TTrading Strategies with Position Limits
Valerii Salov
Abstract
Whether you trade futures for yourself or a hedge fund, your strategyis counted. Long and short position limits make the number of uniquestrategies finite. Formulas of the numbers of strategies, transactions, donothing actions are derived. A discrete distribution of actions, correspond-ing probability mass, cumulative distribution and characteristic functions,moments, extreme values are presented. Strategies time slice distributionsare determined. Vector properties of trading strategies are studied. Algeb-raic not associative, commutative, initial magmas with invertible elementscontrol trading positions and strategies. Maximum profit strategies, MPS,and optimal trading elements can define trading patterns. Dynkin intro-duced the term interpreted in English as "Markov time" in 1963. Neftciapplied it for the formalization of Technical Analysis in 1991.
Technical analysis is anathema to the academic world.
Burton Malkiel, [73, p. 127] ... technical analysis is a broad class of prediction rules with unknown statisticalproperties, developed by practitioners without reference to any formalism.
Salih Neftci, [80, p. 549] ... considerable part of time the particle spends in one semi-plane. Theseparadoxical regularities of particle transition from positive side of line tonegative and vice versa are covered by the theorem called the "arcsine law".
Andrey Kolmogorov, Igor Zhurbenko, Alexander Prokhorov, [63, pp.91 - 92 in Symmetrical Random Walk]
Modern theories of prices exploit mathematics of stochastic processes [81]such as a
Brownian motion . "Brownian paths are wilder than we can imaging [89, p. 19]. They are continuous functions of time [89, p. 10, Theorem 6.1],non-differentiable almost everywhere [89, p. 19 and Theorem 10.1]. Theoreticalcontinuity implies that the frequency of observations can be increased arbitrary.In [48, p. 74 and p. 81], we read "In some markets, second-by-second datais now available, allowing virtually continuous observations of price ..." and "A fundamental property of high frequency data is that observations can occur a r X i v : . [ q -f i n . GN ] D ec t varying time intervals" . The sentences remind the author about school ex-periments determining the gravitational acceleration g ≈ . metersecond . In a darkroom, a steel ball is dropped from three meters along the inverted vertical ruler.A stroboscope flashes four times per second. The ball is visible at centimeterlabels 31, 123, 276, on the floor. While invisible, the ball is still moving.
Engle, estimating the growth of the frequency of observations, says "Thelimit in nearly all cases, is achieved when all transactions are recorded" [36,p. 2]. Time & Sales data distributed by the Chicago Mercantile Exchange,CME, Group for the futures contracts electron-ically traded on the Globex platform is an example. There are no other ticksbetween two neighbors. Each associates time, price, number of traded contracts,and market conditions symbol. The irregular time intervals between neighboringtransactions, waiting times or durations, and corresponding price increments,named by the author a- and b-increments, are studied [100], [103]. Algebraicsums of these elementary, indecomposable further increments, financial atoms ,represent all popular minute, hourly, daily increments and chart price bars.The prices are discrete and occur at irregular times, where financial in-struments with high leverage and large trading positions magnify discreteness.Modern finance attempts to touch the left ear by the right hand . It brings todiscrete markets continuous models to create later sophisticated discretizationschemata, for example, for Monte Caro simulations [43]. Kolmogorov foresaw: "It is quite probable that with the development of modern computing techniques,it will become understood that in many cases it is reasonable to study real phe-nomena without making use of intermediate step of their stylization in the formof infinite and continuous mathematics, passing directly to discrete models" [64].Successes in the numerical integration of Brownian motions, diffusions [77],[62], [85], jump-diffusions [51] applying higher order integration formulas areaccompanied by new evidences of non-Gaussian properties of futures price in-crements and micro-structure of jumps resembling non-equilibrium propertiesof explosions [100, p. 41 "Non-Gaussian atoms", pp. 41 - 43 A Comment onDisequilibrium], [103, pp. 30 - 36 Randomness of Price Increments, pp. 37 - 40Non-Gaussian Relative b-Increments, pp. 50 - 52 Jumps. Chain Reactions].Ignorance of intra-day ticks yields controversial daily price applications. In-deed, absolute daily price increments P − P , P − P or logarithms of priceratios ln( P P ) , ln( P P ) for three days 0, 1, 2 are sums of significantly differentnumbers of summands N and N in day trading sessions 1 and 2, intra-dayprice increments or logarithms of price ratios. Even, if the latter are independ-ent and identically distributed, i.i.d., random variables, the variance of the sum µ S , being under such conditions the sum of the variances, N µ and N µ , [44],makes the sums non-i.i.d. random variables because N (cid:54) = N . This comprom-ises assumption on i.i.d. for daily increments or returns. What a mess! Thepaper considers prices, transaction costs, trading strategies, positions taken withtraded assets as discrete entities and illustrates results using futures contracts. A Portrait of Futures
Leo Melamed [75]: "According to the Bank of International Settlements (BIS),81.3% of all futures traded in 2013 were financial futures and options. Thenotional value of those traded equaled an astounding $1,886,283.4 billion" .From several futures contracts on the same commodity, index, or security, thecontract with the closest settlement date is called the nearby futures contract .Accordingly, there are next , distant , and the most distant futures contracts.While a daily price chain for an individual contract is weird , to escape arbitrage,prices of the mentioned contracts move in sync, Figure 1. The daily prices ofthe neighbors, M June and U September, differ P ESM17 i P ESU17 i (cid:54) = 1 , Figure 2.Figure 1: Time & Sales Globex, , last transactionprices of ESM17 ( nearby in April 2017), and ESU17 ( next in April 2017) fromsession ranges finishing at 15:15:00, Central Standard Time, CST. 185 ESM17and 103 ESU17 sessions, Wednesday April 13, 2016 - Thursday April 13, 2017.Plotted using Microsoft Excel. P ESU17 i is in [minimum 2086.50, Tuesday November 1, 2016; maximum2385.75, Wednesday March 1, 2017]. The price range counted from 2086.50 isless than 15%. Therefore, after a zoom in, the look and feel of P ESM17 i − P ESU17 i is similar to the relative price increment P ESM17 i − P ESU17 i P ESU17 i = P ESM17 i P ESU17 i − , a shiftdown of Figure 2. The latter, for small | P ESM17 i − P ESU17 i | , is close to ln( P ESM17 i P ESU17 i ) .3igure 2: Time & Sales Globex, , ratios of lasttransaction prices of ESM17, and ESU17 from session ranges finishing at15:15:00, CST. Number of sessions N = 103 . Plotted using Microsoft Excel.Figure 3: Time & Sales Globex, , regression oflast transaction prices of ESM17, and ESU17 from session ranges finishing at15:15:00, CST: P ESM17 = (1 . ± . P ESU17 ; coefficient of correlation= 0.999998; intercept is set to zero; confidence two sided probability is 95%; N = 103 . Computed using Microsoft Excel, Data Analysis, Regression.4nder observed conditions, one Figure 2 gives an idea about four quantities.Since P ESM17 i P ESU17 i ≈ , Figure 3 is not a surprise. The empirical regression with103 points and coefficient correlation almost equal to one expresses what weunderstand under "moving in sync".Figure 3 is possible because P ESM17 i and P ESU17 i are linked by dates i . Onthe left side of Figure 1, P ESU17 i circles are missing: the contract was not yettraded. Later, dots are missed due to low liquidity or lost data. For 185 sessionsand prices of ESM17, only 103 "corresponding" points of ESU17 are collected.This is enough to conclude about the almost linear dependence.By eye, tick prices on Monday April 10, 2017 are in sync, Figure 4. Trans-action volumes are too, Figure 5. Ticks {date-time price size} arrive at randomtimes [100], [103]. The depicted discrete properties of the S&P E-Mini futureswith a single tick for ESH18 on April 10, 2017 is a guide behind mathematicalformulas. The paper is about the futures trading strategies and maximum profitstrategies, which can define patterns.
Consider the chains of positive prices P , . . . , P i , . . . , P n , non-negative transac-tion costs C , . . . , C i , . . . , C n , subtracted from profit and losses, P L , making thelatter not better , and integer positive buy , negative sell , and zero do nothingactions U , . . . , U i , . . . , U n , representing the numbers of traded units. Positiveand negative actions are transactions . The chain of actions is a trading strategy .Time & Sales ticks arrive as triplets { t i , P i , V i } , where V is the volume orsize of the trade . The i associates with the order of arrival. V = 0 represents indicative prices or special market conditions rather than trades. The numbersof bought and sold units in a trade are equal and V shows one side. The tradeis a combination of transactions made from the same or different accounts aftermatching prices of the limit orders accepted in different times by a tradingbook . For trading one instrument, yielding Time & Sales ticks, we assume thattransactions are made from one account. The pair of opposite transactions withzero sum is a round-trip trade . The trades associate with adjusting, overlapping,or disjoint time intervals. A net action with transactions and zero sum can bebroken down on round-trip trades in several ways. Accounting may regrouptransactions maximizing realized profits and leaving losses to open marked-to-market positions. With a good luck, the final list will contain only profits. Ifthe good luck turns away, then the natural time order of transactions can yieldsmaller losses or a mixture. All variations must have the same total
P L . Actions affect trading positions - the numbers of securities that are ownedor borrowed and then sold. The long , short , or out of market positions arepositive, negative, or zero integers forming the chain W , . . . , W i , . . . , W n , where W i = W + (cid:80) ij =1 U i . W helps to express U i = W i − W i − for i = 1 , . . . , n . W i will show the position always after the action U i . The marked to market P L ofa trading strategy is equal to [100, Equation 54, page 82]:5igure 4: E-mini S&P 500 Futures Time & Sales Globex, , transaction prices of ESM17, ESU17, ESZ17, and ESH18 forthe time range [Sunday April 9, 2017, 17:00:00 - Monday April 10, 2017,15:15:00], CST. The most distant contract ESH18 had only one tick {date timeprice size} = {2017/04/10 11:18:21 2342 1}. Plotted using custom C++ andPython programs and gnuplot .6igure 5: E-mini S&P 500 Futures Time & Sales Globex, , transaction volumes and cumulative volumes of ESM17,ESU17, ESZ17, and ESH18 for the time range [Sunday April 9, 2017, 17:00:00- Monday April 10, 2017, 15:15:00], CST. ESH18 had one tick {date time pricesize} = {2017/04/10 11:18:21 2342 1}. Plotted using custom C++ and Pythonprograms and gnuplot . P L = k (cid:32) P n n (cid:88) i =1 U i − n (cid:88) i =1 P i U i (cid:33) − n (cid:88) i =1 C i | U i | − C n | n (cid:88) i =1 U i | , (1)where k is the dollar value of a full price point . For instance, for the S&P 500E-Mini futures contract the price value of the full point is 50 US dollars. Theabsolute minimum non-zero price fluctuation is δ = 0 . or 12.5 US dollars.The chain of three prices could be P = (2369 . , . , . . Due to thecurrent S&P 500 E-Mini contract specifications, the prices between these levelsare impossible. The k allows to apply conventional market price quotes and getdollar equivalents. In contrast, transactions costs C i are expressed in dollarsper unit per transaction. The strategy U = (1 , , − with the cost C = (5 , , yields P L = 50 × (2370 . × (1+0+( − − (2369 . × . × . × ( − − (5 × | | + 5 × | | + 5 × | − | ) − × | − | = 15 US dollars.7n futures, costs, fixed per contract per transaction, are common. Brokers mayreduce them to promote large transactions. In equities, commissions can be afixed fraction of price. Then, to continue with the formula, C can be evaluatedon a preliminary step using the commissions discount or P and fraction.The chains of n elements are column-vectors in n -dimensional spaces. Boldsymbols denote vectors. (cid:80) ni =1 P i U i is the scalar product of P and U , W n = (cid:80) ni =1 U i for W = 0 : P L = k ( P n W n − P T U ) − C T abs ( U ) − C n | W n | , where abs() returns a vector with the absolute values of coordinates. T means transpose .For programming these equations, the Standard C++ classes std::vector , std::array [117, pp. 902 - 906, 974 - 977], [56, pp. 897 - 897, 871 - 874], andalgorithms std::inner_product , std::transform , std::accumulate [117, pp.1178 - 1179, 935 - 936, 1177 - 1178], [56, pp. 1131, 1023, 1130] are useful. Theorem 3.1.
There is one and only one strategy U for position W and W .Proof. For W and W , there is unique U since U i = W i − W i − . For U , thereis unique W with W since W i = W + (cid:80) ij =1 U j . (cid:4) Given W and W , U can be computed using the Standard C++ algorithm std::adjacent_difference , while applying std::partial_sum [117, pp. 1179- 1180], [56, pp. 1133, 1137 - 1138] recovers W from U and W . The numberof vectors with n integer coordinates is infinite but positions and actions arelimited . A position limit is a natural number W ∈ N . Position limit W . With the futures account A = $10 , and initial margin IM ESZ = $1 , . covering intraday trading of the single futures December2017 E-mini Standard and Poor’s 500 Stock Price Index contract ESZ17, onecan buy or sell (cid:98) , , . (cid:99) = (cid:98) . . . . (cid:99) = 8 contracts. For a retail trader, constantfees and commissions per contract per one side, buy or sell entering the position W = 8 or − , can be C = $4 . . Due to the costs, right after the transaction,the equity drops in our hypothetical example to A − | W | ∗ C = $10 , − ∗ $4 .
68 = $9 , . . The maintenance margin M M
ESZ = $1 , , typicallysmaller than IM , requires | W | × M M
ESZ = 8 × $1 ,
125 = $9 , . If pricesmove favorably, then the equity increases and eventually position is allowed toadd contracts. If losses reduce the equity below the total maintenance marginrequirement, then funds must be added to the account, or the position or itspart is mandatory liquidated. At closing the position, the remaining cost is × $4 .
68 = $37 . . The difference between the equity after the anticipatedclosing cost and maintenance margin equity is $9 , . − × $4 . − $9 ,
000 =$925 . . This is $925 . = $115 . per contract. Conversion to price pointsyields $115 . k =$50 < . . Depending on market conditions, the ESZ17 price canmove 2.5 points up or down in a matter of a few seconds or minutes [100], [103].Establishing a position size up to the available account equity is too risky andcan quickly ruin an account. The capital limit and margins dictate the positionlimit W . However, due to these factors only, W does not have to be constant. One can voluntary trade small limited | W i | ≤ W or fixed | W i | = W positions.Depending on trading rules , statistics of PL , market conditions , trading fixed8ositions can be too inefficient or risky [61], [121], [122], [58], [93, pp. 55- 79].Still, such strategies can be useful for the evaluation of trading rules generatingindividual trading signals and their separation from the money managementanswering which portion of the capital should be devoted to a next trade.Without consideration of capital limits, position limits are known for theCME futures from the contract specifications, . Forthe E-mini futures the all month limit is 60000 contracts. For Corn, the initialspot-month limit is 600 or 3000000 bushels and the single month limit is 33000or 165000000 bushels. Bushel, a volume measure, is not from the InternationalSystem of Units. It is eight US dry gallons. For corn with 15.5% of moisture thisis ≈ . kg and the limit is equivalent of 4191000 metric tons. The National Ag-ricultural Statistics Service of the United States Department of Agriculture re-ported for 2016: "U.S. corn growers produced 15.1 billion bushels, up 11 percentfrom 2015" , .This is 383.5 million metric tons: the single month limit is ≈ % of the U.S.2016 corn crop. The "romance", when legendary Jesse Livermore could "cornerthe U.S. wheat market" [68], [70], is in the past. 5000 bushels of one contract, ≈ tons, fit two hopper wagons . Details of corn futures ticks are in [103]. Theorem 3.2.
There are S = (2 W + 1) n − unique positions W j and strategies U j for | W i,j | ≤ W , i = 1 , . . . , n ticks, W ,j = W n,j = 0 .Proof. The n th coordinate in W j is zero. The remaining n − coordinatesare W + 1 independent − W, . . . , − , , , . . . , W . Thus, the number of uniquecombinations and W j is S = (2 W + 1) n − . By Theorem 3.1, the number ofcorresponding unique strategies U j is the same. (cid:4) The sets of positions W j with W ,j = W n,j = 0 , | W i,j | ≤ W , i ∈ [1 , n ] , j ∈ [1 , S = (2 W + 1) n − ] and corresponding strategies U j are W and U .Example: W = 1 , n = 3 yield nine pairs W j T ↔ U j T : (0 , , ↔ (0 , , do nothing; (1 , , ↔ (1 , − , , ( − , , ↔ ( − , , ; (1 , , ↔ (1 , , − , ( − , − , ↔ ( − , , ; (0 , , ↔ (0 , , − , (0 , − , ↔ (0 , − , ; (1 , − , ↔ (1 , − , , ( − , , ↔ ( − , , − . Theorem 3.3.
