Fractional Black-Scholes Model and Technical Analysis of Stock Price

Abstract

In the stock market, some popular technical analysis indicators (e.g., Bollinger bands, RSI, ROC, etc.) are widely used to forecast the direction of prices. The validity is shown by observed relative frequency of certain statistics, using the daily (hourly, weekly, etc.) stock prices as samples. However, those samples are not independent. In earlier research, the stationary property and the law of large numbers related to those observations under Black-Scholes stock price model and stochastic volatility model have been discussed. Since the fitness of both Black-Scholes model and short-range dependent process has been questioned, we extend the above results to fractional Black-Scholes model with Hurst parameter 𝐻>1/2, under which the stock returns follow a kind of long-range dependent process. We also obtain the rate of convergence.

1. Introduction

Liu et al. discussed in [1] the Bollinger bands for the Black-Scholes model. They introduced the corresponding statistics 𝑈_𝑡^(𝑛) calculated according to the formulation of the Bollinger bands, which is a stationary process, and then they gave the law of large numbers since {𝑈^(𝑛)_{𝑡+𝑘𝑛}}_𝑘=1,2,… are mutually independent for each fixed 𝑡≥0. Thus the Bollinger bands property which seems unthinkable at first glance was proved. The related results have been extended to stochastic volatility model in [2] and AR-ARCH model in [3].

It has been noted in [4] that “technical analysis is a security analysis discipline for forecasting the direction of prices through the study of past market data, primarily price and volume.” Technical analysis has been widely used among traders and financial professionals in stock markets and foreign exchange markets in recent decades. However, technical analysis has not received the same level of academic scrutiny and acceptance as more traditional approaches such as fundamental analysis, since “a simulated sample is only one realization of geometric Brownian motion” and “it is difficult to draw general conclusions about the relative frequencies” (see [5]). However, given the stock price models, we show here that we can do statistics based on relative frequency of occurrence for some technical analysis indicators.

The fitness of both Black-Scholes model and short-range dependent process has been questioned. Since Willinger et al. [6] gave the empirical evidence of long-range dependence in stock price returns, there have been several empirical studies that lent further support to such property of long-range dependence (see, e.g., [7–10]). We consider the process of alternatives to short-range dependence, a model driven by the fractional Brownian motion (fBm) which is long-range dependent. In the following discussion, we assume that the stock price satisfies the fractional Black-Scholes model (see, e.g., [11]): 𝑆_𝑡=𝑆₀exp𝜇+𝜈𝑡+𝜎𝐵^𝐻_𝑡,𝑡∈[0,𝑇], (1) where 𝜇,𝜈∈ℝ are constants, 𝜎 is a positive real number, 𝐵^𝐻_𝑡 is a fBm with Hurst parameter 𝐻, and 𝐻∈(0,1). The fractional Brownian motion is a continuous-time Gaussian process 𝐵^𝐻_𝑡 on [0,𝑇], which starts at zero with expectation zero for all 𝑡∈[0,𝑇], and has the following covariance function (see, e.g., [12, 13]): 𝐸𝐵^𝐻(𝑡)𝐵^𝐻(𝑠=1)2|𝑡|^2𝐻+|𝑠|^2𝐻−|𝑡−𝑠|^2𝐻,𝑠,𝑡≥0, (2) where 𝐻 is a real number in (0,1), called the Hurst index or Hurst parameter associated with the fractional Brownian motion. In contrast to Brownian motion, the increments of the fBm are not independent if 𝐻≠1/2. The fBm is self-similar, that is, 𝐵^𝐻(𝑎𝑡)(=)|𝑎|^𝐻𝐵^𝐻(𝑡), and the increments are stationary, that is, 𝐵^𝐻(𝑡)−𝐵^𝐻(𝑠)(=)𝐵^𝐻(𝑡−𝑠), and the increments exhibit long-range dependence if 𝐻>1/2, that is, ∑^∞_𝑛=1𝐸[𝐵^𝐻(1)(𝐵^𝐻(𝑛+1)−𝐵^𝐻(𝑛))]=∞, 𝐻>1/2, where 𝑋(=)𝑌 denotes that 𝑋 and 𝑌 have the same distribution. Note that the fBm is in fact a regular Brownian motion if 𝐻=1/2.

