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Comparison to order books. In an order book, trading a basket of multiple assets for another basket of multiple assets requires multiple separate trades. Each of these trades would entail the blockchain fee, increasing the total cost of trading to the trader. In addition, multiple trades cannot be done at the same time with an order book, exposing the trader to the risk that some of the trades go through while others do not, or that some of the trades will execute at unfavorable prices.
In a CFMM, multiple asset baskets are exchanged in one trade, which either goes through as one group trade, or not at all, so the trader is not exposed to the risk of partial execution. Another advantage of CFMMs over order book exchanges is their efficiency of storage, since they do not need to store and maintain a limit order book, and their computational efficiency, since they only need to evaluate the trading function.
Because users must pay for computation costs for each transaction, and these costs can often be nonnegligible in some blockchains, exchanges implementing CFMMs can often be much cheaper for users to interact with than those implementing order books. Previous work. Academic work on automated market makers began with the study of scoring rules within the statistics literature, e. Scoring rules furnish probabilities for baskets of events, which can be viewed as assets or tokens in a prediction market.
The output probability from a scoring rule was first proposed as a pricing mechanism for a binary option such as a prediction market in Unlike CFMMs, these early automated market makers were shown to be computationally complicated for users to interact with. For example. Chen 23 demonstrated that computing optimal arbitrage portfolios in logarithmic scoring rules the most popular class of scoring rules is P-hard.
The first formal analysis of Uniswap was first done in 24 and extended to general concave trading functions in 9. Evans 25 first proved that constant mean market makers could replicate a large set of portfolio value functions. The converse result was later proven, providing a mechanism for constructing a trading function that replicates a given portfolio value function Analyses of how fees 27 28 and trading function curvature 29 30 31 affect liquidity provider returns are also common in the literature.
Finally, we note that there exist investigations of privacy in CFMMs 32 , suitability of liquidity provider shares as a collateral asset 33 , and the question of triangular arbitrage 34 in CFMMs. Convex analysis and optimization Convex analysis. A function f is concave if -f is convex 10 , Chap. The right hand side of this inequality is the first-order Taylor approximation of the function f at x, so this inequality states that for a concave function, the Taylor approximation is a global upper bound on the function.
Convex optimization. We assume the domains of the objective and inequality functions are the same for simplicity. Convex optimization problems are notable because they have many applications, in a wide variety of fields, and because they can be solved reliably and efficiently The list of applications of convex optimization is large and still growing. It has applications in vehicle control 36 37 38 , finance 39 40 , dynamic energy management 41 , resource allocation 42 , machine learning 43 44 , inverse design of physical systems 45 , circuit design 46 47 , and many other fields.
In practice, once a problem is formulated as a convex optimization problem, we can use off-the-shelf solvers software implementations of numerical algorithms to obtain solutions. These solvers can handle problems with thousands of variables in seconds or less, and millions of variables in minutes. Small to medium-size problems can be solved extremely quickly using embedded solvers 50 48 53 or code generation tools 54 55 For example, the aerospace and space transportation company SpaceX uses CVXGEN 54 to solve convex optimization problems in real-time when landing the first stages of its rockets Domain-specific languages for convex optimization.
Convex optimization problems are often specified using domain-specific languages DSLs for convex optimization, such as CVXPY 57 58 or JuMP 59 , which compile high-level descriptions of problems into low-level standard forms required by solvers.
DSLs vastly reduce the engineering effort required to get started with convex optimization, and in many cases are fast enough to be used in production. Using such DSLs, the convex optimization problems that we describe later can all be implemented in just a few lines of code that very closely parallel the mathematical specification of the problems.
Asset n is our numeraire, the asset we use to value and assign prices to the others. CFMM state Reserve or pool. Liquidity provider share weights. The DEX maintains a table of all the liquidity providers, agents who have contributed assets to the reserves. The table includes weights representing the fraction of the reserves each liquidity provider has a claim to.
The weights are nonnegative and sum to one, i. State of the CFMM. The reserves R and liquidity provider weights v constitute the state of the DEX. The DEX state changes over time due to any of the three possible transactions: a trade or exchange , adding liquidity, or removing liquidity. Proposed trade A proposed trade or proposed exchange is initiated by an agent or trader, who proposes to trade or exchange one basket of assets for another. In the sequel we will refer to the vectors that give the quantities, i.