The sample mean
P L of the strategies U does not depend onprice P and equal to a P L = − (cid:80) (2 W +1) n − j =1 C T abs ( U j )(2 W +1) n − .Proof. The number of strategies given by Theorem 3.2 is odd (2 W + 1) n − . Thesingle do nothing strategy , d.n.s, has P L = 0 . The remaining even number formstwo sets with (2 W +1) n − − strategies each: not do nothing strategies U j andtheir inverses − U j . From Equation 1, P L ( U j ) + P L ( − U j ) = − C T abs ( U j ) + C n | W n | ) = − C T abs ( U j ) . Averaging gives a P L . (cid:4) From the market prospective, if U has been applied, no matter by whom,then − U has been applied too and P L ( U ) (cid:54) = − P L ( − U ) . The sum outcomenegative for traders is a payment to the industry. This expresses the knownproperty of a negative non-zero sum game .9 heorem 3.4. There are n (2 W + 1) n − positions and actions in W and U .Proof. By Theorem 3.2, the numbers of positions and strategies are (2 W +1) n − .Each has n coordinates. (cid:4) The unique positions and strategies are indexed by j ∈ [1 , (2 W + 1) n − ] ,yielding n (2 W + 1) n − W i,j and U i,j , i ∈ [1 , n ] . ∀ j , W ,j = W n,j = 0 , U ,j = W ,j − W ,j = W ,j , U n,j = W n,j − W n − ,j = − W n − ,j . ∀ i ∈ [1 , n − , numbersof W i,j = W l ∈ [ − W, W ] , l ∈ [1 , W + 1] are equal. Uniformness follows fromindependence of positions, Theorem 3.2. Then, the numbers of U ,j = W ,j and U n,j = − W n − ,j of each kind W l in all strategies are (2 W +1) n − W +1 = (2 W + 1) n − .Positions are state functions . Actions are transition functions . The do noth-ing action U i,j = 0 does not change the state W i − ,j → W i,j . It is neither a lossnor a profit for a trader and does not pay to the industry. Theorem 3.5.
There are n (2 W + 1) n − do nothing actions in U .Proof. U ,j = 0) + U n,j = 0) = 2(2 W + 1) n − and for i ∈ [2 , n − , each W l is represented by (2 W + 1) n − strategies. U i,j = 0 , if it does not change W i − ,j . Therefore, there is only one do nothing action for each subset yielding ( n − W + 1) n − . Adding for i = 1 and i = n totals n (2 W + 1) n − . (cid:4) Theorem 3.6.
There are nW (2 W + 1) n − transactions in U .Proof. n (2 W + 1) n − − n (2 W + 1) n − = 2 nW (2 W + 1) n − . (cid:4) The Market Universe.
Figure 4 depicts n = 134909 ticks for ESM17. Lowresolution and discreteness hide some. The number of strategies with | W i,j | ≤ is . The Sun mass is . × kg http://solar-center.stanford.edu/vitalstats.html . In photosphere, 73.46% by mass is Hydrogen. Themass of its atom is . × − kg https://en.wikipedia.org/wiki/Unified_atomic_mass_unit . If the fraction is valid for the entire star, then the number ofHydrogen atoms is . × . × . × − ≈ . × . The latter is nothing comparingwith the number of potential strategies for ESM17 traded on Monday April10, 2017, making the a P L formula in Theorem 3.3 not robust. The formulain Theorem 3.6 is robust for the number of transactions, not dollars. Thedistribution of − W ≤ U i,j ≤ W is not uniform. Example, for W = 1 , n = 3 : U i = −
2) = 1 , U i = −
1) = 8 , U i = 0) = 9 , U i = 1) = 8 , U i = 2) = 1 . Theorems 3.4 and 3.5 give the total number of actions 27 and U i = 0) = 9 . The distribution of actions is needed to compute dollars.
There are W +1 action types m ∈ [ − W, W ] , if W ,j = W n,j = 0 , | W i,j | ≤ W .The frequency of do nothing actions p ( m = 0 , W, n ) = n (2 W +1) n − n (2 W +1) n − = W +1 isindependent on n , Theorems 3.4, 3.5. "To guess" formulas for m (cid:54) = 0 , the author10rote a C++ program, reserving memory for (2 W + 1) n − positions vectors ofsize n each using std::vector
18 = 18 × ,
154 = 22 × , × , × , × . The quotients linearly depend on n . The"guessed formula" is (4 n + 6)(2 W + 1) n − . Formulas do not work for n = 1 with single d.n.s. The "guessed formulas" are in Table 1.Table 1: Guessed Counts Formulas, n ∈ [2 , . W m
Count Sum-2 ( n − × (2 W + 1) n − -1 (2 n + 2) × (2 W + 1) n − n × (2 W + 1) n − = 3 n × (2 W + 1) n − n × (2 W + 1) n − = n × n − (2 n + 2) × (2 W + 1) n − ( n − × (2 W + 1) n − -4 ( n − × (2 W + 1) n − -3 (2 n − × (2 W + 1) n − -2 (3 n + 4) × (2 W + 1) n − -1 (4 n + 2) × (2 W + 1) n − n × (2 W + 1) n − = 5 n × (2 W + 1) n − n × (2 W + 1) n − = n × n − (4 n + 2) × (2 W + 1) n − (3 n + 4) × (2 W + 1) n − (2 n − × (2 W + 1) n − ( n − × (2 W + 1) n − -6 ( n − × (2 W + 1) n − -5 (2 n − × (2 W + 1) n − -4 (3 n − × (2 W + 1) n − -3 (4 n + 6) × (2 W + 1) n − -2 (5 n + 4) × (2 W + 1) n − -1 (6 n + 2) × (2 W + 1) n − n × (2 W + 1) n − = 7 n × (2 W + 1) n − n × (2 W + 1) n − = n × n − (6 n + 2) × (2 W + 1) n − (5 n + 4) × (2 W + 1) n − (4 n + 6) × (2 W + 1) n − (3 n − × (2 W + 1) n − (2 n − × (2 W + 1) n − ( n − × (2 W + 1) n − b ( m, W ) × n + a ( m, W ) and (2 W + 1) n − .Distributions are symmetrical. For m (cid:54) = 0 , the frequencies depend on n but lim n →∞ p ( m, W, n ) = lim n →∞ ( b ( m,W ) × n + a ( m,W )) × (2 W +1) n − n × (2 W +1) n − = b ( m,W ) × n + a ( m,W ) n × (2 W +1) = b ( m,W )(2 W +1) does not. a ( m = 0 , W ) = 0 . The b ( m, W ) = 2 W + 1 − | m | , where m ∈ [ − W, W ] , satisfies all formulas in Table 1.The a ( m, W ) is obtained from n = 2 , where the second, last , position iszero and the first action | U ,j | ≤ W . The strategies and inverses count twofor | m | ≤ W , making a ( m, W ) = 2 | m | , and zero for W < | m | ≤ W , yielding a ( m, W ) = 2 | m | − W + 1) . The united formulas are A = { m : | m | ≤ W } , B = { m : W < | m | ≤ W } ; Count A ( m, W, n ) = [(2 W + 1) n − ( n − | m | ] (2 W + 1) n − ; Count B ( m, W, n ) = Count A ( m, W, n ) − W + 1) n − == [(2 W + 1) n − ( n − | m | − W + 1)](2 W + 1) n − ; p A ( m, W, n ) = Count A ( m, W, n ) n (2 W + 1) n − = 12 W + 1 − ( n − | m | n (2 W + 1) ; p B ( m, W, n ) = Count B ( m, W, n ) n (2 W + 1) = p A ( m, W, n ) − n (2 W + 1) . (2)The two counts "contain" all formulas from Table 1, reproduce × (5 + 9 +13) = 216 C++ experimental values, and satisfy Theorem 3.5, since (cid:80) i = ni =0 i = (cid:80) i = ni =1 i = n ( n +1)2 , (cid:80) i =2 ni = n +1 i = (cid:80) i =2 ni =1 i − (cid:80) i = ni =1 i = n (3 n +1)2 , m = W (cid:88) m = − W Count A = n (2 W + 1) n − − ( n − W ( W + 1)(2 W + 1) n − ; m = − W − (cid:88) m = − W Count B + m =2 W (cid:88) m = W +1 Count B = 2 m =2 W (cid:88) m = W +1 Count B == ( W n + W n − W − W )(2 W + 1) n − ; m = W (cid:88) m = − W Count A + 2 m =2 W (cid:88) m = W +1 Count B = n (2 W + 1) n − , Proof.
Vladimir Arnold recollects [2, p. 29] the words of his teacher AndreyKolmogorov (VS’s translation): "Do not look for a mathematical sense in myhydrodynamics achievements. ... I did not derive them from initial axioms ordefinitions (as physicists say, from the "first principals"): my results are notproved but valid and this is much more important!"
The C++ experiments con-vinced the author of the correctness of formulas 2 and left admiration of the12olmogorov’s words. However, Anderzej Pelc’s "Why Do We Believe Theor-ems?" [84] "pressed" not to publish the formulas without a proof.The author could not move from n to n + 1 using mathematical induction . Generating functions [114], [67] require coefficients - a vicious circle . But ...By construction, positions [ − W, W ] are uniformly distributed in the matrix n ticks × [ S = (2 W + 1) n − ] strategies within the first , . . . , n − rows P ositions = W = W , W , . . . W ,S . . . . . . . . . . . .W n − , W n − , . . . W n − ,S . . . Each row, except n th, has (2 W +1) n − W +1 = (2 W + 1) n − positions of each type.The actions are adjacent differences in columns U i,j = W i,j − W i − ,j Actions = U = U , = W , − . . . U ,S = W ,S − U , = W , − W , . . . U ,S = W ,N − W ,S . . . . . . . . .U n − , = W n − , − W n − , . . . U n − ,S = W n − ,S − W n − ,S U n, = 0 − W n − , . . . U n,S = 0 − W n − ,S The matrix (2 W + 1) × (2 W + 1) of all individual transitions | − W − W + 1 . . . . . . W − W − − −− − − −− − − −− −− − − −− −− − − −− − − −−− W → . . . W . . . W − W − W + 1 → − . . . W − . . . . . . W − · · · → . . . . . . . . . . . . . . . . . . . . . → − W − W + 1 . . . . . . W − W · · · → . . . . . . . . . . . . . . . . . . . . .W − → − W + 1 − W + 2 . . . − W + 1 . . . W → − W − W + 1 . . . − W . . . − is applied to the ticks [1 , n − . Due to uniformness of positions in ticks [1 , n − ,each of the (2 W +1) n − types, in moves from ticks [1 , n − , is changed to (2 W +1) n − types: W +1 of actions transfer a position type to another, both from [ − W, W ] . The number of transitions from one type to another is (2 W +1) n − W +1 =(2 W + 1) n − . Therefore, the number of actions of one type m ∈ [ − W, W ] is the length of the diagonal W + 1 − | m | . For the ticks [1 , n − this yields ( n − W + 1 − | m | ) individual actions, which must be multiplied by (2 W +1) n − . The total is Count B = [(2 W +1) n − ( n − | m |− W +1)](2 W +1) n − .Remaining transitions → , ( n − → n add W + 1)(2 W + 1) n − actionsonly for m ∈ [ − W, W ] . Adding it to Count B yields Count A and completes theproof of the next Theorem for the new discrete distribution . (cid:4) Theorem 4.1.
Formulas 2 give the distribution of actions m in U . n = 4 , W = 1 , the illustration of transposed matrices is W T = − − − − − − − − − − − − − − − − − − − − − − − − − − − , U T = − − − − − − − − − − − − − − − − − − − − − − − − − − − − −
10 0 1 − − − − −
10 1 0 −
11 0 0 − . Figure 6 plots the corresponding probability mass function, PMF, of actions p ( m, W = 1 , n = 4) together with PMF for W = 1 and other n , demonstratingthe limit distribution. The distributions are discrete and lines serve only to abetter visibility of dots.The n = 81900 is seconds in the trading session of S&P 500 E-Mini futuressymbolizing the one trade per second frequency. n=81900000 corresponds to ahypothetical case of 1000 trades per second. Figure 7 presents PMF of actionsfor strategies with position limit W = 10 and different numbers of ticks.Dollars paid to the industry as constant costs per contract C for (2 W +1) n − strategies, applied each one time, can be computed using the symmetry of thedistribution as weighted actions and then divided by (2 W + 1) n − to get a P L ± contract. Plotted using Microsoft Excel. $ = C (cid:32) m = W (cid:88) m = − W | m | Count A + m = − W − (cid:88) m = − W | m | Count B + m =2 W (cid:88) m = W +1 | m | Count B (cid:33) = 2 C (cid:32) m = W (cid:88) m =1 mCount A + m =2 W (cid:88) m = W +1 mCount B (cid:33) = 2 C (cid:32)(cid:32) m =2 W (cid:88) m =1 mCount A (cid:33) − W (2 W + 1)(3 W + 1)(2 W + 1) n − (cid:33) = 2 CW (2 W + 1) n − (cid:18) (2 W + 1) n − ( n − W + 1)3 − (3 W + 1) (cid:19) = 2 CW ( W + 1)(2 W + 1)(2 W + 1) n − n − ,a P L = − CW ( W + 1)(2 n − W + 1) ≡ − C (cid:80) (2 W +1) n − j =1 abs ( U j )(2 W + 1) n − . (3)Since (cid:80) i = ni =1 i = n ( n +1)(2 n +1)6 , W ( W + 1)(2 W + 1) is divisible by three and six.The last equation is an identity, where the left side is trivial but the right one15igure 7: Probability mass functions of actions for strategies with position limit ± contracts. Plotted using Microsoft Excel.can "exhaust" any computer. Strategies generating extreme industry gains.
The minimum zero gain isgenerated only by one strategy - d.n.s. Only two strategies create the maximumgain CW ( n − each. Indeed, the maximum action reverses long to short andvice versa extreme positions: W → − W or − W → W . This can be done inticks [2 , n − . The ticks 1 and n add together maximum CW . Distribution function.
Following to Eugene Lukacs [71, pp. 5-6, p. 17], anypurely discrete distribution can be written in the form F ( x ) = (cid:80) j p j ε ( x − ξ j ) ,where x , p j , ξ j are real, p j satisfy p j ≥ , and (cid:80) j p j = 1 , and ε ( x ) = (cid:26) , if x < , if x ≥ .ξ j are discontinuity points of F ( x ) . p j is the saltus at ξ j . Let us enumeratetypes m ∈ [ − W, W ] by j = m + 2 W + 1 , where ξ j = j − W − m , and p A = p A ( m, W, n ) , p B = p B ( m, W, n ) are from Equations 2. Then, the F ( x ) is16 ( x ) = j =4 W +1 (cid:88) j =1 p j ε ( x − ξ j ) = j =4 W +1 (cid:88) j =1 p j ε ( x − j + 2 W + 1) == m = − W − (cid:88) m = − W p B ε ( x − m ) + m = W (cid:88) m = − W p A ε ( x − m ) + m =2 W (cid:88) m = W +1 p B ε ( x − m )= 12 W + 1 m =2 W (cid:88) m = − W ε ( x − m ) − n − n (2 W + 1) m =2 W (cid:88) m = − W | m | ε ( x − m ) − n (2 W + 1) (cid:32) m = − W − (cid:88) m = − W ε ( x − m ) + m =2 W (cid:88) m = W +1 ε ( x − m ) (cid:33) . (4) Characteristic function f ( t ) = (cid:82) ∞−∞ e itx dF ( x ) , i = √− , for a purely dis-crete distribution reduces to the sum f ( t ) = (cid:80) j p j e itξ j [71, p. 17] yielding f ( t ) = 12 W + 1 m =2 W (cid:88) m = − W e itm − n − n (2 W + 1) m =2 W (cid:88) m = − W | m | e itm − n (2 W + 1) (cid:32) m = − W − (cid:88) m = − W e itm + m =2 W (cid:88) m = W +1 e itm (cid:33) . (5)The function is real and even f ( − t ) = f ( t ) , since in sums the formula containsonly pairs e − ix + e ix = 2 cos( x ) . This ensures that the distribution is symmetric[71, p. 30, Theorem 3.1.2], Figures 6, 7. Therefore, f ( t ) = 1 + 2 (cid:80) m =2 Wm =1 cos( tm )2 W + 1 − n − (cid:80) m =2 Wm =1 m cos( tm ) n (2 W + 1) − (cid:80) m =2 Wm = W +1 cos( tm ) n (2 W + 1) . (6) Moments.