In this paper, we discuss the statistical properties of some popular technical indicators such as Bollinger bands, Relative Strength Index (RSI), and Rate of Change (ROC). Under fractional Black-Scholes model (1), we show that the corresponding statistics are stationary and the law of large numbers holds for frequencies of stock prices falling out of normal scope of the technical indicators.

This paper is organized as follows. Section 2 introduces some technical indicators. Section 3 gives the ratios of Bollinger bands, RSI, and ROC falling in the corresponding sets. In Section 4, by constructing a statistic 𝑈_𝑡^(𝑛), we investigate the stationary properties of corresponding statistics. In Section 5, we obtain the law of large numbers for frequencies of the statistics. And we give some comments of the results in Section 6.

2. Definitions of Technical Indicators

Let 𝑆_𝑡 be current stock price and 𝑆_{𝑡−𝑖𝛿} the stock price 𝑖 periods ago, where 𝛿 is the length of the period between two observation spots (the period can be day, minute, etc.). We recall the definitions of technical indicators in the following:

(1) Bollinger Bands. Developed by John Bollinger in the 1980s, Bollinger Bands are volatility bands placed above and below a moving average denoted by 𝑆_𝑡^(𝑛)=1𝑛^𝑛−1_𝑖=0𝑆_{𝑡−𝑖𝛿},𝐵^𝑛,med_𝑡=1∑^𝑛_𝑖=1𝑖^𝑛−1_𝑖=0(𝑛−𝑖)𝑆_{𝑡−𝑖𝛿},𝑠_𝑡^(𝑛)=1𝑛−1^𝑛−1_𝑖=0𝑆_{𝑡−𝑖𝛿}−𝑆_𝑡^(𝑛)2,𝑡≥(𝑛−1)𝛿. (3) The curve 𝐵^𝑛,med_𝑡 is called the middle Bollinger band, the curve 𝐵^𝑛,upp_𝑡=𝐵^𝑛,med_𝑡+2𝑠_𝑡^(𝑛) is called the upper Bollinger band, and 𝐵^𝑛,low_𝑡=𝐵^𝑛,med_𝑡−2𝑠_𝑡^(𝑛) is called the lower Bollinger band, where 𝑛 is the number of selected periods. The Bollinger bands of S&P500 are shown in Figure 1. Usually we take 𝑛=12 or 20, 𝛿=oneday. According to Bollinger [14] and Liu et al. [1], the bands contain more than 88-89% of price action, which makes a move outside the bands significant. Technically, prices are relatively high when above the upper band and relatively low when below the lower band. However, relatively high should not be regarded as bearish or as a sell signal. Likewise, relatively low should not be considered bullish or as a buy signal. As with other indicators, Bollinger bands are seldom used alone. Traders should combine Bollinger bands with basic trend analysis and other indicators for confirmation.

Figure 1 

S&P500 annual Bollinger bands until March 27, 2012.

(2) Relative Strength Index (RSI). The RSI was developed by Wilder [15], and it is classified as a momentum oscillator, measuring the velocity and magnitude of directional price movements. If we denote Δ𝑆⁺_𝑡=𝑆_𝑡+𝛿−𝑆_𝑡∨0,Δ𝑆⁻_𝑡=𝑆_𝑡−𝑆_𝑡+𝛿∨0, (4) the 𝑛-period RSI is defined as RSI_𝑡^(𝑛)∑=100×^𝑛𝑖=1Δ𝑆⁺_{𝑡−𝑖𝛿}∑^𝑛_𝑖=1Δ𝑆⁺_{𝑡−𝑖𝛿}+∑^𝑛_𝑖=1Δ𝑆⁻_{𝑡−𝑖𝛿},𝑡≥𝑛𝛿. (5)

The RSI of S&P500 is shown in Figure 2. Usually we take 𝑛=14, 𝛿=oneday. RSI oscillates between zero and 100, with high and low levels marked at 70 and 30. More extreme high and low levels (80 and 20 or 90 and 10) occur less frequently but indicate stronger momentum. Traditionally, RSI readings greater than the 70 level are considered to be in overbought territory, and RSI readings lower than the 30 level are considered to be in oversold territory. If the RSI is below 50, it generally means that the market is in a week trend. When the RSI is above 50, it generally means that the market is in a strong trend. Zhu [16] discussed the statistical property and the forecasting ability of RSI.

