Buy Signal from Limit Theorem Yiqiao YIN Statistics Department Columbia University November 2, 2017 Abstract This paper studies price actions in capi- tal market as a random walk from limit theo- rems. Through clear construction, we derive algorithms from a series of theorems to cre- ate standardized buy signals given a trader’s committed frequency to participate in the market. 1 Introduction Security prices follow random walk. Although some scholars doubt the concept of efficient market hypoth- esis, there are fruitful amount of previous research ex- ploring and studying this topic. Some notable papers are by Fama and French [2], [3], [4], and [5]. Other scholars such as Malkiel have also provided persua- sive empirical evidence that we do observe data in favor of efficient market hypothesis [6]. An important contribution from Yin (2017) [7] was the idea and theoretical notion of optimal level in se- curity prices. Their work raised a concept that the anomaly prices can be corrected which was a notion not yet discovered in the field of probabilistic price analysis. We took this notion as a foundation and further explore this field of security price random walk. We discovered that a series of constructions can be built to form standard normal distributions. We will further prove these results (see Appendix) and develop a series of trade-able signals from these theories. The hunger for this type of work is necessary for the industry because conventional asset pricing mod- els do not signal buyers when to involve in the mar- ket. Moreover, for retail traders with a fixed trad- ing frequency (assuming rational retail traders), it only makes sense for them to participate in the “low” prices consistent with their frequency. There is cur- rently no models presenting us any algorithms in that sort. This motivates us to formalize these algorithms from theorems we developed and we aim to provide traders a consistent buy strategy so that one can trade, whatever strategy one trades, at a systemati- cally low price. 2 Theoretical Framework In this section, we first present, in §2.1, the architec- ture of the theorems, which are discussed in §2.2. We formalize the theorems based on the notion of Central Limit Theorems, which are proved in respect to the order of theorems in §5 Appendix. Continuing from the proved theorems, we provide the construction of a series of algorithms in §3. 2.1 Architecture Definition 2.1.1. For each company i at a time t, we observe a price, that is, pi,t (2.1.1) Definition 2.1.2. SMAn = 1 n t−n∑ i=n pi,t−n (2.1.2) Definition 2.1.3. Let n be the same value from Def- inition 2.1.2, denote EMAn = (pi,t − EMAn−1)×m+ EMAn−1 (2.1.3) while m = 2 n+1 . 2.2 Theories Theorem 2.2.1. For some n, suppose we have price by Definition 2.1.1 and SMA by Definition 2.1.2, then we have pi,n − SMAn ⇒ χ (2.2.1) while χ is the stand normal distribution. Theorem 2.2.2. Let the distance between price and moving average to be D which is defined as Di := pn − SMAn while i = n, and then we can consider Di to be i.i.d. with EDi = 0 and EDi = σ2 ∈ (0,∞). Then n∑ m=1 Dm /( n∑ m=1 D2 m )1/2 ⇒ χ (2.2.2) while χ is the stand normal distribution. Theorem 2.2.3. Let the distance between price and moving average to be D which is defined as Di := pn − SMAn while i = n, and then we can consider Di to be i.i.d. with EDi = 0 and EDi = σ2 ∈ (0,∞). Let Sn = D1 + · · · + Dn. Let Nn be a sequence of non- negative integer-valued random variables and cn a se- quence of integers with cn → ∞ and Nn/cn → 1 in probability. Then SNn /σ √ an (2.2.3) where χ is a standard normal distribution. 1 Buy Signal from Limit Theorem [by Yiqiao Yin] §4 Theorem 2.2.4. Let the distance between price and moving average to be D which is defined as Di := pn − SMAn while i = n, and then we can consider Di to be i.i.d. with EDi = 0 and EDi = σ2 ∈ (0,∞). Let Sn = D1 + · · · + Dn. Let Nt = sup{m : Sm ≤ t}. Then as t→∞, (µNt − t)/(σ2t/µ)1/2 ⇒ χ (2.2.4) while χ is the stand normal distribution. 