Disjoint support of tender and receive baskets. Intuition suggests that a trade would not include an asset in both the proposed tender and receive baskets, i. We will see later that while it is possible to include an asset in both baskets, it never makes sense to do so.
Two-asset and multi-asset trades. A very common type of proposed trade involves only two assets, one that is tendered and one that is received, i. This is referred to as exchanging asset i for asset j. When a trade involves more than two assets, it is called a multi-asset trade.
Trading function Trade acceptance depends on both the proposed trade and the current reserves. We can interpret the trade acceptance condition as follows. Many existing CFMMs are associated with functions that satisfy the additional property of homogeneity, i. Linear and sum. The CFMM mStable, which held assets that were each pegged to the same currency, was one of the earliest constant sum market makers. Geometric mean. Like the linear and sum trading functions, the geometric mean is homogeneous.
CFMMs that use the geometric mean are called constant mean market makers. Other examples. Because it is a convex combination of the sum and geometric mean functions, which are themselves homogeneous, the resulting function is also homogeneous. Unlike the previous examples, this trading function is not homogeneous. Prices and exchange rates In this section we introduce the concept of asset reported prices, based on a first order approximation of the trade acceptance condition 4.
Unscaled prices. The condition 6 is homogeneous in the prices, i. The reported prices or just prices of the assets are the prices relative to the price of the numeraire, which is asset n. The price of the numeraire is always 1. In general the prices depend on the reserves R. The one exception is with a linear trading function, in which the prices are constant.
Geometric mean trading function prices. Exchange rates. This is approximately how much asset j you get for each unit of asset i, for a small trade. These are first order approximations. We remind the reader that the various conditions described above are based on a first order Taylor approximation of the trade acceptance condition. A proposed trade that satisfies 7 is not quite valid; it is merely close to valid when the proposed trade baskets are small compared to the reserves.
This is similar to the midpoint price average of bid and ask prices in an order book; you cannot trade in either direction exactly at this price. Reserve value. Adding and removing liquidity In this section we describe how agents called liquidity providers can add or remove liquidity from the reserves. Adding or removing liquidity also updates the liquidity provider share weights, as described below. Liquidity change condition. Adding or removing liquidity must be done in a way that preserves the asset prices.
This liquidity change condition is analogous to the trade exchange condition 4. The liquidity change condition 12 simplifies in some cases. Liquidity change condition for homogeneous trading function. Another simplification occurs when the trading function is homogeneous. In words: you can add or remove liquidity by adding or removing a basket proportional to the current reserves. Liquidity provider share update. When liquidity provider j adds or removes liquidity, all the share weights are adjusted pro-rata based on the change of value of the reserves, which is the value of the basket she adds or removes.
These new weights are also nonnegative and sum to one. These proposals are accepted or not, depending on the acceptance conditions given above. Slippage thresholds. One practical and common approach to mitigating this problem during trading is to allow agents to set a slippage threshold on the received basket. Maximal liquidity amounts. While setting slippage thresholds can help with reducing the risk of trades failing, another possible failure mode can occur during the addition of liquidity.
Properties of trades Non-uniqueness. The associated CFMM has the same trade acceptance condition, the same prices, the same liquidity change condition, and the same liquidity provider share updates as the original CFMM. Maximum valid receive basket. A valid trade cannot ask to receive more than is in the reserves. Non-overlapping support for valid tender and receive baskets. The last vector on the right is zero in all entries except k, and positive in that entry.
Assuming the kth asset has value, we would always prefer this. Trades increase the function value. Trading cost is positive. Properties of liquidity changes Liquidity change condition interpretation. One natural interpretation of the liquidity change condition 12 is in terms of a simple optimization problem. Liquidity provision problem. When the trading function is homogeneous, it is easy to understand what baskets can be used to add or remove liquidity: they must be proportional to the current reserves.
Suppose that we add or remove liquidity. Two-asset trades Two-asset trades, sometimes called swaps, are some of the most common types of trades performed on DEXs. In this section, we show a number of interesting properties of trades in this common special case.