(a)

(b)

Figure 2 

S&P500 annual RSI until March 27, 2012.

(3) Rate of Change (ROC). The ROC is a pure momentum oscillator that measures the percent of change in price from one period to the next. The 𝑛-period ROC is defined as ROC_𝑡^(𝑛)𝑆=100×^𝑡−𝑆_{𝑡−𝑛𝛿}𝑆_{𝑡−𝑛𝛿},𝑡≥𝑛𝛿. (6) The ROC of S&P500 is shown in Figure 3. Usually we take 𝑛=12, 𝛿=oneday. Prices are constantly increasing as long as the ROC remains positive. Conversely, prices are falling when the ROC is negative. The ROC has its antennas and grounds which are indefinite and can give identifiable extremes that signal overbought and oversold conditions. In general, it is time to sell out when the ROC rises to the first ultra-buy line (5), and then the rising trend mostly ends when it reaches the second ultra-buy line (10). It is time to buy in when ROC drops to the first ultra-sell line (−5), and then the dropping trend mostly ends when it reaches the second ultra-sell line (−10). Li [17] discussed the empirical evidence of ROC. Like all technical indicators, the ROC oscillator should be used in conjunction with other aspects of technical analysis.

(a)

(b)

Figure 3 

S&P500 annual ROC until March 27, 2012.

3. Some Facts from the Stock Market

Liu et al. [1] traced 15 years of the DOW, S&P500, and NASDAQ daily closing prices and drew the conclusion that in every year more than 94% of daily closing prices are between the Bollinger bands. We give the ratios of Bollinger bands, RSI, and ROC falling in the corresponding sets from January, 2008, to December, 2011, in Tables 1, 2, and 3, where B-B denote the Bollinger bands. We can see that more than 95% of daily closing prices are between the Bollinger bands, more than 81% of RSI are in the interval [20,80], and more than 87% of ROC are in the interval [−10,10]. So we show that the stationary of the indexes is still maintained even since the world economic crisis in 2008. In the following, we give a mathematical proof to this fact under the fractional Black-Scholes model.

Table 1 

Ratio of SPY in 2008–2011.

Table 2 

Ratio of QQQ in 2008–2011.

Table 3 

Ratio of DIA in 2008–2011.

4. Stationary Property

Let 𝑆_𝑡 denote the observed stock price under the model (1). And let ℎ(𝑡,𝑖,𝑛)=exp𝐵^𝐻_{𝑡−𝑖𝛿}−𝐵^𝐻_{𝑡−𝑛𝛿}𝑈,𝑖=0,1,2,…,𝑛−1,_𝑡(𝑛)=𝑓(ℎ(𝑡,0,𝑛),…,ℎ(𝑡,𝑛−1,𝑛)),𝑡≥𝑛𝛿, (7) where 𝑓 is a measurable function: ℝ^𝑛→ℝ. Then we have the following results:

The process {𝑈_𝑡^(𝑛)}_{𝑡≥𝑛𝛿} is stationary.

Remark 1. Let 𝑆_𝑡 be the stock price generated by the model (1), 𝐿_𝑡^(𝑛)=(𝑆_𝑡−𝐵^𝑛,med_𝑡)/𝑠_𝑡^(𝑛)(𝑡≥𝑛𝛿). Then the process {𝐿_𝑡^(𝑛)}_{𝑡≥𝑛𝛿} is stationary.

Remark 2. Let 𝑆_𝑡 be the stock price generated by the model (1). Then the process RSI_𝑡^(𝑛)∑=100×^𝑛𝑖=1Δ𝑆⁺_{𝑡−𝑖𝛿}∑^𝑛_𝑖=1Δ𝑆⁺_{𝑡−𝑖𝛿}+∑^𝑛_𝑖=1Δ𝑆⁻_{𝑡−𝑖𝛿}(𝑡≥𝑛𝛿) (8) is stationary.

Remark 3. Let 𝑆_𝑡 be the stock price generated by the model (1). Then the process ROC_𝑡^(𝑛)=100×(𝑆_𝑡−𝑆_{𝑡−𝑛𝛿})/𝑆_{𝑡−𝑛𝛿}(𝑡≥𝑛𝛿) is stationary.