3 Algorithms This section we take the theorems above, from §2 Theoretical Framework, as given and we introduce a series of algorithms targeting buy signals. Algorithm 3.0.1. Given a buy frequency by an in- vestor c, for all i in a stock pool of companies: Step 1. Observe price pt for each company Step 2. Store pi,t Step 3. Compute SMAn Dn := pi,n − SMAn Step 4. If Dn ≤ c, print ”+1”; else, print ”0”. Print a collection of ”+1” per company i per n. As the first algorithm in the section, it has a very intuitive understanding. One can simply ob- serve price and computes its SMA. Then one needs to look at the difference between price and SMA to know how often should he buy given that he has a fixed frequency. This is and will always be true be- cause the difference of price and SMA follows random walk, as stated in Theorem 2.2.2 and proved in Ap- pendix. This means that this time-series difference we are looking at goes up or down but stay in the middle most often. Such bell-shape curve can give as a precise probability distribution and we can mark down an exact price level given a frequency we want to participate in the market. Algorithm 3.0.2. Given a buy frequency buy an investor c, for all i in a stock pool of companies: Step 1. Observe price pt for each company Step 2. Store pi,t Step 3. Compute Dn := pi,n − SMAn Signaln := n∑ m=1 Dm /( n∑ m=1 D2 m )1/2 Step 4. If Signaln ≤ c, print ”+1”; else, print ”0”. Print a collection of ”+1” per company i per n. Algorithm 3.0.2 takes Algorithm 3.0.1 as a build- ing block and expand the idea and we can normalized the distance (or difference) of summation of a series of distances by square root of its own value to construct buy signals. Algorithm 3.0.3. Given a buy frequency by an in- vestor c, for all i in a stock pool of companies: Step 1. Observe price pt for each company Step 2. Store pi,t Step 3. Compute SNn = D1 + · · · + Dn, σ is the variance of Di, Self-Normn := SNn/σ √ an Step 4. If Self −Normn ≤ c, print ”+1”; else, print ”0”. Print a collection of ”+1” per company i per n. Algorithm 3.0.2 brought up the notion of normal- izing by square of its own value. It is also practical to normalize by itself, which is what Algorithm 3.0.3 was attempting to do. Algorithm 3.0.4. Given a buy frequency buy an investor c, for all i in a stock pool of companies: Step 1. Observe price pt for each company Step 2. Store pi,t Step 3. Compute the mean µ and the variance σ, Renewaln := (µNt − t)/(σ2t/µ)1/2 ⇒ χ Step 4. If Renewaln ≤ c, print ”+1”; else, print ”0”. Print a collection of ”+1” per company i per n. Besides notions of self-normalizing, we can also construct standard normal distribution by taking time, risk, and mean into consideration. Such “re- newal” process can be done without breaking the form of standard normal distribution. For traders who are interested in looking at more parameters, there is such freedom to do so. Algorithm 3.0.5. Given results from the above al- gorithms, that is, Algorithms 3.0.1, 3.0.2, 3.0.3, and 3.0.4, run Step 1. Retreat (Dn), (Signaln), (Self-Normn), and (Renewaln). Step 2. Each i, at any time t, compute Buy : ∑ all signals := val(Dn)+val(Signaln) +val(Self −Normn) +val(Renewaln) Step 3. print(t); print(buy). That is, Buy =  Very Heavy, b = 4 Heavy, b = 3 Not that Heavy, b = 2 Tiny, b = 1 Do Nothing, b = 0 Page 2 Buy Signal from Limit Theorem [by Yiqiao Yin] §5 while b is a discrete time-series function of time t, i.e. b(t) : Z→ {0, 1, 2, 3, 4}. 