We compute beginning moments of order s = 1 , . . . , if they ex-ist, using [44, p. 69, Lemma 2, Equation 11] α s = i s [ d s dt s f ( t )] t =0 and centralmoments using [44, p. 69, Equation 13] µ s = i s [ d s dt s e itα f ( t )] t =0 . Examples, f (cid:48) ( t ) = − (cid:80) m =2 Wm =1 m sin( tm )2 W + 1 + 2( n − (cid:80) m =2 Wm =1 m sin( tm ) n (2 W + 1) + 4 (cid:80) m =2 Wm = W +1 m sin( tm ) n (2 W + 1) , mean = α = f (cid:48) (0) i = 0 . (7)17 (cid:48)(cid:48) ( t ) = − (cid:80) m =2 Wm =1 m cos( tm )2 W + 1 + 2( n − (cid:80) m =2 Wm =1 m cos( tm ) n (2 W + 1) + 4 (cid:80) m =2 Wm = W +1 m cos( tm ) n (2 W + 1) ,α = − f (cid:48)(cid:48) (0) = 2 (cid:80) m =2 Wm =1 m W + 1 − n − (cid:80) m =2 Wm =1 m n (2 W + 1) − (cid:80) m =2 Wm = W +1 m n (2 W + 1)= 2 W (4 W + 1)3 − n − W n − W (7 W + 1)3 n = 2 W ( W + 1)( n − n , variance = µ = α − α = 2 W ( W + 1)( n − n . (8)Let us prove d s dt s cos( mt ) = m s cos( mt + πs ) , useful for computing the mo-ments of higher orders. The induction basis : for s = 0 , , , it is correct cos( mt ) , − m sin( mt ) , − m cos( mt ) . Let us assume it is correct for < s . Then, for s + 1 ,it is m s +1 cos( mt + π ( s +1)) = m s +1 [cos( mt ) cos( πs + π ) − sin( mt ) sin( πs + π )] = m s +1 [ − cos( mt ) sin( πs ) − sin( mt ) cos( πs )] = − m s +1 sin( mt + πs ) . Explicit dif-ferentiation yields the same: ddt m s cos( mt + πs ) = − m s +1 sin( mt + πs ) . The induction step is completed. (cid:4) We get (cid:20) d s dt s f ( t ) (cid:21) t =0 = d s dt s (cid:18) W + 1 (cid:19) + 2 cos( πs ) (cid:80) m =2 Wm =1 m s W + 1 − n −
2) cos( πs ) (cid:80) m =2 Wm =1 m s +1 n (2 W + 1) − πs ) (cid:80) m =2 Wm = W +1 m s n (2 W + 1) ; d s dt s (cid:18) W + 1 (cid:19) = 12 W + 1 for s = 0 or 0 for 0 < s. (9)The right side is zero for odd ≤ s = 2 q + 1 , q = 0 , , . . . , since cos( πs ) =cos( πq + π ) = − sin( πq ) = 0 is the common multiplier. Thus, odd beginningand central, since α = 0 , moments are zeros in agreement with symmetry of S ( m, W, n ) about m = 0 , see [20, p. 183, 15.8 Measures of skewness and excess]. Distributions of actions in time i -slices. Formulas 2 count actions m in U .Slices, by i = 1 , . . . , n , of S = (2 W + 1) n − strategies can be interpreted as time i -slices and divided on two groups 1) i = 1 , n ; 2) i = 2 , . . . , n − . In the 1- and n -slice, each action m ∈ [ − W, W ] has (2 W +1) n − W +1 = (2 W +1) n − entries. In eachslice of the second group, each action m ∈ [ − W, W ] occurs Count B ( m,W,n ) n − = [(2 W +1) n − ( n − | m |− W +1)](2 W +1) n − n − = (2 W + 1 − | m | )(2 W + 1) n − times.Checking: (cid:80) m =2 Wm = − W (2 W + 1 − | m | )(2 W + 1) n − = (2 W + 1) n − [(2 W + 1)(4 W +1) − W (2 W + 1)] = (2 W + 1) n − . The following sums will be needed18 = 1 , . . . , n : j = S (cid:88) j =1 U i,j = 0; i = 1 , n : j = S (cid:88) j =1 | U i,j | = m = W (cid:88) m = − W | m | (2 W + 1) n − = W ( W + 1)(2 W + 1) n − ; i = 1 , n : j = S (cid:88) j =1 U i,j = m = W (cid:88) m = − W m (2 W + 1) n − = W ( W + 1)(2 W + 1) n − i = 2 , . . . , n − j = S (cid:88) j =1 | U i,j | = m =2 W (cid:88) m = − W | m | (2 W + 1 − | m | )(2 W + 1) n − == 2(2 W + 1) n − (cid:32) (2 W + 1) m =2 W (cid:88) m =1 m − m =2 W (cid:88) m =1 m (cid:33) == 4 W ( W + 1)(2 W + 1) n − i = 2 , . . . , n − j = S (cid:88) j =1 U i,j = m =2 W (cid:88) m = − W m (2 W + 1 − | m | )(2 W + 1) n − == 2(2 W + 1) n − (cid:32) (2 W + 1) m =2 W (cid:88) m =1 m − m =2 W (cid:88) m =1 m (cid:33) == 2(2 W + 1) n − (cid:18) W (2 W + 1) (4 W + 1)3 − (2 W ) (2 W + 1) (cid:19) == 2 W ( W + 1)(2 W + 1) n − . (10) Theorem 4.2. ∀ i, l ∈ [1 , n − , (cid:80) j = Sj =1 W i,j W l,j = W ( W +1)(2 W +1) n − δ i,l , wherethe Kronecker delta δ i,l = (cid:26) , if i (cid:54) = l , if i = l . The sum is zero, if i = n ∨ l = n .Proof. In the 1-slice, each position from [ − W, W ] is repeated (2 W +1) n − times.Since we consider S = (2 W +1) n − unique vectors of positions, for n > , any W associates with (2 W +1) n − positions l of each type [ − W, W ] in the 2-slice. Thesum of products of the constant W to these values is zero: (cid:80) l = Wl = − W W l (2 W +1) n − = W (2 W + 1) n − (cid:80) l = Wl = − W l = 0 . W is selected arbitrary making theconclusion valid for any [ − W, W ] : (cid:80) j = Sj =1 W ,j W ,j = 0 . Similar argumentationcan be applied to any pair of distinct i-slices, i = 1 , . . . , n − . Lexicographicalsorting of positions by values in two slices, ignoring others, helps to see it. Fora pair including n -slice, it is trivial because the latter is zero vector. Therefore, ∀ i (cid:54) = l ∨ i = n ∨ l = n, (cid:80) j = Sj =1 W i,j W l,j = 0 . ∀ i = l (cid:54) = n , (cid:80) j = Sj =1 W i,j W l,j = (cid:80) j = Sj =1 W i,j = (2 W + 1) n − (cid:80) l = Wl = − W l = W ( W +1)(2 W +1) n − . To shorten theformula for i, l = 1 , . . . , n − using the Kronecker delta is natural. (cid:4)
19n other words, the columns of the transposed position matrix ( W n × S ) T aremutually orthogonal vectors. The sums (cid:80) j = Sj =1 U i,j for i = 1 , n and i = 2 , ..., n − are given by Equations 10. They play the role of sample variances of actions in i -slices times ( S − or S . In contrast, (cid:80) j = Sj =1 U i,j U i + l,j for i = 1 , . . . , n − and l = 1 , . . . , n − i play the role of sample covariances times ( S − or S . Theorem 4.3.
For i = 1 , . . . , n − , l = 1 , . . . , n − i , (cid:80) j = Sj =1 U i,j U i + l,j = 0 for l > and − W ( W +1)(2 W +1) n − for l = 1 .Proof. (cid:80) j = Sj =1 U i,j U i + l,j = (cid:80) j = Sj =1 ( W i,j − W i − ,j )( W i + l,j − W i + l − ,j ) = − (cid:80) j = Sj =1 W i,j W i + l − ,j + (cid:80) j = Sj =1 W i,j W i + l,j − (cid:80) j = Sj =1 W i − ,j W i + l,j + (cid:80) j = Sj =1 W i − ,j W i + l − ,j . By Theorem 4.2, the last three sums are zeros. The firstsum is zero for l > and − W ( W +1)(2 W +1) n − for l = 1 . (cid:4) Theorem 4.4. ∀ i, l ∈ [1 , n − ∧ i (cid:54) = l , (cid:80) j = Sj =1 | W i,j || W l,j | = W ( W + 1) (2 W +1) n − . The sum is zero, if i = n ∨ l = n .Proof. In a pair of i -, l -slices, i, l ∈ [1 , n − , i (cid:54) = l , the number of unique pairs ( W i,j , W l,j ) taken once is (2 W +1) . For them, (cid:80) W i,j = WW i,j = − W (cid:80) W l,j = WW l,j = − W | W i,j || W l,j | = (cid:80) W i,j = WW i,j = − W | W i,j | (cid:80) W l,j = WW l,j = − W | W l,j | = 4 (cid:80) W i,j = WW i,j =1 W i,j (cid:80) W l,j = WW l,j =1 W l,j = W (2 W +1) . Each pair is repeated (2 W +1) n − (2 W +1) = (2 W + 1) n − times, making the total W ( W + 1) (2 W + 1) n − or zero, if i = n ∨ l = n : the n -slice is zero vector. (cid:4) Theorem 4.5.
For ≤ W , the following formulas take place A : n = 2 , j = S (cid:88) j =1 | U ,j || U ,j | = 13 W ( W + 1)(2 W + 1);B : 2 < n, j = S (cid:88) j =1 | U ,j || U ,j | = 32 W ( W + 1) (2 W + 1) n − ;C : 2 < n, j = S (cid:88) j =1 | U ,j || U n,j | = W ( W + 1) (2 W + 1) n − ;D : 3 < n, < i < n, j = S (cid:88) j =1 | U ,j || U i,j | = 43 W ( W + 1) (2 W + 1) n − ;E : 1 < i < n − j = S (cid:88) j =1 | U i,j || U i +1 ,j | = 115 W (28 W + 56 W + 27 W − W + 1) n − ;F : 1 < i < n − , i + 1 < r < n, j = S (cid:88) j =1 | U i,j || U r,j | = 169 W ( W + 1) (2 W + 1) n − . roof. ∀ j ∈ [1 , S ] ∧ ≤ n, U ,j = W ,j . For n = 2 , U ,j = − U ,j = − W ,j , A : j = S (cid:88) j =1 | U ,j || U ,j | = j = S (cid:88) j =1 W ,j = l = W (cid:88) l = − W l = 2 l = W (cid:88) l =1 l = W ( W + 1)(2 W + 1)3 . For n > , there are (2 W + 1) n − values of U ,j = W ,j of each type [ − W, W ] and − W ≤ U ,j + U ,j ≤ W or − W − W ,j ≤ U ,j ≤ W − W ,j . The (2 W +1) n − values U ,j = − W are followed once by each U ,j ∈ [0 , W ] . The (2 W + 1) n − values U ,j = − W + 1 are followed once by each U ,j ∈ [ − , W − . . . . The (2 W + 1) n − values U ,j = W are followed once by each U ,j ∈ [ − W, . Thus, B : j = S (cid:88) j =1 | U ,j || U ,j | = (2 W + 1) n − (cid:32) m =2 W (cid:88) m =0 | − W || m | + · · · + m =0 (cid:88) m = − W | W || m | (cid:33) == (2 W + 1) n − l =2 W (cid:88) l =0 m =2 W − l (cid:88) m = − l | − W + l || m | == 2(2 W + 1) n − l = W − (cid:88) l =0 ( W − l ) m =2 W − l (cid:88) m = − l | m | == 2(2 W + 1) n − l = W − (cid:88) l =0 ( W − l ) (cid:32) m = l (cid:88) m =1 m + m =2 W − l (cid:88) m =1 m (cid:33) == 2(2 W + 1) n − l = W − (cid:88) l =0 ( W − l ) (cid:18) l ( l + 1)2 + (2 W − l )(2 W − l + 1)2 (cid:19) == 2(2 W + 1) n − l = W − (cid:88) l =0 ( − l + 3 W l − (4 W + W ) l + W (2 W + 1)) == − W + 1) n − ( W − W W (2 W + 1) n − ( W − W (2 W − −− W + 1) n − W (4 W + 1) ( W − W W (2 W + 1) n − == 32 W ( W + 1) (2 W + 1) n − . Since U n,j = − W n − ,j is uniformly distributed, each of W + 1 values [ − W, W ] of U ,j represented by (2 W + 1) n − times associates with (2 W + 1) n − actionsfrom [ − W, W ] and for < n C : j = S (cid:88) j =1 | U ,j || U n,j | = (2 W + 1) n − l = W (cid:88) l = − W m = W (cid:88) m = − W | l || m | == (2 W + 1) n − l = W (cid:88) l = − W | l | m = W (cid:88) m = − W | m | = W ( W + 1) (2 W + 1) n − . i -slices, we get for < n , and < i < n D : j = S (cid:88) j =1 | U ,j || U i,j | = l = W (cid:88) l = − W | l | (cid:80) m =2 Wm = − W | m | (2 W + 1 − | m | )(2 W + 1) n − W + 1 == (2 W + 1) n − W ( W + 1) (cid:32) (2 W + 1) m =2 W (cid:88) m = − W | m | − m =2 W (cid:88) m = − W m (cid:33) == 43 W ( W + 1) (2 W + 1) n − . For < i < n − , each i -slice, containing (2 W + 1 − | l | )(2 W + 1) n − actions l , is followed by a ( i + 1) -slice with the same actions and counts. Actionsassociations between neighboring slices are not arbitrary. For W = 1 , U i,j = − is followed by U i +1 ,j = 0 , , . U i,j = − creates W i,j = − or W i,j = 0 with U i +1 ,j = 0 , , or − , , . Lexicographical sorting of strategies by i - and ( i +1) -actions uncovers the association pattern repeated (2 W + 1) n − times. Known (cid:80) i = ni =1 n = n ( n + 1)(2 n + 1)(3 n + 3 n − and the sums of powers 1, 2, 3of the natural numbers are applied. An elegant method for arbitrary powers isexplained by Etherington [37]. E : j = S (cid:88) j =1 | U i,j || U i +1 ,j | = (2 W + 1) n − l =2 W (cid:88) l = − W | l | i =2 W −| l | (cid:88) i =0 m =2 W − i (cid:88) m = − i | m | == 2(2 W + 1) n − l =2 W (cid:88) l =1 l i =2 W − l (cid:88) i =0 (cid:32) m = i (cid:88) m =1 m + m =2 W − i (cid:88) m =1 m (cid:33) == 2(2 W + 1) n − l =2 W (cid:88) l =1 l i =2 W − l (cid:88) i =0 (cid:0) ( i − W ) + W ( W + 1) (cid:1) == 13 (2 W + 1) n − l =2 W (cid:88) l =1 l (2 W + 1 − l )[2 l − (2 W + 1) l + 8 W ( W + 1)] == 115 W (28 W + 56 W + 27 W − W + 1) n − . For a pair of non-neighboring slices < i < n − , i + 1 < r < n , actions de-pendence is "forgotten". Again, lexicographical sorting of strategies by actionsin i - and r -slice uncovers the pattern repeated (2 W + 1) n − times F : j = S (cid:88) j =1 | U i,j || U r,j | = (2 W + 1) n − l =2 W (cid:88) l = − W | l | (2 W + 1 − | l | ) m =2 W (cid:88) m = − W | m | (2 W + 1 − | m | ) == 83 W ( W + 1)(2 W + 1) n − l =2 W (cid:88) l =1 l (2 W + 1 − l ) = 169 W ( W + 1) (2 W + 1) n − . (cid:4) { n = 1 .. }×{ W = 1 .. } , and { n = 7 }×{ W = 1 .. } ,a C++ program directly building the strategies and counting their actions andproducts has computed the sums of Theorem 4.5 without exceptions corres-ponding to the formulas A - F. A few illustrations are for your attention.Theorem 4.5 A, n = 2 , ( W, (cid:80) j = Sj =1 | U ,j || U ,j | ) : (1 , , (2 , , (3 , , (4 , , (5 , , (6 , , (7 , , (8 , , (9 , , (10 , .For n = 4 , W = 1 with the formula letter following the sum value U ,j U ,j U ,j U ,j
18 B 16 D 12 C U ,j
22 E 16 D U ,j
18 B
See also W T and U T presented earlier for this case . For n = 7 , W = 3 , U ,j U ,j U ,j U ,j U ,j U ,j U ,j U ,j U ,j U ,j U ,j U ,j There is an interpretation for remembering location of Formulas A - F. FormulaA is applied only for n = 2 . For ≤ n , the first row (B, D, ..., D, C) rotatesaround the "origin" C 90 degrees counterclockwise making a symmetrical rightangle. For n = 3 , the angle B-C-B has no D. For ≤ n , the second row (E, F,..., F, D) rotates around the "origin", right most F, 90 degrees counterclockwisealso making a symmetrical right angle. For n = 4 , there is no inner angle butsingle E, see above. Creation of nested angles is repeated until the single E, foreven n , or last angle E-F-E, for odd n . The sum of these n ( n − elements isequal to
2B + C + 2( n − n − ( n − n − F . The n × n matrix issymmetric and the sum of the off-diagonal elements is the double:
4B + 2C +4( n − n − n − n − . The diagonal is in Equations 10. The system of vectors U is linearly dependent: one, d.n.s., is [123, p. 46,Lemma 14.3], and < n < (2 W + 1) n − [123, p. 51, Basis], [50, p. 14, Theorem2]. We can find in U a linearly independent system [123, p. 45, Lemma 14.1]. Lemma 5.1.