Forward exchange function. We will now show that the function F is concave. Reverse exchange function. In a similar way to the forward trade function, the reverse exchange function is nonnegative and increasing, but this function is convex rather than concave. This follows from a nearly identical proof. Forward and reverse exchange functions are inverses. The forward and reverse exchange functions are inverses of each other, i.
Analogous functions for a limit order book market. There are analogous functions in a market that uses a limit order book. They are piecewise linear, where the slopes are the different prices of each order, while the distance between the kink points is equal to the size of each order. The associated functions have the same properties, i. Evaluating F and G. In some important special cases, we can express the functions F and G in a closed form.
On the other hand, when the forward and reverse trading functions F and G cannot be expressed analytically, we can use several methods to evaluate them numerically It can be shown that the convergence is monotone decreasing. We note that one of the largest CFMMs, Curve, uses a trading function that is not homogeneous and uses this method in production 1. Figure 1: Left. Forward exchange functions for two values of the reserves.
Reverse exchange functions for the same two values of the reserves. Slope at zero. Exchanging multiples of two baskets Here we discuss a simple generalization of two-asset trade, in which we tender and receive a multiple of fixed baskets.
The same analysis holds in this case as in the simple two-asset trade. We can introduce the forward and reverse functions F and G, which are inverses of each other. There is also an inequality analogous to 19 , using this definition of the exchange rate. We mention two specific important examples in what follows. Liquidating assets. We can also show that the liquidation value is at most as large as the discounted value of the basket; i.
Purchasing a basket. This follows from a nearly identical argument to that of the liquidation value. Multi-asset trades We have seen that two-asset trades are easy to understand; we choose the amount we wish to tender or receive , and we can then find the amount we will receive or tender. In the multi-asset case, there are more degrees of freedom.
There are many valid tendered baskets, which are shown in figure 2. We will assume that U is increasing and concave. Increasing means that the trader would always prefer to have a larger net change than a smaller one, which comes from our assumption that all assets have value. Thus we can globally and efficiently solve the non-convex problem 21 by solving the convex problem No-trade condition. We can give simple conditions under which this happens for the case when U is differentiable.
We can interpret the set of prices p for which this is true, i. It is easy to see that K is a convex polyhedral cone. The GOU process also referred to as the exponential Ornstein-Uhlenbeck process has been used to model commodity prices. Maxim likelihood parameter estimation for single-variable GBM and GOU processes Here, we attempt to find parameters that maximize the likelihood that the given time series is a realization of our governing equation.
This parameter estimation scheme allows one to extract the noise from the individual time series via Eqs. In what follows, we will use the correlations between the finite-time noise variables as our numerical scheme directly utilizes those. By construction, these normal variables have zero mean and unit variance.
This follows from the fact that the MLE for multivariate normal variables again are equal to the sample means and sample covariances Johnson and Wichern ; Rayner see also Appendix 3 for some details. Further, for the two-point correlation functions of the scaled GOU processes in the stationary regime one has Gardiner ; Singh et al.
The Cholesky decomposition is advantageous as it provides a lower triangle matrix, which minimizes the number of non-zero elements and is computationally efficient to invert. Let us now assume that at some time tl a time series is masked. Without loss of generality, we take that time series to be last in the index set. As we will see next, it is useful to solve Eq.
Because in this equation the left hand side is a normal random variable with unit variance, the sum of the variances for the variables on the right hand side of the equation must sum to unity. Given a sufficient initial period of complete information, we can estimate the SDE equation parameter space using our described MLE method Franco ; Tang and Chen ; Johnson and Wichern ; Rayner assuming the parameters are time invariant. We then use Eq. In order to predict information we are lacking in Eq.
Results To test the models posed, we have two testing environments. Hence, parameters and correlations are easily controlled in order to test the feasibility robustness of the prediction scheme. In this environment, we are able to use multiple realizations to average results and obtain important observations that can not be seen in one realization.
The second environment is application on a cryptocurrency market and social media containing several major cryptocurrencies including Bitcoin, Ethereum, Litecoin, Monero and XRP. The trends of these time series can be seen in Fig. While cryptocurrencies have no restrictions on when a trade can be placed, we find it useful to define two daily metrics to use as a daily time series.