4 Conclusion This paper starts with a strong motivation §1 and introduced the background of why we study security prices in such construction. Next, we present a clear architecture, in §2, and a series of theorems under such building blocks. Continuing with the results from theorems which we can collect empirically, we develop trade-able algorithms §3. In summary, we believe such algorithms can build a capital market with less risk (an anomaly corrector). For future reference, we believe our attempts also opened up a lot more potential research problems. For example, what would happen if everyone starts to use this strategy? Another great question can be, what would be an ideal (although state-of-art) game plan after the algorithm tells traders to buy? In macro point of view, how would an economy per- form in long run if one implements this strategy in a larger scale? What if an unknown outside monetary force enter the market and act as an anomaly, how would this algorithm deal with such situation? 5 Appendix 5.1 Proof of Theorem 2.2.1 This is a relatively easy proof since the definition fol- low the premises of the Central Limit Theorem. That is, we have pi,n and SMAn that are i.i.d.. Then by C.L.T., pi,n−SMAn ⇒ χ while χ stands for standard normal distribution. Q.E.D. 5.2 Proof of Theorem 2.2.2 From weak law we know that n∑ m=1 D2 m/nσ 2 → 1. Also note y−1/2 s continuous at 1, then we have( σ2n / n∑ m=1 D2 m )1/2 → 1, in prob., see ?∑n m=1 Dm σ √ n ( σ2n∑n m=1 D2 m )1/2 ⇒ χ · 1, from ? = χ Notice that the ? is because in Weak Convergence, there is a theorem stated that Xn ⇒ X∞ if and only if for every bounded continuous function g we have Eg(Xn)→ Eg(X∞). Since we discussed the continu- ity of function y−1/2 at 1, this line is valid. Q.E.D. Remark 5.2.1. From [1], Section 2, the theorem stated the following. Suppose Xn ⇒ X, Yn ≥ 0, and Yn ⇒ c, where c > 0 is a constant, then XnYn ⇒ cX. 5.3 Proof of Theorem 2.2.3 From Kolmogorov’s inequality we know P ( max (1−ε)cn≤m≤(1+ε)cn |Sm − S[(1−ε)cn]| ) ≤ 2ε/δ2 If Dn = SNn /σ √ cn and Yn = Scn/σ √ cn, then it follows that lim sup n→∞ P (|Dn − Yn| > δ) ≤ 2ε/δ2,∀ε then we have P (|Dn − Yn| > δ) → 0 for each δ > 0, i.e., Xn − Yn → 0 in probability. This is because of the Cnverging together lemma stated in Weak Con- vergence part of [1]. We state the theorem in remark below. Q.E.D. Remark 5.3.1. Suppose Xn ⇒ X and Yn ⇒ c, where c is a constant then Xn + Yn ⇒ X + c. A useful con- sequence is that if Xn ⇒ X and Zn −Xn ⇒ 0 then Xn ⇒ X. 5.4 Proof of Theorem 2.2.3 From convergence theorem, we know that Nt tµ → 1 so from Theorem 2.2.3 we have SN − µNt σ √ t/µ → 0 then it is sufficient to show (Sn − t)/ √ t→ 0 since it follows that (µNt−t)√ σ2t/µ ⇒ χ. We have given finite variance, that is, σ2 < ∞, so by D.C.T. (Dominated Convergence Theorem), we have P ( max 1≤m≤2tµ Ym > ε √ t ) = 2t µ P (Y1 > ε √ t) = 2 µε2E(Y 2 1 ;Y1 > ε √ t)→ 0 which proves that (Sn − t)/ √ t → 0 is true. Hence, this completes the proof. Remark 5.4.1. This is because of the Converging to- gether lemma stated Weak Convergence. Please see Remark 5.3.1. Q.E.D. Page 3 Buy Signal from Limit Theorem [by Yiqiao Yin] §5 References [1] Durrett, Rick, “Probability: Theory and Exam- ples ”, 4E. [2] Fama, Eugene F. (1992), “The cross-section of expected stock returns,” Journal of Finance, 47, 427-465. [3] Fama, Eugene F. (1993), “Common risk factors in the returns on stocks and bonds,” Journal of Financial Economics, 33, 3-56. [4] Fama, Eugene F. (1996), “Multifactor portfolio efficiency and multifactor asset pricing,” Journal of Financial and Quantitative Analysis, 31, 441- 465. [5] Fama, Eugene F. (2006), “Profitability, invest- ment, and average returns,”, Journal of Financial Economics, 82, 491-518. [6] Malkiel, Burton G. (2007), “Reflections on the Ef- ficient market Hypothesis: 30 Years Later,” The Financial Review, 40, 1-9. [7] Yin, Yiqiao (2017), “Anomaly Correction by Op- timal Trading Frequency”, CUSJ, Page 4 Introduction Theoretical Framework Architecture Theories Algorithms Conclusion Appendix Proof of Theorem 2.2.1 Proof of Theorem 2.2.2 Proof of Theorem 2.2.3 Proof of Theorem 2.2.3