The rank of the span of U is less than n .Proof. ∀ j , (cid:80) i = ni =1 U i,j = 0 . Multiplying it by < P yields P (cid:80) i = ni =1 U i,j = (cid:80) i = ni =1 P U i,j = P T U j = 0 . The n -dimensional P is orthogonal to each U j ∈ U , P ⊥ U [123, p. 92 - 94, Orthogonality], and to the span of U . The rank of thespan is less than n . (cid:4) The P = ( P = P, . . . , P n = P ) T is interpreted as a flat price.23 rthogonal vectors of U . By Lemma 5.1, for n = 2 , all W + 1 vectors of U are collinear. For n = 3 , all (2 W + 1) vectors of U are coplanar with the ortho-gonal basis { (1 , , − T , (1 , − , T } ∈ U . Let ( ∗ ) κ is a chain empty for κ = 0 : (0) = (0 , , , (1 , − = (1 , − , , − , ((0) , , (0) , − , (0) ) = (1 , , − .For ≤ n , η = 0 , . . . , (cid:98) n − (cid:99) , the (cid:98) n − (cid:99) +1 η -vectors ((0) η , , (0) n − − η , − , (0) η ) T ∈ U are mutually orthogonal . For ≤ n , λ = 0 , . . . , (cid:98) n (cid:99) − , the (cid:98) n (cid:99) λ -vectors ((0) λ , , − , (0) n − − λ , − , , (0) λ ) T ∈ U are mutually orthogonal togetherwith the η -vectors. For ≤ n , ν = 0 , ..., (cid:98) n − (cid:99) , the (cid:98) n − (cid:99) + 1 ν -vectors ((0) ν , , − , , (0) n − − ν , , − , , (0) ν ) T ∈ U is an orthogonal alternative tothe λ -vectors. The θ -vector ((0) (cid:98) n − (cid:99) , , − , , (0) (cid:98) n − (cid:99) ) T ∈ U for odd ≤ n =2 l + 1 , l = 1 , . . . is orthogonal to the η -vectors. Examples: n n − (cid:98) n − (cid:99) (cid:98) n (cid:99) (cid:98) n − (cid:99) η λ ν U T n/a n/a n/a n/a η : (1 , − n/a n/a n/a n/a η : (1 , , − θ : (1 , − , n/a n/a η : (1 , , , − η : (0 , , − , λ : (1 , − , − , n/a n/a η : (1 , , , , − η : (0 , , , − , λ : (1 , − , , − , θ : (0 , , − , , η : (1 , , , , , − η : (0 , , , , − , η : (0 , , , − , , λ : (1 , − , , , − , ν : (1 , − , , , − , η : (1 , , , , , , − η : (0 , , , , , − , η : (0 , , , , − , , λ : (1 , − , , , , − , ν : (1 , − , , , , − , θ : (0 , , , − , , , η : (1 , , , , , , , − η : (0 , , , , , , − , η : (0 , , , , , − , , η : (0 , , , , − , , , λ : (1 , − , , , , , − , λ : (0 , , , − , − , , , ν : (1 , − , , , , , − , { η } ⊥ { λ } , { η } ⊥ { ν } , { θ } ⊥ { η } , { θ } ⊥ { λ } , but { λ } (cid:54)⊥ { ν } , { θ } (cid:54)⊥ { ν } . Rank of U . Lemma 5.1 limits the rank of U from above. For n = 2 , , itis n − , , . The proofs are the orthogonal bases in U : { η : (1 , − T } ,24 η : (1 , , − T , θ : (1 , − , T } , { η : (1 , , , − T , η : (0 , , − , T , λ :(1 , − , − , T } . For n = 6 , the rank is n − : { η : (1 , , , , , − T , η :(0 , , , , − , T , η : (0 , , , − , , T , (1 , , − , − , , T , (1 , − , , , − , T } .The latter two are not η, λ, ν, θ -strategies. For n = 8 , the rank is n − : { η : (1 , , , , , , , − T , η : (0 , , , , , , − , T , η : (0 , , , , , − , , T , η : (0 , , , , − , , , T , (0 , , − , , , − , , T , (1 , − , − , , , − , − , T , (1 , , , − , − , , , T } . The latter four are not η, λ, ν, θ -strategies. For n =5 , , U contains maximum n − , orthogonal vectors. This is proved bychecking all mutually orthogonal combinations using a C++ program.
Let us notice that ( W , W , . . . , W n − , Tj → (0 , W , . . . , W n − , W n − ) Tj is acyclic permutation of coordinates and does not change the length of the vector.This is a rotation expressed by R W j , where the matrix R is orthogonal R = . . . . . . . . . . . . . . . . . . . . . . . . . . . , RR T = . . . . . . . . . . . . . . . . . . . . . . . . . . . = I . An example using the × R T is found in [50, p. 69, Exercise 10, matrix A ].Since W ,j = W n,j = 0 , U j = W j −R W j = ( I −R ) W j , where I is the identitymatrix with ones on the diagonal. Determinant det( R ) = 1 . Thus, the Gramianmatrix for U is G U = U T U = W T ( I −R ) T ( I −R ) W = W T (2 I −R−R T ) W . Thesquare n × n matrix I −R−R T has the main diagonal with twos, sub and superdiagonals with -1, and two symmetric corner elements -1. This guarantees thatfor ≤ n sums of rows and columns in the matrix are zero vectors. ApplyingTheorem 1 about the matrices product rank from [7, p. 76] twice, we concludethat U T U has the rank less than n . This is another proof of Lemma 5.1.
Atthe same time, it is exactly n − for n = 2 , , , , and due to η - and λ -vectorsnot less than (cid:98) n − (cid:99) + 1 + (cid:98) n (cid:99) for < n . Theorem 5.1.
The rank of U is n − .Proof. For ≤ n and any W , U has n − strategies with only two orderedtransactions: buy one at ≤ i < n followed by sell one at the last n th tick: (cid:100) . . . (cid:101) − | . . . | − | . . . | − . . . . . . . . . . . . . . . . . . (cid:98) . . . (cid:99) − . . . . . . . . . . . . . . . . . . . The top left submatrix of U T , always obtainable after a suitable rearrangementof rows, is diagonal ( n − × ( n − identity matrix with determinant one. Anygreater minor of order n is zero due to Lemma 5.1. Then, the highest order ofnon-zero minor of U is n − . This is the rank [123, p. 132]. (cid:4) heorem 5.1 means that the span of the trading strategies U , containingd.n.s., is the hyperplane of the linear space created, spanned, by arriving ticks. Buy and hold.
This strategy is a popular investment benchmark. "Holding"is "never selling" a purchased security or real estate. All futures, and manybonds and options expire. Examples of perpetual and long paying interestbonds are the Dutch Water Bonds dated by 1624, the British Consoles issuedfirst time in 1751 [1], [78], some perpetual debt in France [5]. The perpetualdebt financial instruments is not only a history [3], [115]. The lookback Russianput option [105], [25] with "reduced regret" has no expiration date.In practice, "never selling" is "holding for a long time". For futures, with wellknown expiration date and time, "hold" might mean "up to the expiration". Fora chain of prices "hold" might mean "until the last tick". Even, if one "holds"or does not sell what has been purchased, one way to estimate its value is toassume that it is sold at the price P n and cost C n - mark-to-market or "fair"value. This adds an artificial sell transaction to each valuation tick of interestafter buying. All strategies j in U exit the market with W n,j = 0 , if they enterit. The d.n.s with W n,d.n.s = 0 never enters the market.For comparison with an investment, the buy is assumed coinciding with thebeginning of the investment. Here, the single strategy buying at the beginningand marking-to-market at the end U b.a.h. = (1 , , . . . , , − T is "buy and hold",b.a.h. This is the η -strategy. The strategies from the proof of Theorem 5.1buying at i and selling at the end is "buy, hold, and sell", b.h.s. The b.a.h. isb.h.s but not necessarily vice versa. The system of n − b.h.s. is the base of U :it is linearly independent, and any other strategy in U is a linear combinationof b.h.s. For instance, (1 , − , T = (1 , , − T − , , − T ; (1 , − , T =(1 , , − T − (0 , , − T . A system of vectors may have several bases. All ofthem are equivalent [123, pp. 47 - 50] and in our case have n − vectors. The n − b.h.s. are not mutually orthogonal. Each has the Euclidean length √ .The U is a subsystem of a linear space over the fields of rational, real, or complexnumbers. Its span is a hyperplane in one of these spaces with the n − b.h.s.serving as hyperplane non-orthogonal basis. Entry-wise operations.
The abs ( U j ) is the entry-wise absolute value func-tion. The author did not find a suitable notation to express this. [53, p. 88]: "The Hadamard product of two matrices A = [ a ij ] and B = [ b ij ] with the samedimensions (not necessarily square) with entires in a given ring is the entry-wiseproduct A ◦ B = [ a ij b ij ] , which has the same dimensions as A and B ." This isalso known as
Schur product [18]. The history of names Schur and Hadamardproduct is in [53, pp. 92 - 95, Historical remarks]. Entry-wise Hadamard powersand square roots are denoted A ◦ , A ◦ , A ◦ [86]. If we take the non-negativesquare root values, then abs ( U j ) = ( U j ◦ U j ) ◦ and PL q × S = − k ( P n × q ) T U n × S − ( C n × q ) T ( U n × S ◦ U n × S ) ◦ , (11)26here P n × q is the price matrix with q scenarios, price column-vectors of size n , U n × S =(2 W +1) n − is the strategies matrix for the set U , PL q × S is the profit andloss matrix with S columns of size q corresponding to q price scenarios and theset U , C n × q is the cost matrix with q scenarios, cost column-vectors of size n .If the cost per share is a fixed non-negative fraction f of price, "equity case",then ( C n × q ) T = kf ( P n × q ) T . If the cost per contract is the constant C , "futurescase", then ( C n × q ) T = ( C J n × q ) T , where J n × q is the Hadamard identity matrix with all elements equal to one. For q = 1 , the "full" matrix form is reduced to PL × S , the row-vector P L of size S . This is a sample of S = (2 W + 1) n − P L values for U and one price scenario P . The distribution of actions in U , Equations 2, Figures 6, 7, mean a P L , Equations3, Theorem 3.3, do not depend on P . A distribution of P L for U depends on P and C , Equation 11. Without losing generality, the n − basis strategiescan be ordered as U b.h.s. = (1 , , . . . , , − T , U b.h.s. = (0 , , . . . , , − T , . . . , U b.h.s.n − = (0 , , . . . , , − T , where the first n − coordinates are zeros, except 1at i and -1 at n . Due to these and hyperplane properties of U , any strategy U j =( U ,j , U ,j , . . . , U n − ,j , U n,j ) T ∈ U in the basis is U j = (cid:80) i = n − i =1 U i,j U b.h.s.i =( U ,j , U ,j , . . . , U n − ,j , − (cid:80) i = n − i =1 U i,j ) T . Multiplication and summation yieldcorrect n th coordinate because (cid:80) i = ni =1 U i,j = 0 .The first component P L I of the P L distribution is values for j = 1 , . . . , (2 W +1) n − : − k P T U j = − k P T (cid:80) i = n − i =1 U i,j U b.h.s.i = k (cid:80) i = n − i =1 U i,j ( P n − P i ) . Theirsum is zero, Theorem 3.3. The second component P L II of the P L distribution isvalues for j = 1 , . . . , (2 W +1) n − : − C T ( U j ◦ U j ) ◦ = − C T ( | U ,j | , . . . , | U n,j | ) T .Their mean sum for constant cost C is a P L , Equation 3. The P L II list has re-peated values. Depending on P , the P L I list may have repeated values too.Thus, each P L I value is a linear combination of k ( P n − P i ) , i = 1 , . . . , n − .Each P L II value is a corresponding linear absolute combination of − C . It iseither zero for single d.n.s. or even negative multiple of C . For fixed P and C ,corresponding values in two lists relate each to other due to integer coefficientsof linear combinations. For q = 1 , Equation 11 converts a sample distributionof n prices P i into a sample distribution of (2 W + 1) n − values P L j . Sample distributions of P i and ∆ P i . A chain of n numbers { P , . . . , P i , . . . , P n } creates chains of n − adjacent absolute differences { ∆ P = P − P , . . . , ∆ P i = P i − P i − , . . . , ∆ P n = P n − P n − } , relative differences { ∆ P P , . . . , ∆ P i P i − , . . . , ∆ P n P n − } ,and popular log-returns { ln( P P ) , . . . , ln( P i P i − ) , . . . , ln( P n P n − ) } . The latter two,well defined for < P i P i − , are close for ∆ P i P i − → . Futures prices are positive.Many theories and speculations are devoted to these chains, when P i are pricesor rates. Some focus on increments. Other pay attention to prices.Indeed, a Brownian motion is about increments, their independence, Gaus-sian properties, fundamental proportionality of their variance to elapsed time274], [35], [81], [89]. Its sophisticated combinations are popular in pricing deriv-atives [81], [54], [55], [26]. In contrast, the trading pattern head and shoulders [80, Figure 3a, pp. 559 - 561], [24, p. 236], [79, pp. 74, 76, 108 - 110, 153 -155], [60, pp. 108 - 110] appearing also in coin-tossing experiments, not possess-ing predictive power [73, p. 131], cares about price levels in time combinationsresembling a top of a human body. For one, trading on such patterns is as-trology . However, markets are people and programs created by people tradingsomething. If participants "believe" into such matters and trade based on their"beliefs", then a trading feedback can affect markets transforming "beliefs" toreality. The impact should depend on the fraction of trading "believers". Howthis fraction may form, based on the properties of random prospects , is sug-gested in [102, p. 33, Hypothesis]. For a pragmatist, a "working pattern" ismore important than "why it is working". Dynkin-Neftci times decompose asituation on 1) evaluation that an event has occurred using available, "not fromthe future", information, and 2) gathering statistics how frequently the event isfollowed by a certain scenario in the past.The