We define the closing price of a cryptocurrency as the market price at GMT each day and the daily volume as the amount of currency traded in terms of USD in the 24 hours prior to GMT. Here all time series have been normalized such that their mean is unity Full size image Our social media data sets contain tweets from Twitter, comments from Reddit, and events from GitHub that all pertain to any cryptocurrency, including minor, less traded cryptocurrencies.
While the minor coins are included in the data set, they have relatively low number of tweets, comments, or events. We define daily statistics, even though this approach is generalizable to hour or even minute resolution. If a given time step returns zero due to the increased resolution, this becomes an absorbing state for both stochastic equations and parameters can not be estimated. While increased resolution is possible, a sufficient density of activity at the increased resolution is required from a technical standpoint.
Indeed, the underlying equations which best describe these time series are likely more complex than what we have posed here. However, at the lowest order of improvement, any unaccounted effects for interactions could be simply cast as a more complex noise term. If these interactions persist between sets of time series, our correlated noise will allow these interactions to take place, even though we have not explicitly engineered these interactions in our underlying equation.
The duration of these data sets spans from January 1st to August 31st An important assumption was that the parameters can be approximated as time independent. Indeed, the parameters do have some time variance over long periods Bassler et al. For this reason, we limit the training window for estimating correlations and parameters to the previous days. Examples of the parameter estimation over time can be seen in Appendix 4. In both synthetic and empirical data environments, we tested how well the method preforms at predicting market value and social media activity levels.
We measured this in three ways. For the first, we used a sliding window approach where a single day at a time will be predicted using the previous days. For the second method, we measured the accuracy for increased prediction duration. Here, we selected evenly spaced initial times and predicted the time series 14 days into the future. This will test how much uncertainty builds up as we extend our prediction further in time.
This provides an estimate for how much the prediction deviates from the ground truth. In principle, all realizations are technically valid, even though some are increasingly unlikely. We show this in Fig. Because of the nature of these stochastic equations and that we only have one realization to work with in our data set, we expect some significant variations in the accuracy of our predictions.
Our prediction with correlation uses 11 other correlated time series, and we made a prediction assuming no correlation with those time series. A training window of time steps was used in each case. Estimated parameters and correlations for these examples can be seen in Table 4 Appendix 4 Full size image To properly test our model, we use many realizations to show its robustness under ideal conditions.
In this case, we generate data using known equations with set parameters. While our algorithm will estimate the parameters, the equation used will be known. This produces an idealization over our cryptocurrency data, where we can not say for certain what their underlying equation actually is.
We also define the correlations and parameters to be constant in time, another idealization. We choose our parameters to produce significant variance in signal, but so that we do not lose accuracy due to our time step size. For our correlation matrix, we define all off-diagonal elements to be equal. We also explore the case where we add an exception to our correlation definition to observe its impact.
Figure 3 shows the accuracy of our prediction methods using various combinations of inputs and compares their result to a prediction assuming the time series is uncorrelated with the other time series. Figure 3 a explores predicting the sign of the slope of the data. This is compared to a prediction without using correlated time series. This is done by setting the correlation of the added time series and the predicted time series to the square root of the original correlation Full size image Figure 3 b shows the MAPE value when a single data point is masked.
As expected, our prediction greatly exceeds the prediction without correlation. A more important observation is a small decrease as the number of time series increase. One would expect as the number of time series increase that the unaccounted noise in the prediction method would decrease, however we see a less significant decrease in MAPE value than expected. This may be a result of the homogeneity of our definition of correlations.
To minimize the amount of unaccounted noise, each time series must be highly correlated with the time series intended to be predicted, but relatively uncorrelated with other time series used in the prediction method.
LISTING UNIQUE VALUES IN STATA FOREX
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| 5mbtc to btc | In this work we put forth a universally composable treatment of the Bitcoin protocol. The scheme uses the superposition of security keys and privacy keys in both the placement and delivery phases to guarantee content security and demand privacy, respectively. We then use Eq. This problem can be solved using several methods. For small sample sizes, this difference is small enough that many continue to use GBM and other variants for modeling. Hence, we use the GOU process for the social media activity and volume traded as when a fluctuation happens, they typically cannot be sustained for extended periods of time, and in time they return to some long-run mean. While outsourcing storage is convenient, this data is often sensitive, making data breaches a serious concern. |
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