5. Law of Large Numbers

Let 𝐾^(𝑛)_Γ,𝑖=𝐼_{𝑈^(𝑛)_{𝑖𝛿∈Γ}}, 𝑖≥𝑛, where Γ is a subset of ℝ. And let 𝑉^(𝑛)_𝑁,Γ=1𝑁+1^𝑁_𝑖=0𝐾^(𝑛)_Γ,𝑛+𝑖, (9) which is the observed frequency of the events [𝑈^(𝑛)_{(𝑛+𝑖)𝛿}∈Γ](𝑖=0,1,…,𝑁).

It is natural to assume 𝛿<1; that is, the length between two observation spots is less than one year. From the above discussion, we can let 𝑝=𝑃(𝑈_𝑡^(𝑛)∈Γ), 𝑡≥𝑛𝛿. We denote 𝑈^(𝑛)_𝑖𝛿 by 𝑈_𝑖^(𝑛), 𝑖≥𝑛, in the following discussion for convenience. Denote by ℝ^𝑚×𝑛 the set of 𝑚×𝑛 real matrices, and set 𝑍_𝑘,𝑗=𝐾^(𝑛)_{Γ,(𝑘+1)𝑛+𝑗}−𝑃𝑈_𝑛^(𝑛)∈Γ (10) for each fixed 𝑗 and 𝑘; we give the following lemma.

Lemma 4. For all Γ⊂ℝ, there exist 𝛼∈(0,1) and a constant 𝐶>0; when 𝑁 is large enough and |𝑘₁−𝑘₂|−𝑛>𝑁^𝛼, one has 𝐸𝑍_𝑘1,𝑗𝑍_𝑘2,𝑗≤𝐶|2𝐻−1|𝑁^{𝛼(𝐻−1)/2}. (11)