Ornstein-Uhlenbeck process [120] combines variable’s increments withits single value playing a special role. When the variable crosses zero line thedrift for increments reverses its sign. The farer from the level, the greater driftmagnitude is. Being accompanied by random shocks, it directs the variable backto the level. All continues on the opposite side. This ability to fluctuate aroundan attractor level is reused by the mean-reversion models of interest rates, wherethe level is shifted from zero to a positive value [54, pp. 418 - 419]. A care shouldbe taken to avoid rates going to the negative territory similar to the originalBachelier’s price assuming Gaussian properties for absolute increments. Cox,Ingersoll, Ross standard deviation of random term proportional to the squareroot of rate and log-normal properties of latter is one way to ensure positivity[54, p. 418]. The author has enjoyed the brief and thorough review [57, p.271]: "... his [VS: Laplace’s] formidable intuition has led him to a differentialequation which is entirely justifiable, and is in fact the Fokker-Plank equationfor a one-dimensional Ornstein-Uhlenbeck process, which appears as the weaklimit of the Bernoulli-Laplace urn models" .The author has formulated the "chicken and egg question" of what is morefundamental prices or their increments [95], [100, pp. 34 - 35]. A hybrid ap-proach relies on both. How many price levels should be taken into consideration?Are these levels permanent? One of the problems is non-stationarity of markets[69, pp. 194]: "There are many reasons for considering nonstationary markets,the most obvious of which is that the economic conditions keep changing andthis change cannot be adequately captured by stationary models" . In [100, p.35] the author has expressed his view: "... markets have many modes replacingeach other in time, where prices or increments get varying accents" .Sample means of prices a P = (cid:80) i = ni =1 P i n and increments a ∆ P = (cid:80) i = ni =2 ∆ P i n − withfixed P are interdependent [103, p. 11, Equation 4]: a P = P + n − n a ∆ P − (cid:80) i = ni =2 i ∆ P i n . While each statistics does not depend on the order of sample num-28ers, together they are bound by the term (cid:80) i = ni =2 i ∆ P i , where products i ∆ P i are sensitive to the order due to the multiplier i . We expect that other samplestatistics for both sets are interdependent, in general. For sample variances ( S Pn − ) = (cid:80) i = ni =1 ( P i − a P ) n − and ( S ∆ Pn − ) = (cid:80) i = ni =2 (∆ P i − a ∆ P ) n − , we get ( S Pn − ) = (cid:80) i = ni =1 ( P i − a P ) n − P + (cid:80) i = ni =2 P i − n ( a P ) n − S ∆ Pn − ) = (cid:80) i = ni =2 (∆ P i − a ∆ P ) n − (cid:80) i = ni =2 (∆ P i ) − ( n − a ∆ P ) n − P − P n + 2 (cid:80) i = ni =2 P i ∆ P i − ( n − a ∆ P ) n − S Pn − ) = ( n − S ∆ Pn − ) + P n + (cid:80) i = ni =2 ( P i − P i ∆ P i ) − n ( a P ) n − a ∆ P ) . (12) Distributions of
P L I , P L II , and P L = P L I + P L II can be consideredseparately. Their sample means depend neither on P i nor ∆ P i , i = 1 , . . . , n , and,for constant C , are a P L I = 0 , a P L = a P L II = − CW ( W +1)(2 n − W +1) . Variances of P L and
P L I should depend on P . For S = (2 W + 1) n − , since | U i,j | ≤ W , ≤ ( S P L I n − ) = (cid:80) j = Sj =1 ( P L Ij − a P L I ) S − (cid:80) j = Sj =1 ( P L Ij ) S − (cid:80) j = Sj =1 ( − k (cid:80) i = ni =1 P i U i,j ) S − ≤ k W SS − i = n (cid:88) i =1 P i ) < k W ( i = n (cid:88) i =1 P i ) . (13)It is known and easy to prove using mathematical induction that ( (cid:80) i = ni =1 a i ) = (cid:80) i = ni =1 a i + 2 (cid:80) l = nl =1 (cid:80) i = l − i =1 a i a l = (cid:80) i = ni =1 a i + 2 (cid:80) l = nl =1 a l (cid:80) i = l − i =1 a i = (cid:80) i = ni =1 a i +2 (cid:80) l = nl =2 a l (cid:80) i = l − i =1 a i = (cid:80) i = ni =1 a i + 2 (cid:80) l = n − l =1 a l (cid:80) i = ni = l +1 a i . With Equations 10, ( S P L I n − ) = (cid:80) j = Sj =1 ( − k (cid:80) i = ni =1 P i U i,j ) S − k S − j = S (cid:88) j =1 (cid:32) i = n (cid:88) i =1 P i U i,j + 2 l = n (cid:88) l =2 P l U l,j i = l − (cid:88) i =1 P i U i,j (cid:33) == k S − i = n (cid:88) i =1 P i j = S (cid:88) j =1 U i,j + 2 j = S (cid:88) j =1 l = n (cid:88) l =2 P l U l,j i = l − (cid:88) i =1 P i U i,j == k S − W ( W + 1) S (cid:32) i = n (cid:88) i =1 P i − P − P n (cid:33) + 2 j = S (cid:88) j =1 l = n − (cid:88) l =1 P l U l,j i = n (cid:88) i = l +1 P i U i,j .
29n the first summand, (cid:80) i = ni =1 P i is the square of the price vector P length. Thesecond summand can be expressed as the double sum (cid:80) j = Sj =1 of n ( n − terms j = S (cid:88) j =1 P P U ,j U ,j + P P U ,j U ,j + . . . + P P n U ,j U n,j ++ P P U ,j U ,j + . . . + P P n U ,j U n,j ++ . . . + . . . ++ P n − P n U n − ,j U n,j . By Theorem 4.3, all sums above the bottom diagonal are zeros. After summationby j , the diagonal terms get the common multiplier − W ( W +1)(2 W +1) n − , and ( S P L I n − ) = k W ( W + 1) S S − (cid:32) i = n (cid:88) i =1 P i − i = n − (cid:88) i =1 P i P i +1 − P − P n (cid:33) == k W ( W + 1)(2 W + 1) n − W + 1) n − − i = n (cid:88) i =2 (∆ P i ) . (14)The variance of P L II for constant C is ( S P L II n − ) = (cid:80) j = Sj =1 ( P L
IIj − a P L II ) S − (cid:80) j = Sj =1 ( P L
IIj ) − S ( a P L II ) S − C (cid:80) j = Sj =1 ( (cid:80) i = ni =1 | U i, j | ) − S ( a P L II ) S − , where j = S (cid:88) j =1 ( i = n (cid:88) i =1 | U i, j | ) = j = S (cid:88) j =1 (cid:32) i = n (cid:88) i =1 U i,j + 2 l = n (cid:88) l =1 | U l,j | i = l − (cid:88) i =1 | U i,j | (cid:33) == i = n (cid:88) i =1 j = S (cid:88) j =1 U i,j + 2 l = n (cid:88) l =1 i = l − (cid:88) i =1 j = S (cid:88) j =1 | U l,j || U i,j | . Equations 10 "evaluate" the left term-sum: two equal sums for i = 1 , n , plus n − equal intermediate sums for i = 2 , . . . , n − : W ( W + 1)(2 W + 1) n − + ( n − W ( W + 1)(2 W + 1) n − = n − W ( W + 1)(2 W + 1) n − . Theorem 4.5"evaluates" the right term-sum: for n = 2 , or
4B + 2C + 4( n − n − n − n − for ≤ n . Thus, n = 2 : ( S P L II n − ) = 43 W ( W + 1) (2 W + 1) ; 3 ≤ n : ( S P L II n − ) =4 C W ( W + 1)(2 W + 1) n − (6 n (2 W + 2 W + 1) − W ( W + 1) − W + 1) n − − . (15) Theorem 6.1.
For U , ( S P Ln − ) = ( S P L I n − ) + ( S P L II n − ) .Proof. P L j = P L Ij + P L
IIj = − k (cid:80) i = ni =1 P i U i,j − (cid:80) i = ni =1 C i | U i,j | . a P L = a P L I + a P L II = 0 + a P L II = a P L II . The a P L II does not assume a particular constantcase C i = C but more general vector C .30 S P Ln − ) = (cid:80) j = Sj =1 ( P L j − a P L ) S − (cid:80) j = Sj =1 ( P L j ) − S ( a P L ) S − (cid:80) j = Sj =1 ( P L Ij + P L
IIj ) − S ( a P L II ) S − (cid:80) j = Sj =1 [( P L Ij ) + 2 P L Ij P L
IIj + (
P L
IIj ) ] − S ( a P L II ) S − S P L I n − ) + ( S P L II n − ) + 2 (cid:80) j = Sj =1 P L Ij P L
IIj S − , where j = S (cid:88) j =1 P L Ij P L
IIj = − k j = S (cid:88) j =1 ( P U i,j + · · · + P n U i,j )( C | U i,j | + · · · + C n | U i,j | ) . Opening brackets under the sum yields the terms − k (cid:80) j = Sj =1 P i U i,j C l | U l,j | = − kP i C l (cid:80) j = Sj =1 U i,j | U l,j | . However, ∀ i,l , (cid:80) j = Sj =1 U i,j | U l,j | = 0 . Indeed, U consistsof a d.n.s. and pairs ( U j , U j (cid:48) = − U j ) , strategies and their "mirror reflections"in the index i , time, axis. U i,d.n.s. | U l,d.n.s. | = 0 . In any pair, U i,j | U l,j | +( − U i,j ) |− U l,j | = ( U i,j − U i,j ) | U l,j | = 0 . (cid:4) Similar to U , the set of positions W consists of the do nothing position , d.n.p, W d.n.p. = (0 , . . . , T and pairs of mirror reflections ( W j , − W j ) in index i ,time, axis. Let us define on W the binary operation denoted ⊕ W , a pairwisearithmetic addition of coordinates W j ⊕ W W l = ( W ,j ⊕ W W ,l , . . . , W n,j ⊕ W W n,l ) T , so that each coordinate sum > W is replaced with W and < − W with − W . This, otherwise ordinary addition, ensures that for any pair of positionvectors the vector-result belongs to W , the closure property .Following to Cayley [16, p. 41], [17, pp. 144 - 153], we illustrate the operationfor W = 3 using the table, named today after him, for coordinates of positions ⊕ | − − − −− −− −− −− −− −− −− −−− | − − − − − − − | − − − − − − | − − − − | − − − | − − | − | The first, left, and second, right, elements are selected by column and row. Theresult is on the row and column intersection. Italic numbers show "underflows"31 ⊕ − − and "overflows" ⊕ . The bold numbers correspond tousual addition of integers. The number of table entries, pairs of the Cartesianproduct [ − W, W ] × [ − W, W ] , is (2 W + 1) . The number of "underflows" and"overflows" is W ( W + 1) . The number of ordinary additions is (2 W + 1) − W ( W + 1) = 3 W + 3 W + 1 . For ≤ W , the greatest W +3 W +1(2 W +1) = is for W = 1 . lim W →∞ W +3 W +1(2 W +1) = . The ratio monotonically decreases with thegrowing W because of the negative "derivative" − W +1) . The "drop" to theasymptotic level is − = .The operation is commutative : W j ⊕ W W l = W l ⊕ W W j . Its, symmetricwith respect to the main diagonal, Cayley table shows this well. Such tables are"less friendly" for conclusions about associativity requiring three elements andtwo sequential operations. For some values, associativity holds: (1 ⊕ ⊕ ⊕ (1 ⊕
1) = 1 . In general, ⊕ W is not associative : (1 ⊕ ⊕ − but ⊕ (1 ⊕ −
1) = 1 . An example of a commutative not associative operation isthe mean: a + b . The game Rock–Paper–Scissors also illustrates a commutativenot associative operation: ( RP ) S = S , R ( P S ) = R , RP = P R = P , etc. ∀ W ∈ W , W ⊕ W W d.n.p. = W d.n.p. ⊕ W W = W . Hence, W d.n.p. in W isthe two-sided identity element or simply identity [72, p. 67]. Every W ∈ W is invertible with the unique inverse element − W ∈ W : W ⊕ W − W = − W ⊕ W W = W d.n.p. . The identity is own inverse.There might be several solutions of W j ⊕ W X = W l or W i,j ⊕ W X i = W i,l .If W i,l = ± W , then, depending on W i,j , several X i can be good: 1) ⊕ x = 3 , x = 1 , , ; 2) ⊕ x = 3 , x = 2 , ; 3) − ⊕ x = − , x = − , − , − . Choosinga solution by absolute minimum value ensures uniqueness: 1) 1; 2) 2; 3) -1.Not every equation W i,j ⊕ W X i = W i,l has a solution: − ⊕ x = 3 requires forbidden x = 5 (cid:54)∈ [ − , . In spite of unique invertibility of every W ∈ W ,the equivalent, due to commutativity of ⊕ W , equations W i,j ⊕ W X i = W i,l and X i ⊕ W W i,j = W i,l have no, or | W i,j | + 1 (one or several) solutions: DIAGRAM W i,j ⊕ W X i = W i,l | W i,l − W i,j | > W (cid:46) (cid:38) | W i,l − W i,j | ≤ W { X i } = Ø { X i } (cid:54) = Ø | W i,l | < W (cid:46) ↓ | W i,l | = W unique solution | W i,j | + 1 solutions : X i = W i,l − W i,j X i ∈ [ W − W i,j , W ]for W i,l = W ; X i ∈ [ − W, − W − W i,j ]for W i,l = − W. With several solutions X i , X i = W i,l − W i,j has the least absolute value. Financial sense behind ( W , ⊕ W ) is: applying strategies to an account andsingle futures type the sums of corresponding positions cannot exceed by absolutevalue a level determined by margin requirements and/or position limits .32 lassification. Due to the closure, the algebraic structure ( W , ⊕ W ) is a magma [11, p. 1, LAWS OF COMPOSITION, Definition 1]: not associative [11, p. 4,ASSOCIATIVE LAWS, Definition 5], commutative [11, p. 7, PERMUTABLEELEMENTS, COMMUTATIVE LAWS, Definitions 7, 8], initial , because hasthe identity [11, p. 12, IDENTITY ELEMENT; CANCELLABLE ELEMENTS;INVERTIBLE ELEMENTS, Definition 2], and with a unique inverse element for each element in W . A term, interchangeably applied with magma, is groupoid [72, p. 67], [90, p. 90], [8, p. 6, Definition 1], [15, p. 1], [91, p. 1].If W i,j ⊕ W X i = W i,l with the commutative ⊕ W would have a unique solution X i for any pair ( W i,j , W i,l ) of the Cartesian product [ − W, W ] × [ − W, W ] , thenthe groupoid ( W , ⊕ W ) would be a quasigroup [72, p. 72], [15, p. 9], [8, p. 6,Definition 1], [91, p. 23, 1.3. Definition]. Moreover, since ( W , ⊕ W ) has theidentity W d.n.p. , it would be a loop [72, p. 73], [15, p. 15], [8, p. 8, Definition4], [91, p. 24, 1.6. Definition]. For completeness, an associative loop is a group and associative commutative loop is an Abelian group . Our "loop" differs.
