Competitive Algorithms for VWAP and Limit Order Trading

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6 as: REV M (S, S ) q tw t, REV A(S, S ) r tx t T Note that both these quantities are solely determined by the execution sequences exec M (S, S ) and exec A(S, S ), respectively. For an algorithm A which is constrained to sell exactly N shares, we define the OWT competitive ratio of A, R OWT(A), as the maximum ratio (under any S Σ) of the revenue obtained by A, as compared to the revenue obtained by an optimal offline algorithm A. More formally, for A which is constrained to sell exactly N shares, we define R OWT(A) = max max REV A (SS ) S Σ A REV A(S, S ) where S is the limit order sequence placed by A on S. If the algorithm A is randomized then we take the appropriate expectation with respect to S A. We define the VWAP competitive ratio, R VWAP(A), as the maximum ratio (under any S Σ) between the market and algorithm VWAPs. More formally, define VWAP M (S, S ) as REV M (S, S )/ T wt, where the denominator is just the total executed ume of orders placed by the market. Similarly, we define VWAP A(S, S )asrev A(S, S )/N, since we assume the algorithm sells no more than N shares (this definition implicitly assumes that A gets a 0 price for unsold shares). The VWAP competitive ratio of A is then: R VWAP(A) = max S Σ {VWAPM (S, S )/VWAP A(S, S )} where S is the online sequence of limit orders generated by A in response to the sequence S. 3.3 OWT Results in the Order Book Model For the OWT problem in the order book model, we introduce a more subtle version of the price variability assumption. This is due to the fact that our algorithm s trading can impact the high and low prices of the day. For the assumption below, note that exec M (S, ) is the sequence of executions without the interaction of our algorithm Order Book Price Variability Assumption.. Let 0 <p min p max be known positive constants, and define R = p max/p min. For all intraday trading sequences S Σ, the prices p t in the sequence exec M (S, ) satisfy p t [p min,p max], for all t =1,...,T. Note that this assumption does not imply that the ratios of high to low prices under the sequences exec M (S, S )or exec A(S, S ) are bounded by R. In fact, the ratio in the sequence exec A(S, S ) could be infinite if the algorithm ends up selling some stocks at a 0 price. Theorem 6. In the order book model under the order book price variability assumption, there exists an online algorithm A for selling N shares achieving sell OWT competitive ratio R OWT(A) = 2 log(r) log(n). Proof. The algorithm A works by guessing a price p in the set {p min2 i :1 i log(r)} and placing a sell limit order for all N shares at the price p at the beginning of the day. (Alternatively, algorithm A can place log(r) sell T limit orders, where the i-th one has price 2 i p min and ume N/ log(r).) By placing an order at the beginning of the day, the algorithm undercuts all sell orders that will be placed during the day for a price of p or higher, meaning the algorithm s N shares must be filled first at this price. Hence, if there were k shares that would have been sold at price p or higher without our activity, then A would sell at least kp shares. We define {p j} to be the multiset of prices of individual shares that are either executed or are buy limit order shares that remained unexecuted, excluding the activity of our algorithm (that is, assuming our algorithm places no orders). Assume without loss of generality that p 1 p 2... Consider guessing the kth highest such price, p k. Ifanorder for N shares is placed at the day s start at price p k, then we are guaranteed to obtain a return of kp k. Let k = argmax k {kp k }. We can view our algorithm as attempting to guess p k, and succeeding if the guess p satisfies p [p k /2,p k ]. Hence, we are 2 log(r) competitive with the quantity max 1 k N kp k. Note that ρ = N i=1 N i=1 p i 1 i ipi max 1 k N kp k log(n) max kp k 1 k N where ρ is defined as the sum of the top N prices p i without A s invement. Similarly, let {p j} be the multiset of prices of individual executed shares, or the prices of unexecuted buy order shares, but now including the orders placed by some selling algorithm A. We now wish to show that for all algorithms A which sell N shares, REV A N i=1 p i ρ. Essentially, this inequality states the intuitive idea that a selling algorithm can only lower