changes if we allow for the HFT to (randomly) miss the opportunity to intermediate some trades between LT and MMs. While adding such a feature will make the model more realistic, it complicates the analysis substantially without much additional insight. 8

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