While uniqueness of X i is achieved by selecting from a finite set of solutionsthe one with the absolute minimum, not every pair ( W i,j , W i,l ) has a solution:pairs for which | W i,l − W i,j | > W require forbidden values (cid:54)∈ [ − W, W ] .Malcev [72], Belousov [8], and Sabinin [91] define quasigroup not only asa groupoid with a need to solve a system of two (or one for two-sided case)equations but alternatively as an algebra . The latter includes the set, the mainbinary operation, and two (or one for two-sided case) binary inverse operations.In our case, the algebraic structure includes: the set W , the total (defined forall pairs of elements) not associative commutative binary operation ⊕ W , theidentity element W d.n.p. , the inverse element − W for each element W . It canbe added a partial inverse binary operation (cid:9) W . The adjective "partial" hasthe traditional meaning: "defined for some but not all pairs of elements". Theoperation ⊕ W is total. The inverse operation (cid:9) W is partial. Malcev illustratessuch a possibility using the set of natural numbers including zero, binary arith-metic addition defined for each pair of numbers, and partial binary arithmeticsubtraction defined only for pairs ( a, b ) , where a ≥ b [72, p. 30].The Cayley table for the coordinate i subtraction W i,l (cid:9) W W i,j is (cid:9) | − − − −− −− −− −− −− −− −− −−− | − − − n / a n / a n / a − | − − − n / a n / a − | − − − n / a0 | − − − | n / a − − | n / a n / a − | n / a n / a n / a The operation (cid:9) W is not commutative (the table is antisymmetric with respectto zero diagonal), not associative (for instance, (1 (cid:9) ( − (cid:9) ( −
1) = 3 but (cid:9) (( − (cid:9) ( −
1) = 1 ), partial. The number of pairs for which the result is33ot available is W ( W + 1) . The total number of pairs is (2 W + 1) . The numberof pairs with the defined subtraction is the difference W + 3 W + 1 .The terms partial magma , partial loop are applied in cases, where the ma-jor operation is partial [82]. While the main properties of the algebraic system ( W , ⊕ W , (cid:9) W ) defined on the finite set of trading positions W with total bin-ary not associative commutative addition ⊕ W , and partial not associative, notcommutative subtraction (cid:9) W , including domain pairs counting, are described,the author feels uncomfortable to name it a partial loop because ⊕ W is totaland viewed as the main operation.Non-associativity of the main algebraic operation creates a link to works onnon-associative algebras including the contribution of Etherington [38]. Theirfocus is on different and often more complicated algebraic structures than onediscussed. Following to the Schafer’s remark [104, p. 1], emphasizing that nonassociative algebra does not assume associativity, while not associative al-gebra means that associativity is not satisfied, we say not associative ⊕ W . Let W = W j ⊕ W W l . By Theorem 3.1, with W = W ,j = W ,l = 0 , W ↔ U , W j ↔ U j , W l ↔ U l . What is a corresponding binary operation U = U j ◦ U l ?The following recipe exists. Using W i,j = (cid:80) r = ir =1 U r,j and W i,l = (cid:80) r = ir =1 U r,l ,convert U j to W j and U l to W l . Then, "add" positions W = W j ⊕ W W l andapplying adjacent difference convert W to U . The latter should be recognizedas the result of U j ◦ U l .We have seen from distributions of positions and actions that, while positionsin steps i − and i are combined independently and uniformly, for actions itis not so because they have to ensure that positions are within the limits. Theauthor believes that it is impossible in a general case < i < n to compute U i given U i,j and U i,l . Information from step i − is needed.For ⊕ W , the Cayley table is antisymmetric , × ( − , with respect to thesecond zero diagonal. Theorem 8.1. ∀ a, b ∈ [ − W, W ] , − ( a ⊕ W b ) = ( − a ) ⊕ W ( − b ) .Proof. Multiplying both a and b by − corresponds to the reflection in thesecond zero diagonal of the Cayley table. But the table is antisymmetric, × ( − ,with respect to this reflection. (cid:4) We can write U i = W i − W i − = ( W i,j ⊕ W W i,l ) − ( W i − ,j ⊕ W W i − ,l ) =([ W i − ,j + U i,j ] ⊕ W [ W i − ,l + U i,l ]) − ( W i − ,j ⊕ W W i − ,l ) . For i = 1 , this yields U = U ,j ⊕ W U ,l and ◦ ≡ ⊕ W . For i = n , U n,j = 0 − W n − ,j = − W n − ,j , U n,l = 0 − W n − ,l = − W n − ,l , and U n = − ( W n − ,j ⊕ W W n − ,l ) = { by Theorem8.1 } = (( − W n − ,j ) ⊕ W ( − W n − ,l )) = U ,j ⊕ W U ,l and ◦ ≡ ⊕ W . Thus, thecoordinate wise operation of U = U j ◦ U l for ≤ i ≤ n is U i = ([ W i − ,j + U i,j ] ⊕ W [ W i − ,l + U i,l ]) − ( W i − ,j ⊕ W W i − ,l ) . (16)34 The maximum profit trading strategies
Today, for given P , C , W = 1 , computing P L values by Equation 1, inorder to select the maximum profit strategy, MPS, is impossible. A quantumcomputer [40], [23] would need (cid:100) log (3 ) (cid:101) = (cid:100) ln(3)ln(2) (cid:101) = 213825 qbits to represent the corresponding coherent superposition quantum states . Plus,quantum algorithms are required [112], [113]. Recent successes are 2000 qbits DWave 2000Q computer for annealing simulation , an analog computer , [13], and51 qbits generic computational device created at Harvard University [41], [88].Since d.n.s. with
P L = 0 is available, MPS cannot lose. In [93], the authorhas developed the l- and r-algorithms (left and right) with linear complexity O ( n ) . This is faster than genetic algorithms [46], which do not guarantee max-imum. Given P , C , and W ∈ N , it returns U with maximum P L . Withoutloosing generality, W = 1 . This strategy, denoted MPS0 and not reinvestingprofits, is a foundation for MPS1 and MPS2 reinvesting them. Similar to MPS0,MSP1 reverses long and short positions. In contrast with MPS0, MPS1 addsto positions while switching, if initial and maintenance futures margins permit.MPS2 extracts the absolute maximum reinvesting immediately, if it is profitable.Discrete MPS0, MPS1, MPS2 have been studied [93], [94], [95], [96], [97], [100],[103]. MPS0, MPS1, MPS2 are objective market properties. Not all elements ofMPS are Markov times . "Markov time". Neftci is, probably, first who applied Markov times to form-alize Technical Analysis [80, p. 553]: "... one contribution this article makesis to recognize the importance of Markov times as a tool to pick well-definedrules for issuing signals at market turning points. Let { X t } be an asset price... Let { I t } be the sequence of information sets (sigma-algebras) generated bythe X t and possibly by other data observed up to time t . ... a random vari-able τ is a Markov time if the event A t = { τ ≤ t } is I t -measurable - that is,whether or not τ is less than t can be decided given I t " . Giles says [42, p. 175]: "... Neftci’s Markov Times approach..." . This does not mean that Neftci intro-duced "Markov times" but suggested an approach using them. The definitionof a Markov time is in the first English edition of the Shiryaev’s textbook [110],cited by Neftci [80, p. 556, Theorem] with a typo: Springer’s year is 1984 butnot 1985. The primary source defines "Markov time" [110, p. 469, Definition 3,Russian 1979]. Who introduced the term "Markov time"? Which Markov?
Let us review two phrases. 1) Carl Boyer about "Bernoulli": "No familyin the history of mathematics has produced as many celebrated mathematiciansas did the Bernoulli family ..." [12, p. 415], [101]. 2) August Wilhelm vonHofmann about "Nikolay Nikolaevich Zinin": "If Zinin has nothing more thanto convert nitrobenzene into aniline, even then his name should be inscribed ingolden letters in the history of chemistry" [52]. Admirers of American indigoblue, 2,2’-Bis(2,3-dihydro-3-oxoindolyliden) C H N O , jeans are indebted toZinin for synthesis of benzeneamine C H N H . It is well known that Zininwas a private teacher of chemistry to young Alfred Nobel. It is less frequently35ited that Zinin had brilliant mathematical skills remarked by astronomer IvanMikhailovich Simonov and geometer Nikolay Ivanovich Lobachevsky. Zinin wastheir pupil at the mathematics branch of the philosophical department of theKazan University, 1830 - 1833. His graduation thesis "Perturbation Theory"written on "On perturbations of elliptic motions of planets", the topic suggestedby Simonov, was awarded by the golden medal [49, p. 25 - 33].Boyer’s phrase reminds about celebrities with non-unique names. AndreyAndreevich Markov, father (06/2/(14)/1856 - 07/20/1922), Vladimir AndreevichMarkov, younger brother of father (05/07(19)/1871 - 01/18/(30)/1897), AndreyAndreevich Markov, son of father (09/09(22)/1903 - 10/11/1979), and Alexan-der Alexandrovich Markov, related by profession (03/24/1937 - 10/23/1994) arefirst class mathematicians. Neglect of history is the road to misunderstanding: "Markov chains" honors the father and "Markov algorithms" is about the son’scontribution [66]. To emphasize achievements of Markov father, the author re-phrases Hofmann’s words and believes that chemist Zinin, being also the firstclass mathematician, would agree: "If Markov [father] has nothing more than tocreate chains named after him, even then his name should be inscribed in goldenletters in the history of mathematics" .Markov: "In my opinion, the cases of variables linked into a chain so thatwhen the value of one of them becomes known, subsequent variables appear in-dependent on the variables preceding it, deserve attention" [74, p. 365, VS’stranslation]. Interesting facts about Markov are in Oscar Borisovich Sheynin’s[106], [107]. From "Theory of Probability. An Elementary Treatise against aHistorical Background" (English and Russian manuscripts kindly provided byO.B.S. to the author in an email), the author has known: "’Markov chain’first appeared (in French) in 1926 (Bernstein 1926, first line of § 16)" , [108].1926 and 1927 are the years of submission and publication of [9, p. 40], Figure8. Reading Sheynin’s manuscripts, the author was thinking about losses of themathematical community following from the fact that they are unpublished.
These are
Markov chains - not times . Howard Taylor III applies "Markovtimes" since 1968 [118, p. 1333 "Markov time or stopping time" ] but not in 1965[119]. The former article cites Dynkin’s [30]. The 1965’s paper has no referencesto Russian works. Dynkin says "Markov moment" in [33, p. 150]. This isnot a "moment of distribution" routinely occurring in statistical literature butsynonym of "time" , best expressed by English nouns "time" or "instant" . Inthe monograph [32, p. 54], Dynkin and Yushkevich write "Markov moment"with the meaning of "Markov time".Three independent significant works of 1963 on stopping times are [30], [109],[19]. The latter two have no the words "Markov times" but Dynkin’s paper istranslated in English using the "Markov instant". "Doklady ..." received it onDecember 12, 1962. His monograph [31, p. 142] with Preface dated by March31, 1962 defines a random variable independent on the future and names it"Markov moment" with English synonyms "Market instant", "Market time".While the fundamental properties of the Markov and strong Markov processesdescribed in the latter monograph were presented by Dynkin earlier [27], [28],[29], the author did not find in there the words "Markov moment", "Markov36igure 8: University Library. University of Illinois at Urbana-Champaign. Aphotocopy of the fragment of page 40 of Sergei Natanovich Bernstein’s paper[9] (LEXILOGOS’s translation ): "I have dwelt quite a long time on the study of the chainsof A. Markoff, not only because they provide a rather rare example, where theexplicit formulas which intervene in the calculations are not very complicatedand thus allow better to glimpse the nature of things, but also because I believethat there are several real phenomena that can be interpreted mathematically, byintroducing directly or indirectly chains similar to those we have just studied" .instant", "Markov time". This investigation suggests that it was Dynkin whofirst published the name "Markov time" in the paper [30] and monograph [31]in 1963, Figure 9.
Figure 9: A photocopy of the fragment of page 142 of Eugene BorisovichDynkin’s monograph [31,
Strictly Markov Processes ] (VS’s translation): "Let X = ( x t , ζ, M t , P x ) - Markov process. A real valued function τ ( ω ) ( ω ∈ Ω) weshall call Markov moment (or random variable, independent on future), if ..." "Markov time"? An inventor has the right to name his or her invention.Markov chains and times imply commonality due to the probability spaces setupbased on the theories of sets and measure. Accents differ. The Markov propertyof chains associate with conditional probabilities of future events dependent onthe current realized state and independent on the past. The Markov timesassociate with the past and current events, which can be determined from theinformation already available. Neftci considers the latter property essential forthe signals evaluation. The only reason, why current I t but not past informationis needed, is that I t includes the past information.Let us review the mature "Let an expanding sequence of σ -algebras F ⊆ F ⊆ · · · ⊆ F n ⊆ . . . isgiven. Let us name the random variable τ , taking non-negative integer valuesand the value + ∞ , Markov moment, if for any finite n { τ = n } ∈ F n . If, inaddition, τ < + ∞ with probability , then let us speak that τ - stopping moment.Let X n - a random variable, measurable relative to F n . The stopping time τ isnamed optimal, if the value M X τ is maximal". Here, the Russian "moment"is a synonym of "time", "instant". This definition implies a set of elementaryevents and its nesting sigma algebras forming expanding, due to F , measurablespaces. It adds measures - probabilities. These are traditional expanding prob-ability spaces. Is there something in this definition binding τ to the Markovchains except the common setup? The author does not see it. Imagine a study that highlights interesting moments in Markov processes.Naming them "Markov times" is reasonable, if other
X processes are also inscope. Then, "Markov times", "X times" indicate that the instants are fromdifferent processes. Dynkin’s definition has a wider sense but the word "Markov"associates with Bernstein’s "Markov chains" and narrows it.
It is Dynkin time .Neftci selects an existing wider concept with a narrower name to extractinstants in price time series independent on the future.
Contribution of SalihNeftci [80] is in that, he has suggested to apply Markov times, defined to distin-guish the class of strictly Markov processes in works of Dynkin and Yushkevich[27], [28] and used by the modern theory of stochastic processes [111], to formal-ize Technical Analysis widely exploited by traders [79] and ignored [73] or studied[14] by academicians.
In this paper, the words
Dynkin-Neftci time are chosen toavoid an impression that prices are considered a priori as Markov chains. Thetime helps to detect trading patterns of Technical Analysis algorithmically bycomputers and statistically estimate their significance or uselessness.
Will suchstatistics continue in the future is assumptions: differential equations describea ballistic trajectory but an anti-missile system can change it unpredictably .While Markov chains are widely applied, Markov himself writes [74, p. 397]: "Our conclusions can be expanded also on the complex chains in which eachnumber is directly connected not with one but several preceding it numbers" . Letus notice that Dynkin times can be used within such a framework and likelyfor non-Markov processes. The assumption about dependence of a next stateonly on the last one undoubtedly simplifies simulations.
However, a statementaxiomatically and a priori postulating independence of the future on the past canbe a speculation influencing on proper understanding markets.
What may happen, if a model applies non-Dynkin-Neftci times?
Sometrading simulators get delayed quotes . They are useful for training as long as astudent has no access to non-delayed ticks. "Looking in the future" 10 minutesahead makes liquid S&P 500 E-mini futures a "boring money machine". Thehindsight, for which the simulator is not responsible, does not teach. Afterswitching back to a non-delayed mode, emotions return. Without mathematicsit is clear that information from the future creates arbitrage in time illustratedin
Back to the Future Part II, created by Robert Zemeckis, Bob Gale, 1989 .38here is a concept related to Dynkin-Neftci times. Discussing a filtration F i , a history of the stock until time i on the tree of prices states, [6, p. 32]defines "a previsible process ... on the same tree whose value at any given nodeat time-tick i is dependent only on the history up to one time-tick earlier, F i − " .It is considered imperative, that mathematics of pricing must eliminate the-oretical arbitrage. The same theorem, proving a necessary and sufficient con-dition of the absence of arbitrage, is in [55, pp. 7 - 9, Theorem 1.7], [26,p.4, Theorem]. This rationally completes otherwise insufficient stochastic pricemodels yielding a unique value of a derivative. Combining a strategy replicatingportfolio with the absence of arbitrage yields option values [55, p. 3].The insider trading of the frozen orange juice futures with tremendous profitsfor the main personages of the American comedy Trading Places, directed byJohn Landis, 1983 , is an unlawful arbitrage.
In the fiction , the "instrument ofrevenge" is a nearby April contract. Currently, the expiration months are Janu-ary F, March H, May K, July N, September U, November X. Since the eventswere developing during the holiday season in December, January or March con-tracts could be realistic. The wall clock is approaching 9:00:00 am - the open-ing. Currently, the market opens at 8:00:00 am. After opening at 102 centsper pound, prices move up: → → → → → → → → , also due to Duke brothers buying on a stolen but falsified cropreport. Winthorpe: "Now. Sell thirty April - one forty two!" or in anotherhearing "Now. Sell two hundred April at one forty two!" This triggers the opposite trend : → → → → → → → → → right before the orange crop TV report. "Ladies and gentlemen, theorange crop estimates for the next year." Silence. The report is bearish: thecold winter is not apparently affecting the orange harvest. The real panic is: → → → → → → → → last. One contract is 15,000pounds: one point is $150. The move is (142 − ∗ $150 = $16 , . At the end,Winthorpe is busy closing the short positions and three times saying "hundred".300 contracts could profit $5,085,000 before commissions, taxes, and in 1983. Itis worth noticing that during 1979 - 1983 there were no such low frozen orangejuice prices. In real life, it would attract attention of the existing since 1974Commodity Futures Trading Commission, CFTC.