executed or unmatched buy order share prices. To prove this, we use induction to show that the removal of the activity of a selling algorithm causes these prices to increase. First, remove the last share in the last sell order placed by either A or the market on an arbitrary sequence merge(s, S ) by this we mean, take the last sell order placed by A or the market and decrease its ume by one share. After this modification, the top N prices p 1...p N will not decrease. This is because either this sell order share was not executed, in which case the claim is trivially true, or, if it was executed, the removal of this sell order share leaves an additional unexecuted buy order share of equal or higher price. For induction, assume that if we remove a share from any sell order that was placed, by A or the market, at or after time t then the top N prices do not decrease. We now show that if we remove a share from the last sell order that was placed by A or the market before time t, then the top N prices do not decrease. If this sell order share was not executed, then the claim is trivially true. Else, if the sell order share was executed, then claim is true because by removing this executed share from the sell order either: i) the corresponding buy order share (of equal or higher value) is unmatched on the remainder of the sequence, in which case the claim is true; or ii) this buy N i=1 1 i

7 order matches some sell order share at an equal or higher price, which has the effect of removing a share from a sell order on the remainder of the sequence, and, by the inductive assumption, this can only increase prices. Hence, we have proven that for all A which sell N shares REV A ρ. We have now established that our revenue satisfies 2 log(r)e S A[REV A(S, S )] max k} 1 k N ρ/ log(n) max{rev A }/ log(n), A where A performs an arbitrary sequence of N sell limit orders. 3.4 VWAP Results in the Order Book Model The OWT algorithm from Theorem 6 can be applied to obtain the following VWAP result: Corollary 7. In the order book model under the order book price variability assumption, there exists an online algorithm A for selling N shares achieving sell VWAP competitive ratio R VWAP(A) =O(log(R) log(n)). We now make a rather different assumption on the sequences S Bounded Order Volume and Max Price Assumption. The set of sequences Σ satisfies the following two properties. First, we assume that each order placed by the market is of ume less than γ, which we view as a mild assumption since typically single orders on the market are not of high ume (due to liquidity issues). This assumption allows our algorithm to place at least one limit order at a time interleaved with approximately γ market executions. Second, we assume that there is large ume in the sell order books below the price p max, which means that no orders placed by the market will be executed above the price p max. The simplest way to instantiate this latter assumption in the order book model is to assume that each sequence S Σ starts by placing a huge number of sell orders (more than V max ) at price p max. Although this assumption has a maximum price parameter, it does not imply that the price ratio R is finite, since it does not imply any lower bound on the prices of buy or executed shares (aside from the trivial one of 0). Theorem 8. Consider the order book model under the bounded order ume and max price assumption. There exists an algorithm A in which after exactly γn market executions have occurred, then A has sold at most N shares and REV A(S, S ) N = VWAP A(S, S ) (1 ɛ)vwap M (S, S ) pmax ɛn where S is a sequence of N sell limit orders generated by A when observing S. Proof. The algorithm divides the trading day into ume intervals whose real-time duration may vary. For each period i in which γ shares have been executed in the market, the algorithm computes the market VWAP of only those shares traded in period i; let us denote this by VWAP i. Following this ith ume interval, the algorithm places a limit order to sell exactly one share at a price close to VWAP i. More precisely, the algorithm only places orders at the discrete prices (1 ɛ)p max, (1 ɛ) 2 p max,... Following ume interval i, the algorithm places a limit order to sell one share at the discretized price that is closest to VWAP i, but which is strictly smaller. For the analysis, we begin by noting that if all of the algorithm s