Proof. Let 𝑊_𝑘,𝑖=𝐵^𝐻_{[(𝑘+1)𝑛+𝑗−𝑖]𝛿}−𝐵^𝐻_{[(𝑘+1)𝑛+𝑗−(𝑖+1)]𝛿}, 𝑘∈ℤ⁺, 𝑖=0,1,2,…,𝑛−1. And set 𝑋=(𝑊_𝑘1,0,…,𝑊_{𝑘1,𝑛−1})^𝑇≜(𝑋₁,…,𝑋_𝑛)^𝑇, 𝑌=(𝑊_𝑘2,0,…,𝑊_{𝑘2,𝑛−1})^𝑇≜(𝑌₁,…,𝑌_𝑛)^𝑇, 𝑍=(𝑋^𝑇,𝑌^𝑇)^𝑇, where 𝑋^𝑇 is the transpose of 𝑋; then 𝐸𝑋=𝐸𝑌=𝟎, 𝐸𝑍=𝟎. We denote by 𝐴=(𝑎_𝑖𝑗) the covariance matrix of 𝑋, denote by 𝐷=(𝑑_𝑖𝑗) the covariance matrix of 𝑌, and denote by 𝐵=(𝑏_𝑖𝑗) the covariance matrix of 𝑋 and 𝑌. Let Σ be the covariance matrix of 𝑍; then Σ=_𝐵^𝐴𝐵_𝑇𝐷, |𝐴|>0,|𝐷|>0. Since the fBm has stationary increments, we can get 𝐴=𝐷, and 𝑎_𝑖𝑗=𝑑_𝑖𝑗=12𝑖−𝑗−1𝛿^2𝐻+12𝑖−𝑗+1𝛿^2𝐻−(𝑖−𝑗)𝛿^2𝐻,𝐻∈(0,1). (12) Similar to the conclusion given by Deng and Barkai [18], we can get from simple calculation 𝑏_𝑖𝑗=𝐸𝑋_𝑖𝑌_𝑗∼𝛿²𝐻(2𝐻−1)𝑘₁−𝑘₂+𝑗−𝑖^2𝐻−2,𝑘₂−𝑘₁⟶+∞, (13) where 𝑝_𝑘∼𝑞_𝑘 means lim_𝑘→∞(𝑝_𝑘/𝑞_𝑘)=1. So we have 𝐵=(𝑏_𝑖𝑗)→𝟎.
When 𝐻=1/2, we can easily derive that the conclusion of Lemma 4 holds. We assume 𝐻≠1/2 in the following proof. Let 𝑝(𝑧) be the probability density function of 𝑍 and 𝐹(𝑧) the distribution function of 𝑍, and let the marginal distributions for 𝑍 be 𝐹_𝑖(𝑧_𝑖), 𝑖=1,2,…,2𝑛, where 𝑧∈ℝ^2𝑛×1, 𝑧=(𝑧₁,𝑧₂,…,𝑧_2𝑛)^𝑇. Take the notation Θ=[−𝑀,𝑀],𝑀>0, and Γ^′=Θ×Θ×⋯×Θ≜Θ^𝑛. Furthermore, we put 𝑔(𝑍)=𝑍_𝑘1,𝑗𝑍_𝑘2,𝑗. (14)
First we will consider the integral of 𝑔(𝑍) on Γ^′×Γ^′. Referring to Bernstein [19], we have Σ⁻¹=Σ₁+Σ₂, where Σ=𝐷−𝐵^𝑇𝐴⁻¹𝐵, Σ₁=𝐴⁻¹𝟎𝟎𝐷⁻¹,Σ₂=𝐴⁻¹𝐵Σ⁻¹𝐵^𝑇𝐴⁻¹−𝐴⁻¹𝐵Σ⁻¹−Σ⁻¹𝐵^𝑇𝐴⁻¹𝐷⁻¹𝐵^𝑇𝐴−𝐵𝐷⁻¹𝐵^𝑇−1𝐵𝐷⁻¹. (15) We take the notation 𝑑𝑧=𝑑𝑧₁,𝑑𝑧₂,…,𝑑𝑧_2𝑛. Then we obtain ⋯^2𝑛_Γ′×Γ′𝑔(𝑧)𝑝(𝑧)𝑑𝑧=⋯^2𝑛_Γ′×Γ′𝑔(𝑧)(2𝜋)^−𝑛(|𝐴||𝐷|)^−1/2−1×exp2𝑧^𝑇Σ₁𝑧×𝐷−𝐵^𝑇𝐴⁻¹𝐵^−1/2|𝐷|^−1/2−1exp2𝑧^𝑇Σ₂𝑧−1𝑑𝑧, (16) where lim_{|𝑘1−𝑘2|→∞}(|𝐷−𝐵^𝑇𝐴⁻¹𝐵|/|𝐷|)=(1/|𝐷|)(|𝐷−lim_{|𝑘1−𝑘2|→∞}(𝐵^𝑇𝐴⁻¹𝐵)|)=1. Assume lim_{𝑘2−𝑘1→∞}(𝑀/|𝑘₁−𝑘₂|^1−𝐻)=0; then 𝑧^𝑇Σ₂𝑧→0, so we get ∀𝜖>0, ∃𝑁₁, ∀|𝑘₁−𝑘₂|>𝑁₁, −1exp2𝑧^𝑇Σ₂𝑧=1−11−2𝑧^𝑇Σ₂𝑧+𝑜𝑧^𝑇Σ₂𝑧=1−12𝑧^𝑇Σ₂𝑧+𝑜𝑧^𝑇Σ₂𝑧. (17) Choose 𝛼∈(0,1) satisfying 𝑁^𝛼>𝑁₁, where 𝑁<(𝑁+1)/𝑛; then if |𝑘₁−𝑘₂|−𝑛>𝑁^𝛼 and 𝑁 is large enough, there exist 𝐶₀>0, and 𝐶₀ has relation with 𝑛, such that 12𝑧^𝑇Σ₂𝑧+𝑜𝑧^𝑇Σ₂𝑧≤𝐶₀𝑀²|𝐻(2𝐻−1)|𝑁^{2𝛼(𝐻−1)}. (18) Therefore, by (16), (17), and (18), we can derive that there exist 𝐶₀>0 satisfying ⋯^2𝑛_Γ′×Γ′𝑔(𝑧)𝑝(𝑧)𝑑𝑧≤𝑝²𝜖+𝐶₀𝑀²|𝐻(2𝐻−1)|𝑁^{2𝛼(𝐻−1)}≤𝐶₀|2𝐻−1|𝑁^{𝛼(𝐻−1)}. (19)
Then we will consider the integral of 𝑔(𝑍) on the complementary set of Γ^′×Γ^′ in the following. Let Ξ be a random variable satisfying 𝑃(Ξ=(−∞,−𝑀))=1/2, 𝑃(Ξ=(𝑀,∞))=1/2. Let Γ_𝑖 be the set that contains all elements of the following form: Ξ×⋯×Θ×⋯×Ξ×⋯×Θ≜Ξ^(𝑖), (20) where Θ occurs 𝑖 times and Ξ occurs 2𝑛−𝑖 times in Ξ^(𝑖), 𝑖=0,1,…,2𝑛−1. So we can see that Γ_𝑖 is composed of _𝑖^2𝑛⋅2^2𝑛−𝑖 mutually disjoint elements. Therefore, the complementary set of Γ^′×Γ^′ should be (Γ^′×Γ^′)^𝑐=⋃^2𝑛−1_𝑖=0Γ_𝑖 and Γ_𝑖⋂Γ_𝑗=Φ, 𝑖≠𝑗; that is, the complementary set of Γ^′×Γ^′ is the union of (∑^2𝑛−1_𝑖=0𝑖^2𝑛⋅2^2𝑛−𝑖=3^2𝑛−1) mutually disjoint sets.
Since ⋯^2𝑛_Ξ(𝑖)𝑔(𝑧)𝑝(𝑧)𝑑𝑧≤⋯^2𝑛_Ξ(𝑖)𝑝(𝑧)𝑑𝑧≤𝐹𝑧₁,𝑧₂,…,𝑧_2𝑛≤min_{1≤𝑖≤2𝑛}𝐹_𝑖𝑧_𝑖≤𝐹₁()≤𝛿−𝑀^𝐻𝑒^{−𝑀2/2𝛿2𝐻}𝑀, (21) where 𝑀 occurs 𝑖 times and −𝑀 occurs 2𝑛−𝑖 times within 𝑧₁,𝑧₂,…,𝑧_2𝑛, the second inequality holds because |𝑔(𝑧)|≤1, and the last inequality holds because ∫_𝑀^+∞𝑒^−𝑥2/2𝑑𝑥=∫^−𝑀_−∞𝑒^−𝑥2/2𝑑𝑥≤𝑒^−𝑀2/2/𝑀.
Take 𝑀=𝑁^{𝛼(1−𝐻)/2}. We conclude from (19) and (21) that 𝐸𝑍_𝑘1,𝑗𝑍_𝑘2,𝑗=⋯^2𝑛_Γ′×Γ′𝑔(𝑧)𝑝(𝑧)𝑑𝑧+⋯^2𝑛_{Γ′×Γ′𝑐}𝑔(𝑧)𝑝(𝑧)𝑑𝑧≤𝐶₀|2𝐻−1|𝑁^{𝛼(𝐻−1)}+^2𝑛−1_𝑖=0𝑖2𝑛⋅2^2𝑛−𝑖𝛿^𝐻𝑒^{−(𝑀)2/2𝛿2𝐻}𝑀≤𝐶₀|2𝐻−1|𝑁^{𝛼(𝐻−1)}+3^2𝑛−1𝛿^𝐻𝑁^{𝛼(𝐻−1)/2}. (22) Take 𝐶=𝐶₀+(3^2𝑛−1)𝛿^𝐻/|2𝐻−1|; then we have 𝐶₀|2𝐻−1|𝑁^{𝛼(𝐻−1)}+3^2𝑛−1𝛿^𝐻𝑁^{𝛼(𝐻−1)/2}≤𝐶|2𝐻−1|𝑁^{𝛼(𝐻−1)/2} (23) from which the proof immediately follows.