While the no-arbitrage theoretically links stochastic price processes withunique options values, the author believes that for trading futures with highleverage and large positions, a model accurately simulating discrete prices ismore practical than the condition of no-arbitrage needed to rationally price de-rivatives. After reviewing this section, the author’s younger son-student Dmitrihas "invented" the joke: "To be a trader, one does not have to be successful" . Optimal trading elements, OTE.
Between entering and exiting the mar-ket, MPS0 reverses long to short positions [93, pp. 25 - 26, Property 4]. SeveralMPS0 may generate the same
P L . For constant C , times of reversal transac-tions and some local price minimums and maximums coincide. The net action (cid:80) i = ni =1 U i,MP S = 0 , MPS0 ∈ U . The time of the last transaction of MPS0 is39on-Dynkin-Neftci. It can change after arriving new information. This artificialtransaction marks P L to market. All transactions before the last one associatewith Dynkin-Neftci times and can be used as signals for building real tradingrules [94].
The later can lose money.
MPS0 with W = 1 creates optimal trades adjacent in time. [96, p. 39] definesthe optimal trading element : "a collective name for properties associated withan optimal trade returned by an MPS" . This does not limit the number ofproperties. The key is their association with the MPS0 optimal trades. Thelatter depend on the filtering cost F C . Perspective properties are [96, p. 39]:a) trade direction - a buy or sell to initiate the trade, b) profit - optimal tradesalways profit, c) duration - time length of the trade, d) number of ticks includingthe first and last transaction of the trade, e) volume - total market volume duringthe trade, f) empirical distribution of a-increments - waiting times betweenneighboring ticks, g) empirical distribution of b-increments - price incrementsbetween neighboring ticks, h) empirical distribution of price and/or volume.Once a MPS0 with a filtering cost as a tool is applied to a chain of ticks andOTE are evaluated, the next analytical step is computing statistics of OTE.Due to reversal properties, a buying OTE, BOTE, is followed by a selling OTE,SOTE, and vice versa. The mean b-increment (price increment) of a BOTE isalways positive. The mean b-increment of a SOTE is always negative.
OTE by example.
Figure 10 represents eight OTEs for filtering cost $100found in trading sessions on April 10, 2017 for ESM17, ESU17, and ESZ17. Thered up and blue down parallel lines shadow areas above time intervals of theoptimal trades. Table 2: ESM17, Session OTEs,
F C = $100, C = $4.68, W = 1 . t start P start t end P end ∆ t, s P L , $ Type1 2017-04-09 17:02:54 2350.75 2017-04-09 20:04:00 2359.00 10866 403.14 BOTE2 2017-04-09 20:04:00 2359.00 2017-04-10 05:03:55 2349.75 32395 453.14 SOTE3 2017-04-10 05:03:55 2349.75 2017-04-10 09:35:48 2363.25 16313 665.64 BOTE4 2017-04-10 09:35:48 2363.25 2017-04-10 11:14:41 2347.50 5933 778.14 SOTE5 2017-04-10 11:14:41 2347.50 2017-04-10 12:37:14 2360.00 4953 615.64 BOTE6 2017-04-10 12:37:14 2360.00 2017-04-10 13:04:45 2354.50 1651 265.64 SOTE7 2017-04-10 13:04:45 2354.50 2017-04-10 14:06:28 2360.25 3703 278.14 BOTE8 2017-04-10 14:06:28 2360.25 2017-04-10 15:00:07 2351.00 3219 453.14 SOTEThe eight profits and durations from Table 2 form two sample distributions PL distributionMean = 489.0775Samples size = 8 , transaction prices of ESM17, ESU17, andESZ17 for the time range [Sunday April 9, 2017, 17:00:00 - Monday April 10,2017, 15:15:00], CST. Plotted using custom C++ and Python programs andgnuplot . Maximum value = 778.14Maximum value count = 1 inimum value = 265.64Minimum value count = 1Variance = 33590.9598Std. deviation = 183.278367Skewness = 0.322282476Excess kurtosis = -1.395385240 (233.442, 311.256] 21 (311.256, 389.07] 02 (389.07, 466.884] 33 (466.884, 544.698] 04 (544.698, 622.512] 15 (622.512, 700.326] 16 (700.326, 778.14] 1Trade time distributionMean = 9879.125Samples size = 8Maximum value = 32395Maximum value count = 1Minimum value = 1651Minimum value count = 1Variance = 105625105Std. deviation = 10277.4075Skewness = 1.82711153Excess kurtosis = 1.686830950 (0, 3239.5] 21 (3239.5, 6479] 32 (6479, 9718.5] 03 (9718.5, 12958] 14 (12958, 16197.5] 05 (16197.5, 19437] 16 (19437, 22676.5] 07 (22676.5, 25916] 08 (25916, 29155.5] 09 (29155.5, 32395] 1 Start and birth times of OTE.
When the first tick arrives, nothing isknown with respect to the MPS0 and OTE, unless the previous trading sessionsare considered. The
OTE start time t OT Es remains unknown until the first pricearrived will mark a move exceeding F C at least by one δ ESM = 0 . . Thisevent, the OTE birth time t OT Eb , is in the future making t OT Es non-Dynkin-Neftci. The F C are counted from a local minimum or maximum price. t OT Eb is Dynkin-Neftci. Just only such a price drop or rise occurs, t OT Es is fixed.After this, the start time is Dynkin-Neftci but the current OTE end time t OT Ee coinciding with the next OTE start time are unknown - non-Dynkin-Neftci.42otice, all MPS0 start, birth, and end times prior the current just fixed starttime cannot change. They are Dynkin-Neftci times. Figure 11 is a zoom into Figure 10 for ESM17 SOTE , Table 2. Again, the end time can bedetermined only after arriving the birth time. | ∆ P | = × $100$50 + 0 .
25 = 4 . . The last maximum (our case) price is 2363.25observed at 09:35:48. Would the price go higher, it would become the newtrailing high. Subtracting | ∆ P | yields the birth price ( P S s =2363 . , t S s = 09 : 35 : 48; P S b = 2359 . , t S e = 09 : 59 : 13; P S e =? , t S e =?) .Figure 11: Time & Sales Globex, , ESM17, SundayApril 9, 2017, 17:00:00 - Monday April 10, 2017, 15:15:00. MPS0, F C = $100 ,transaction prices, SOTE .By definition, OTEs include arbitrary properties associated with optimalMPS trades. Figure 12 illustrates four properties of SOTE . Let us notice,that if price b-increments would be i.i.d log-normal (or normal), then distri-bution of prices would be the same but with a different mean and variance.The sample distribution density of prices has three maximums and does notcorrespond to a unimodal log-normal (or normal) distribution.43igure 12: Time & Sales Globex, , ESM17, SundayApril 9, 2017, 17:00:00 - Monday April 10, 2017, 15:15:00. MPS0,
F C = $100 ,sample distributions of a-increments, b-increments, prices, and volumes, SOTE . Three scenarios for OTE.
Once a new OTE is born, three exclusive scen-arios exist: 1) the profit of the current OTE will grow at least by one δ , absoluteminimal price fluctuation ; 2) a next opposite type OTE will replace the current;3) the trading session will terminate. For a non-intraday trader, the chains ofprices and OTEs are continued. Technically, one can apply MPS, as a tool,using any chain of prices such as daily last prices.
Figure 11 illustrates the first scenario for SOTE4, S , from Table 2. Afterthe birth time t b = .
25 = 17 δ ESM points up on the right of the last lowest price until the B birth price.Table 3 collects basic OTE properties after switching to F C = $74 . .ESM17 prices are the same. S on Figures 13, 14 presents the second scenario.The influence of F C under other equal conditions on the number of OTEs, theirdurations and
P L distributions, spectra is studied in [100], [103].44able 3: ESM17, Session OTEs,
F C = $74.99, C = $4.68, W = 1 . t start P start t end P end ∆ t, s P L , $ Type1 2017-04-09 17:02:54 2350.75 2017-04-09 20:04:00 2359.00 10866 403.14 BOTE2 2017-04-09 20:04:00 2359.00 2017-04-10 05:03:55 2349.75 32395 453.14 SOTE3 2017-04-10 05:03:55 2349.75 2017-04-10 06:37:43 2355.50 5628 278.14 BOTE4 2017-04-10 06:37:43 2355.50 2017-04-10 07:59:57 2352.50 4934 140.64 SOTE5 2017-04-10 07:59:57 2352.50 2017-04-10 09:35:48 2363.25 5751 528.14 BOTE6 2017-04-10 09:35:48 2363.25 2017-04-10 10:42:06 2350.75 3978 615.64 SOTE7 2017-04-10 10:42:06 2350.75 2017-04-10 10:53:17 2354.00 671 153.14 BOTE8 2017-04-10 10:53:17 2354.00 2017-04-10 11:14:41 2347.50 1284 315.64 SOTE9 2017-04-10 11:14:41 2347.50 2017-04-10 12:37:14 2360.00 4953 615.64 BOTE10 2017-04-10 12:37:14 2360.00 2017-04-10 13:04:45 2354.50 1651 265.64 SOTE11 2017-04-10 13:04:45 2354.50 2017-04-10 13:15:52 2358.25 667 178.14 BOTE12 2017-04-10 13:15:52 2358.25 2017-04-10 13:40:14 2354.50 1462 178.14 SOTE13 2017-04-10 13:40:14 2354.50 2017-04-10 14:06:28 2360.25 1574 278.14 BOTE14 2017-04-10 14:06:28 2360.25 2017-04-10 15:00:07 2351.00 3219 453.14 SOTE15 2017-04-10 15:00:07 2351.00 2017-04-10 15:02:13 2354.25 126 153.14 BOTE16 2017-04-10 15:02:13 2354.25 2017-04-10 15:14:30 2351.25 737 140.64 SOTEFigure 13: MPS0, Filtering cost $74.99, E-mini S&P 500 Futures Time & SalesGlobex, , transaction prices of ESM17 for the timerange [Sunday April 9, 2017, 17:00:00 - Monday April 10, 2017, 15:15:00], CST.Plotted using custom C++ and Python programs and gnuplot .In the second scenario, the price does not go a δ in the profit direction ofthe OTE type. Selling short S at the OTE birth price 2352.50, 07:59:57 and45igure 14: Time & Sales Globex, , ESM17, SundayApril 9, 2017, 17:00:00 - Monday April 10, 2017, 15:15:00. MPS0, F C = $74 . ,transaction prices, SOTE .buying at the next B birth price 2355.50, 08:31:51 loses ( − . . × $50 − $9 .
36 = − $159 . . A simple trading rule - enter/exit and revert positionby buying BOTE and selling SOTE at the OTE birth price - resembles the Alexander’s filter , see details in [100, p. 71, pp. 92 - 93]. Applying it to ( S , B ,with the B ’s profit ([2363 . − . − . × $50 − × $4 .
68 = $228 . , wouldcompensate the S ’s loss − $159 . and be profitable $228 . − $159 .
36 = $68 . ,Figure 14. This rule can start losing, if the second scenario continues in a chainof consecutive OTEs. Empirical distributions of the OTE P L [97, p. 27, FigureProfit Frequencies], [100, p. 95, Figure 35] help estimating the mean
P L . It wasfound negative in a range of
F C and C = $4 . . A mathematical expectationof P L can be not the only criterion influencing on trading decisions [102].The third scenario implies that the current OTE did not get any developmentrelative to
F C with regard to the profit given the time prior the trading sessionsis closed. The SOTE , ESM17, SundayApril 9, 2017, 17:00:00 - Monday April 10, 2017, 15:15:00. MPS0,
F C = $74 . ,transaction prices, after SOTE . The maximum loss strategy, MLS.
Such a strategy could be a risk estim-ator for given P and C . Here, we only mention that U MLS ∈ U is not necessarily − U MP S ∈ U . For instance, for P = ( P, . . . , P ) T , P > , C = ( C, . . . , C ) T , C > , U MP S = U d.n.s. , losing nothing, while many other strategies losemore due to transactions costs. From section "Strategies generating extremeindustry gains", it follows that under such conditions, for even < n , U MLS =( W, − W, . . . , W, − W ) T with the loss − CW ( n − (valid for any < n ).Another and only candidate would be − U MLS = ( − W, W, . . . , − W, W ) T . MPS studies, 2007 - 2017.
Potential profit, as a number computed withoutaccounting transaction costs, and its simple, under this condition, algorithmwere suggested by Robert Pardo [83, pp. 125 - 126]. His words "the measurementof the potential profit that a market offers is not a widely understood idea" attracted the author, who thought that transaction costs would complicate thealgorithm, yield a rich concept of the maximum profit strategy, vector, expandresearch and application horizons: MPS is another face of the same market.47he unpublished formulation of MPS and related algorithms were developed bythe author in 1994. They were not applied or widely reported in that time andeventually have been followed by the studies and publications listed below.1) The l- and r-algorithms for MPS0, and algorithms for the first MPS1 andsecond MPS2 P&L reserve strategies [93], [100, p. 203, Appendix F, AMinor Correction].2) Fist published evaluations of MPS0, MPS1, MPS2 [93, Chapter 7]. UsingMPS0 as a performance benchmark [93, pp. 151 - 152], for comparingdifferent intervals of a single market [93, p. 152], for comparing differentmarkets [93, p. 153, Chapter 10], as moving indicators [93, p. 153].3) First published statistics of MPS0 optimal trades [93, Chapter 9].4) Proposal to apply MPS0 for options on potential profit [93, p. 155].5) Proposal to consider MSP0 as a quantitative alternative to not well definedtrend and volatility [93, pp. 153 - 154], [96, Alternative analysis].6) Using MPS0 for filtering events [93, pp. 154 - 155], and defining tradingsignals of real trading rules and strategies [94].7) Introducing the a-b-c-increments classification [95]. Statistical studies ofthe a-b-c-increments including MPS0 optimal trades [95], [96], [97], [100],[103]. Chain reactions [100, pp. 101 - 105], [103, pp. 52 - 56, Extremeb-increments].8) Proposal of new notation for iteration of functions and iteral of functionsto describe a-b-c-processes [98], [99].9) Introducing OTE, BOTE, SOTE [96]. Studying OTE statistics and ex-pectations for trading based on the OTE birth time and price [96], [97],[100, p. 95, Figure 35].10) New (not in [93], [94], [95], [96]) discrete, spectral MPS0 properties andsoftware framework [100], [103]. Relationships between MPS0 and tradingvolume [103, pp. 26 - 30, Figures 18 - 23]. Mathematical expectations asnot the only reason for trading decisions [101], [102, p. 12, Livermoreabout the hope and fear, pp. 28 - 31, The role of time. Trading andspeculation].
10 Patterns
Strategies vs. rules. "Trading strategies" have two meanings: 1) records oftrading actions like chains - vectors of U , and 2) reasons causing the actions.Here, the latter are named "trading rules". Automation of trading depends onthe formalization of rules. Formalization of a rule is valuable, if it yields an48lgorithm or program, which can be evaluated by a human being or computer.For this, it can rely on Dynkin-Neftci times distinguishing events, which can bedetermined without looking to the future.Given available information required by a rule, the latter is evaluated and aconclusion is made whether an event takes place. The number of events withina time interval, associates with the frequency of the market offers connected tothe rule. The next is to estimate what may happen after the event. This es-timation can be subjective or imply a preliminary objective statistical research,a search of dependencies between an event and market follow up, if any . Sucha research assumes evaluation of the past up to day information and may havemeaning, if the future repeats the past in something. Finding this somethingis one of the goals. Due to non-stationary market conditions, once determineduseful regularities may stop working and trigger a new research. The questiondiscussed in the last section, especially logical if prices are totally unpredictable,is why do speculative markets exist ?MPS0 indicates local price minimums and maximums on a given time inter-val. The last extreme depends on the future, non-Dynkin-Neftci time. Otherare Dynkin-Neftci times. A MPS0 builds a chain of OTEs such as on Figure10 ( B , S , B , S , B , S , B , S . An OTE is characterized by the start andend OTE times and prices t s , P s , t e , P e . Its birth time and price t b , P b arepractically important being deal with a Dynkin-Neftci time. How can MPS0 and OTE define patterns?