limit orders are executed during the day, the total revenue received by the algorithm would be at least (1 ɛ)vwap M (S, S )N. To see this, it suffices to note that VWAP M (S, S ) is a uniform mixture of the VWAP i (since by definition they each cover the same amount of market ume); and if all the algorithm s limit orders were executed, they each received more than (1 ɛ)vwap i dollars for the interval i they followed. We now count the potential lost revenue of the algorithm due to unexecuted limit orders. By the assumption that individual orders are placed with ume less than γ, then our algorithm is able to place a limit order during every block of γ shares have been traded. Hence, after γn market orders have been executed, A has placed N orders in the market. Note that there can be at most one limit order (and thus, at most one share) left unexecuted at each level of the discretized price ladder defined above. This is because following interval i, the algorithm places its limit order strictly below VWAP i,soifvwap j VWAP i for j>i, this limit order must have been executed. Thus unexecuted limit orders bound the VWAPs of the remainder of the day, resulting in at most one unexecuted order per price level. A bound on the lost revenue is thus the sum of the discretized prices: i=1 (1 ɛ)i p max p max /ɛ. Clearly our algorithm has sold at most N shares. Note that as N becomes large, VWAP A approaches 1 ɛ times the market VWAP. If we knew that the final total ume of the market executions is V, then we can set γ = V/N, assuming that γ>>1. If we have only an upper and lower bound on V we should be able to guess and incur a logarithmic loss. The following assumption tries to capture the market ume variability Order Book Volume Variability Assumption. We now assume that the total ume (which includes the shares executed by both our algorithm and the market) is variable within some known region and that the market ume will be greater than our algorithms ume. More formally, for all S Σ, assume that the total ume V of shares traded in exec M (S, S ), for any sequence S of N sell limit orders, satisfies 2N V min V V max. Let Q = V max /V min. The following corollary is derived using a constant ɛ =1/2 and observing that if we set γ such that V γn 2V then our algorithm will place between N and N/2 limit orders. Corollary 9. In the order book model, if the bounded order ume and max price assumption, and the order book ume variability assumption hold, there exists an online algorithm A for selling at most N shares such that VWAP A(S, S ) 1 4 log(q) VWAPM (S, S ) 2pmax N

8 100 QQQ: log(q)=4.71, E= JNPR: log(q)=5.66, E= x MCHP: log(q)=5.28, E= x x 10 6 CHKP: log(q)=6.56, E= x 10 6 Figure 3: Here we present bounds from Section 4 based on the empirical ume distributions for four real stocks: QQQ, MCHP, JNPR, and CHKP. The plots show histograms for the total daily umes transacted on Island for these stocks, in the last year and a half, along with the corresponding values of log(q) and E(P bins ) (denoted by E ). We assume that the minimum and maximum daily umes in the data correspond to V min and V max, respectively. The worst-case competitive ratio bounds (which are twice log(q)) of our algorithm for those stocks are 9.42, 10.56, 11.32, and 13.20, respectively. The corresponding bounds on the competitive ratio performance of our algorithm under the ume distribution model (which are twice E(P bins )) are better: 7.54, 7.72, 7.94, and 9.00, respectively (a 20 40% relative improvement). Using a finer ume binning along with a slightly more refined bound on the competitive ratio, we can construct algorithms that, using the empirical ume distribution given as correct, guarantee even better competitive ratios of 2.76, 2.73, 2.75, and 3.17, respectively for those stocks (details omitted). 4. MACROSCOPIC DISTRIBUTION MODELS We conclude our results with a return to the price-ume model, where we shall introduce some refined methods of analysis for online trading algorithms. We leave the generalization of these methods to the order book model for future work. The competitive ratios defined so far measure performance relative to some baseline criterion in the worst case over all market sequences S Σ. It has been observed in many online settings that such worst-case metrics can yield pessimistic results, and various relaxations have been considered, such as permitting a probability distribution over the input sequence. We now consider distributional models that are considerably weaker than assuming a distribution over complete market sequences S Σ. In the ume distribution model, we assume only that there exists a distribution P over the total ume V traded in the market for the day, and then examine the worst-case competitive ratio over sequences consistent with the randomly chosen ume. More precisely, we define R VWAP(A, P )=E V P [ max S seq(v ) ] VWAP M (S). VWAP A(S) Here V P denotes that V is chosen with respect to distribution P, and seq(v ) Σ is the set of all market sequences (p 1,v 1),...,(p T,v T ) satisfying T vt = V. Similarly, for OWT, we can define [ ] p R OWT(A, P maxprice) =E p Pmaxprice max. S seq(p) VWAP A(S) Here P maxprice is a distribution over just the maximum price of the day, and we then examine worst-case sequences consistent with this price (seq(p) Σ is the set of all market sequences satisfying max 1 t T p t = p). Analogous buy-side definitions can be given. We emphasize that in these models, only the distribution of maximum ume and price is known to the algorithm. We also note that our probabilistic assumptions on S are considerably weaker than typical statistical finance models, which would posit a detailed stochastic model for the step-by-step eution of (p t,v t). Here we instead permit only a distribution over crude, macroscopic measures of the entire day s market activity, such as the total ume and high price, and analyze the worst-case performance consistent with these crude measures. For this reason, we refer to such settings as the macroscopic distribution model. The work of El-Yaniv et al. [3] examines distributional assumptions similar to ours, but they emphasize the worst-

9 case choices for the distributions as well, and show that this leads to results no better than the original worst-case analysis over all sequences. In contrast, we feel that the analysis of specific distributions P and P maxprice is natural in many financial contexts and our preliminary experimental results show significant improvements when this rather crude distributional information is taken into account (see Figure 3). Our results in the VWAP setting examine the cases where these distributions are known exactly or only approximately. Similar results can be obtained for macroscopic distributions of maximum daily price for the one-way trading setting. 4.1 Results in the Macroscopic Distribution Model We begin by noting that the algorithms examined so far work by binning total umes or maximum prices into bins of exponentially increasing size, and then guessing the index of the bin in which the actual quantity falls. It is thus natural that the macroscopic distribution model performance of such algorithms (which are common in competitive analysis) might depend on the distribution of the true bin index. In the remaining, we assume that Q is a power of 2 and the base of the logarithm is 2. Let P denote the distribution of total daily market ume. We define the related distribution P bins over bin indices i as follows: for all i =1,...,log(Q) 1, P bins (i) is equal to the probability, under P, that the daily ume falls in the interval [V min2 i 1, V min2 i ), and P bins (log(q)) is for the last interval [V max /2, V max ]. We define E as E(P bins ) = ( [ E i P bins log(q) i=1 P bins 1/P bins (i) 2 (i). ]) 2 Since the support of P bins has only log(q) elements, E(P bins can vary from 1 (for distributions P that place all of their weight in only one of the log(q) intervals between V min, V min2, V min4,...,v max ) to log(q) (for distributions P in which the total daily ume is equally likely to fall in any one of these intervals). Note that distributions P of this latter type are far from uniform over the entire range [V min, V max ]. ) Theorem 10. In the ume distribution model under the ume variability assumption, there exists an online algorithm A for selling N shares that, using only knowledge of the total ume distribution P, achieves R VWAP(A, P ) 2E(P bins ). All proofs in this section are provided in the appendix. As a concrete example, consider the case in which P is the uniform distribution over [V min, V max ]. In that case, P bins is exponentially increasing and peaks at the last bin, which, having the largest width, also has the largest weight. In this case E(P bins ) is a constant (i.e., independent