Let us consider a hypotheticalchain ( B , S , B , S , B , S . By the MPS0 properties, P B e = P S s , P S e = P B s , P B e = P S s , P S e = P B s , P B e = P S s . Since S is current, its birth price P S b < P S s is realized and P S s − P S b ≥ F C + δ , where F C is converted to fullpoints. The known head and shoulders pattern can be defined by the condition: ( P B s < P B s )&&( P B s == P B s )&&( P B e < P B e )&&( P B e < P B e )&&( P == P B b ) . The logical equality == and less < comparisons assume tolerances. Thesetolerances together with F C are the pattern optimization parameters.
Algorithmic optimization.
Since B , . . . , B , P S s , and P S b are fixed, thelogical expression in the previous paragraph requires only a one time evaluationof all individual comparisons but ( P == P B b ) , which must be monitored. Arecognition of this pattern can be efficient and does not require reevaluation allsix OTEs for each arriving price P . Interval [ t s , t b ] . In contrast with theories of continuous prices, real prices arediscrete [100, pp. 32 - 33], [103, pp. 3 - 10]. In particular, futures prices areproducts of natural numbers and δ . For constant F C , the price change to mon-itor is expressed in δ as | P OT Es − P OT Eb | = | δN OT Es − δN OT Eb | = δ | ∆ N OT E | > F Ck and N OT Es,b = (cid:98) F Cδk (cid:99) + 1 . Example, (cid:98) × $1000 . × $50 (cid:99) + 1 = 17 deltas or 4.25 fullpoints; (cid:98) × $74 . . × $50 (cid:99) + 1 = 12 deltas or 3 full points. Notice, that for F C = 0 the formula returns 1: MPS extracts only profitable but not break even optimaltrades. 49t is possible that due to a price gap the price will jump over P OT Eb . Thisstill indicates that the new OTE is born and actual price P OT Es is fixed in theprice chain. The time of the gapped price is the born time t OT Eb .Now, the current OTE segment [ t s , t b ] is fixed. The [ t b , t current ] is developingfollowing one of the three scenarios. The l-, r-algorithms routinely extract allprevious times t s and t b and allow to study what happens at these times andin the past adjacent intervals [ t s , t b ] and [ t b , t e ] or entire OTE intervals [ t s , t e ] .The next paragraph is an example of a study. Empirical Cumulative Distribution Functions, ECDF, of OTE profits.
Once t b is detected, the profit of [ t s , t b ] cannot be realized since it is in the past.It is interesting how far the price can move in the profit direction after t b .In other words, with fixed C = $4 . and F C how many OTEs exceed thisinterval by δ, δ, δ, . . . etc. Figure 16 depicts ECDF of OTE profits dependingFigure 16: Time & Sales Globex, , ESZ17,Wednesday January 4, 2017 - Friday December 8, 2017, 227 trading ses-sions with the time range 17:00:00 (previous day) - 15:15:00 (closing day),6,241,260 transaction ticks, C = $4 . . MPS0 OTE ECDF for F C =6 . , . , . , . , . , . , . , . , . , . dollars. Plot-ted using custom C++, Python, AWK programs, and Microsoft Excel.on F C . The results are for individual trading sessions assuming that a traderdoes not leave open positions for next sessions. The ranges 15:30:00 - 16:00:00are ignored. Table 4 summarizes sample statistics.50able 4: ESZ17 OTE Profit Sample Statistics, C = $4 . . F C N
Mean Min N Min
Max N Max
StDev Skew. E-Kurt.6.24 860375 5.41 3.14 751179 1340.64 1 10.0 20.8 -0.8212.49 69715 31.15 15.64 36654 1340.64 1 32.3 7.14 10.524.99 12342 92.88 40.64 2601 1340.64 1 68.5 3.51 25.237.49 5816 146.40 65.64 793 1665.64 1 96.5 3.38 25.449.99 3542 195.03 90.64 439 1665.64 1 119.9 2.88 16.374.99 1749 288.59 140.64 130 1790.64 1 156.6 2.43 11.499.99 1034 379.66 190.64 68 1790.64 1 198.4 2.18 7.03124.99 734 450.82 240.64 52 1815.64 1 223.2 2.07 6.00149.99 534 524.11 290.64 31 1828.14 1 243.5 1.94 5.05199.99 317 658.23 390.64 11 2090.64 1 280.7 1.95 4.61Let us review Table 4 and
F C = $49 . as an example. Buying BOTE orselling SOTE at t b makes the part of the OTE profit × $49 .
99 = $99 . unavailable. At the first glance, this is attractive because the mean OTE profitis $195 . and the difference $195 . − $99 .
98 = $95 . is positive. However,"on the way back to a next t b " additional $99 . has to be subtracted creatingthe mean loss $95 . − $99 .
98 = − $4 . . At the same time, taking the OTEmean profit always at a corresponding price ignores less frequent but moreprofitable offers. In addition, not all OTE for a given F C reach the mean OTEprofit. These factors will negatively influence on the mean
P L results. For oneinterested in details, Table 5 contains empirical profits and corresponding massand cumulative frequencies for the curve
F C = $49 . on Figure 16.Table 5: ESZ17 OTE EPMF, ECDF, C = $4 . , F C = $49 . . i Profit, $ N i (cid:80) j = ij =1 N j N i (cid:80) j =66 j =1 N j (cid:80) j = ij =1 N j (cid:80) j =66 j =1 N j i Profit, $ N i (cid:80) j = ij =1 N j N i (cid:80) j =66 j =1 N j (cid:80) j = ij =1 N j (cid:80) j =66 j =1 N j
13 240.64 107 2752 0.030 0.77714 253.14 67 2819 0.019 0.79615 265.64 86 2905 0.024 0.82016 278.14 61 2966 0.017 0.83717 290.64 63 3029 0.018 0.85518 303.14 53 3082 0.015 0.87019 315.64 52 3134 0.015 0.88520 328.14 36 3170 0.010 0.89521 340.64 40 3210 0.011 0.90622 353.14 35 3245 0.010 0.91623 365.64 30 3275 0.008 0.92524 378.14 26 3301 0.007 0.93225 390.64 18 3319 0.005 0.93726 403.14 27 3346 0.008 0.94527 415.64 19 3365 0.005 0.95028 428.14 16 3381 0.005 0.95529 440.64 19 3400 0.005 0.96030 453.14 10 3410 0.003 0.96331 465.64 18 3428 0.005 0.96832 478.14 9 3437 0.003 0.97033 490.64 11 3448 0.003 0.97334 503.14 9 3457 0.003 0.97635 515.64 10 3467 0.003 0.97936 528.14 6 3473 0.002 0.98137 540.64 9 3482 0.003 0.98338 553.14 7 3489 0.002 0.98539 565.64 3 3492 0.001 0.98640 578.14 4 3496 0.001 0.98741 590.64 3 3499 0.001 0.98842 603.14 1 3500 0.000 0.98843 615.64 3 3503 0.001 0.98944 628.14 1 3504 0.000 0.98945 640.64 3 3507 0.001 0.99046 653.14 5 3512 0.001 0.99247 665.64 3 3515 0.001 0.99248 678.14 3 3518 0.001 0.99349 690.64 1 3519 0.000 0.99450 715.64 2 3521 0.001 0.99451 728.14 2 3523 0.001 0.99552 740.64 3 3526 0.001 0.99553 765.64 2 3528 0.001 0.99654 778.14 1 3529 0.000 0.996Continued on next page52able 5 – continued from previous page i Profit, $ N i (cid:80) j = ij =1 N j N i (cid:80) j =66 j =1 N j (cid:80) j = ij =1 N j (cid:80) j =66 j =1 N j
55 803.14 1 3530 0.000 0.99756 815.64 1 3531 0.000 0.99757 840.64 1 3532 0.000 0.99758 853.14 2 3534 0.001 0.99859 878.14 1 3535 0.000 0.99860 903.14 1 3536 0.000 0.99861 965.64 1 3537 0.000 0.99962 1028.14 1 3538 0.000 0.99963 1140.64 1 3539 0.000 0.99964 1315.64 1 3540 0.000 0.99965 1340.64 1 3541 0.000 1.00066 1665.64 1 3542 0.000 1.000We say an "empirical probability mass function", EPMF, of the OTE profits:with fixed
C < F C the profits are discrete due to discrete prices P i = N i δ ."Permitted" profits are [100, p. 93 Formulas 55]: P L
OT Emin = kδ ( (cid:98) F Ckδ (cid:99) +1) − C ; P L
OT Ei = P L
OT Emin + kδi , i = 0 , , . . . . Indeed, Equation 1 is for U with W = 0 but not only U ∈ U with W n = 0 . If W n (cid:54) = 0 , then the position is marked tothe market using P n . This allows to rewrite Equation 1 for growing jP L j = kδ i = j (cid:88) i =1 U i ( N j − N i ) − i = j (cid:88) i =1 C i | U i | − C n | i = j (cid:88) i =1 U i | , j = 1 , . . . , n. (17)For n = j = i = 1 , it returns − C | U | and for U (cid:54) = 0 the fees are paid attaking the position and due to marking-to-market. For an MPS0 OTE [ t s , t e ] , U i = 0 for s < i < e , | U i = s | = | U i = e | = W , U s = − U e and P L
OT Es,e = kδW | N e − N s | − W ( C s + C e ) . (18)With C s = C e = C < F C , W = 1 , and | N e − N s | ≥ (cid:98) F Ckδ (cid:99) + 1 , we always canexpress | N e − N s | = (cid:98) F Ckδ (cid:99) + 1 + i and come to the discrete P L
OT Ei .Table 6: ESZ17 BOTE and SOTE Profit Sample Statistics, C =$4 . , F C = $49 . .Type N Mean Min N Min
Max N Max
StDev Skew. E-Kurt.BOTE 1786 199.3 90.64 221 1340.64 1 120.5 2.52 11.1SOTE 1756 190.68 90.64 218 1665.64 1 119.2 3.26 22.1BOTH 3542 195.03 90.64 439 1665.64 1 119.9 2.88 16.353igure 17 illustrates BOTE and SOTE ECDFs. Deviations are small. Bothare close to ECDF on Figure 16 for
F C = $49 . . See statistics in Table 6.The author notices that after identical scaling of Figure 16 and Figure 35 fromFigure 17: Time & Sales Globex, , ESZ17, Wed-nesday January 4, 2017 - Friday December 8, 2017, 227 trading sessions withthe time range 17:00:00 (previous day) - 15:15:00 (closing day), 6,241,260 trans-action ticks, C = $4 . . MPS0 BOTE and SOTE ECDF for F C = $49 . .Plotted using custom C++, Python, AWK programs, and Microsoft Excel.[100, p. 95], the corresponding ECDF curves almost coincide. This indicatesthat empirical statistics of the OTE profits observed for ESZ17 in 227 sessionsin 2017 and ESZ13 in 184 sessions in 2013 are close.
11 Why do speculative markets exist?
A mortgage-backed security value may depend on dynamics of 360 monthlyinterest rates [21, pp. 199 - 209], [39, pp. 705 - 709, Exhibit 1], while themajority of them is well correlated [47, pp. 93 - 107, 3.2 Principal ComponentAnalysis], PCA. Theoretical throwing between an increasing number of randomfactors in a model and further attempts to reduce the dimension of its space bymeans of PCA resemble incessant fluctuations of prices laughing at the idea ofmarket equilibrium. Significant and repeated price fluctuations define marketopportunities. MPS is their objective measure [100, pp. 6 - 7].54 ne may like, hate, or ignore speculation but whether the price moves arerandom, chaotic, trendy, or not, substantial and recurrent market offers, whichcan be objectively accounted for, applying MPS, are an essential condition of theexistence of speculative markets and the thriving interest in them. Emphasizingthis quantitative role of MPS, the author understands that we may talk aboutan essential but insufficient condition of existence and that free markets andrelation to them assume a certain economic, political, and cultural society basis.
MPS and OTE express market states in terms of positions, actions, profitsunderstood by both traders and academicians. MPS explains trader’s aspiration.The "universe" of strategies is measured by the "galactic" number (2 W + 1) n − routinely reaching . Importance of MPS for the economy is in combiningtransactions costs, prices, and actions for measuring market offers up to thisday.
For instance, it would be interesting to see more evidences, if the ECDFsof OTE profits, expressed in one currency, Figures 16, 17, persist in years.Steven Strahler cites [116] Leo Melamed, the legendary founder of the finan-cial futures and Globex, about exchanges and electronic technology behind: "Wehave not yet gone into the galaxy, but we’re thinking about it" . After 11 years,Melamed, congratulating the Futures Magazine with the 500th issue, confirms[75]: "... the future of futures markets is limited only by our own imagination" .The scale of the MPS is suitable for this journey.
Acknowledgments.
I am grateful to Oscar Sheynin for sharing [108] andvaluable opinions on the history of probability theory and statistics.
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New Scientist, Daily News , July18, 2017, .[89] Rogers, L. Chris G., Williams, David.
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Commentationes Mathematicae Universitatis Carolinae ,Volume 41, No. 2, 2000, pp. 377 - 400.[93] Salov, Valerii, V.
Modeling Maximum Trading Profits with C++: NewTrading and Money Management Concepts . Hoboken, NJ: John Wiley &Sons, Inc., 2007.[94] Salov, Valerii, V. Idealized models for real profits.
Futures Magazine ,Volume XXXVII, No. 5, May 2008, pp. 36 - 39.[95] Salov, Valerii, V. Market Profile and the distribution of price.
FuturesMagazine , Vol. XL, No. 6, June 2011, pp. 34 - 36.[96] Salov, Valerii, V. Trading system analysis: Learning from perfection.
Fu-tures Magazine , Vol. XL, No. 11, November, 2011, pp. 34 - 39, 43.[97] Salov, Valerii, V. High-frequency trading in live cattle futures.
FuturesMagazine , Vol. XL, No. 6, May 2012, pp. 26 - 27, 31.[98] Salov, Valerii, V. Notation for Iteration of Functions, Iteral.
Arxiv, Math-ematics, Dynamical Systems , June 30, 2012, pp. 1 - 23, available at http://arxiv.org/abs/1207.0152 [99] Salov, Valerii, V. Inevitable Dottie Number. Iterals of cosine and sine.
Arxiv, Quantitative Finance, General Finance , December 1, 2012, pp. 1 -17, available at http://arxiv.org/abs/1212.1027
Arxiv, Quantitative Finance, General Finance , December 6,2013, pp. 1 - 222, http://arxiv.org/abs/1312.2004 .[101] Salov, Valerii. "The Gibbon of Math History". Who Invented the St.Petersburg Paradox? Khinchin’s resolution.
Arxiv, Mathematics, Historyand Overview , March 11, 2014, pp. 1 - 17, http://arxiv-web3.library.cornell.edu/abs/1403.3001 .[102] Salov, Valerii. The Role of Time in Making Risky Decisions and the Func-tion of Choice.
Arxiv, Quantitative Finance, General Finance , December 27,2015, pp. 1 - 52, http://arxiv.org/abs/1512.08792 .[103] Salov, Valerii. The Wandering of Corn.
Arxiv, Quantitative Finance, Gen-eral Finance , April 3, 2017, pp. 1 - 65, https://arxiv.org/abs/1704.01179?context=q-fin .[104] Schafer, R., D.
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Theory of Probability. An Elementary Treatise againsta Historical Background . Unpublished manuscript the English and Russianversions of which were provided by O.B.S. in a private email.[109] Shiryaev, Albert, N. On Optimum Methods in Quickest Detection Prob-lems.
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Valerii Salov received his M.S. from the Moscow State University, Departmentof Chemistry in 1982 and his Ph.D. from the Academy of Sciences of the USSR,Vernadski Institute of Geochemistry and Analytical Chemistry in 1987. He isthe author of the articles on analytical, computational, and physical chemistry,the book Modeling Maximum Trading Profits with C++,
John Wiley and Sons,Inc., Hoboken, New Jersey , 2007, and papers in
Futures Magazine and
ArXiv . [email protected]@comcast.net