of Q), leading to a constant competitive ratio. On the other hand, if P is exponential, then P bins is uniform, leading to an O(log(Q)) competitive ratio, just as in the more adversarial price-ume setting discussed earlier. In Figure 3, we provide additional specific bounds obtained for empirical total daily ume distributions computed for some real stocks. We now examine the setting in which P is unknown, but an approximation P is available. Let us define ][ ] C(P bins bins log(q), P )= P bins P (j) bins (i). [ log(q) j=1 i=1 P bins(i) C is minimized at C(P bins,p bins )=E(P bins ), and C may bins be infinite if P (i) is0whenp bins (i) > 0. Theorem 11. In the ume distribution model under the ume variability assumption, there exists an online algorithm A for selling N shares that using only knowledge of an approximation P of P achieves R VWAP(A, P ) 2C(P bins, P bins ). As an example of this result, suppose our approximation obeys (1/α)P bins bins (i) P (i) αp bins (i) for all i, for some α > 1. Thus our estimated bin index probabilities are all within a factor of α of the truth. Then it is easy to show that C(P bins bins, P ) αe(p bins ), so according to Theorems 10 and 11 our penalty for using the approximate distribution is a factor of α in competitive ratio. 5. REFERENCES [1] B. Awerbuch, Y. Bartal, A. Fiat, and A. Rosén. Competitive non-preemptive call control. In Proc. 5 th ACM-SIAM Symp. on Discrete Algorithms, pages , [2] A. Borodin and R. El-Yaniv. Online Computation and Competitive Analysis. Cambridge University Press, [3] R. El-Yaniv, A. Fiat, R. M. Karp, and G. Turpin. Optimal search and one-way trading online algorithms. Algorithmica, 30: , [4] M. Kearns and L. Ortiz. The Penn-Lehman automated trading project. IEEE Intelligent Systems, To appear. 6. APPENDIX 6.1 Proofs from Subsection 2.3 Proof. (Sketch of Theorem 3) W.l.o.g., assume that Q = 1 and the total ume is V. Consider the time t where the fixed schedule f sells the least, then f t N/T. Consider the sequences where at time t we have p t = p max, v t = V, and for times t t we have p t = p min and v t = 0. The VWAP is p max and the fixed schedule average is (N/T)p max +(N N/T)p min. Proof. (Sketch of Theorem 4) The algorithm simply sells u t = (v t/v min)n shares at time t. The total number of shares sold U is clearly more than N and U = u t = (v t/v min)n =(V/V min)n QN t t The average price is VWAP A(S) =( p tu t)/u = p t(v t/v )=VWAP M (S), t t where we used the fact that u t/u = v t/v.

10 Proof. (of Theorem 5) We start with the proof of the lower bound. Consider the following scenario. For the first T time units we have a price of Rp min, and a total ume of V min. We observe how many shares the online algorithm has bought. If it has bought more than half of the shares, the remaining time steps have price p min and ume V max V min. Otherwise, the remaining time steps have price p max and negligible ume. In the first case the online has paid at least Rp min/2 while the VWAP is at most Rp min/q + p min. Therefore, in this case the competitive ratio is Ω(Q). In the second case the online has to buy at least half of the shares at p max,so its average cost is at least p max /2. The market VWAP is Rpmin = p max / R, hence the competitive ratio is Ω( R). For the upper bound, we can get a R competitive ratio, by buying all the shares once the price drops below Rp min. The Q upper bound is derive by running an algorithm that assumes the ume is V min. The online pays a cost of p, while the VWAP will be at least p/q. 6.2 Proofs from Section 4 Proof. (Sketch of Theorem 10) We use the idea of guessing the total ume from Theorem 1, but now allow for the possibility of an arbitrary (but known) distribution over the total ume. In particular, consider constructing a distribution G bins over a set of ume values using P and use it to guess the total ume V. Let the algorithm guess ˆV = V min2 i with probability G bins (i). Then note that, for any price-ume sequence S, ifv [V min2 i 1, V min2 i ], VWAP A(S) G bins (i)vwap M (S)/2. This implies an upper bound on R VWAP(A, P ) in terms of G bins. We then get that G bins (i) P bins (i) minimizes the upper bound, which leads to the upper bound stated in the theorem. Proof. (Sketch of Theorem 11) Replace P with P in the expression for G bins in the proof sketch for the last result.

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