Then we obtain the law of large numbers.

Theorem 5. There exist 𝛽∈(0,1) and a constant 𝐶>0 such that 𝐸𝑉^(𝑛)_𝑁,Γ−𝑃𝑈_𝑛^(𝑛)∈Γ²≤𝐶𝑁^𝛽. (24)

Proof. To simplify notation, we put Λ={𝑘∶0≤𝑘𝑛+𝑗≤𝑁} and set for each fixed 𝑗 and 𝑘, 𝑌_𝑗=_𝑘∈Λ𝐾₍₎^(𝑛)_{Γ,𝑘+1𝑛+𝑗}−𝑃𝑈_𝑛^(𝑛)∈Γ. (25) Since the process {𝑈_𝑡^(𝑛)}_{𝑡≥𝑛𝛿} is stationary, we have 𝑃(𝑈_𝑡^(𝑛)∈Γ)=𝐸[𝐼_{𝑈^(𝑛)_{(𝑘+1)𝑛+𝑗∈Γ}}] holds for all 𝑘>0. In addition, by C-r inequality and Lemma 4, it follows that 𝐸𝑉^(𝑛)_𝑁,Γ−𝑃𝑈_𝑛^(𝑛)∈Γ²1=𝐸𝑁+1^𝑛−1_𝑗=0𝑌_𝑗²≤𝑛(𝑁+1)^2𝑛−1_𝑗=0𝐸𝑌²_𝑗=𝑛(𝑁+1)^2𝑛−1_{𝑗=0𝑘∈Λ}𝐸𝑍²_𝑘,𝑗+^𝑛−1_{𝑗=0𝑘12≠𝑘𝑘1,𝑘2∈Λ}𝐸𝑍_𝑘1,𝑗𝑍_𝑘2,𝑗≤𝑛(𝑁+1)²𝑁+1+^𝑛−1_{𝑗=0𝑘12𝛼0<−𝑘≤𝑁𝑘1,𝑘2∈Λ}𝐸𝑍_𝑘1,𝑗𝑍_𝑘2,𝑗+^𝑛−1_{𝑗=0𝑘12𝛼−𝑘>𝑁𝑘1,𝑘2∈Λ}𝐸𝑍_𝑘1,𝑗𝑍_𝑘2,𝑗≤𝑛(𝑁+1)²𝑁+1+2𝑛𝑁+1𝑛𝑁^𝛼+2𝑛𝑁+1𝑛𝑁−𝑁^𝛼𝐶|2𝐻−1|𝑁^{𝛼(𝐻−1)/2}. (26) Let 𝛽=min{1−𝛼,𝛼(1−𝐻)/2} and 𝐶=3𝑛+2𝑛𝐶|2𝐻−1|; then 𝐸|𝑉^(𝑛)_𝑁,Γ−𝑃(𝑈_𝑛^(𝑛)∈Γ)|²≤𝐶/𝑁^𝛽.

Remark 6. From Theorem 5, it is reasonable to use the stationary distribution of 𝑈_𝑛^(𝑛) to calculate 𝑉^(𝑛)_𝑁,Γ, which is the relative frequency of the technical indicators falling in the corresponding set.

Corollary 7. Let 𝐻_𝑖^(𝑛)=𝐼_{𝐿^(𝑛)_{𝑖𝛿≥2}}, 𝑖≥𝑛; then 𝐸𝐻_𝑖^(𝑛)=𝑃(𝐿^(𝑛)_𝑖𝛿≥2). Let 𝐽_𝑁^(𝑛)=1𝑁+1^𝑁_𝑖=0𝐻^(𝑛)_𝑛+𝑖, (27) which is the observed frequency of the events [𝐿^(𝑛)_{(𝑛+𝑖)𝛿}≥2](𝑖=0,1,…,𝑁), that is, the frequency of stock falling out of the Bollinger bands. Then there exist 𝛽∈(0,1) and a constant 𝐶>0 such that 𝐸𝐽_𝑁^(𝑛)−𝑃𝐿^(𝑛)_𝑛𝛿≥2²≤𝐶𝑁^𝛽, (28)

Corollary 8. Let 𝐻_𝑖^(𝑛)=𝐼_{{𝑅𝑆𝐼}^(𝑛)_{𝑖𝛿∈Γ}}, 𝑖≥𝑛, where Γ=[0,20]∪[80,100]. Then 𝐸𝐻_𝑖^(𝑛)=𝑃(𝑅𝑆𝐼^(𝑛)_𝑖𝛿∈Γ). Let 𝐽_𝑁^(𝑛)=1𝑁+1^𝑁_𝑖=0𝐻^(𝑛)_𝑛+𝑖, (29) which is the observed frequency of the events [𝑅𝑆𝐼^(𝑛)_{(𝑛+𝑖)𝛿}∈Γ](𝑖=0,1,…,𝑁). Then there exist 𝛽∈(0,1) and a constant 𝐶>0 such that 𝐸𝐽_𝑁^(𝑛)−𝑃(𝑅𝑆𝐼^(𝑛)_𝑛𝛿∈Γ)²≤𝐶𝑁^𝛽. (30)

Corollary 9. Let 𝐻_𝑖^(𝑛)=𝐼_{{𝑅𝑂𝐶}^(𝑛)_{𝑖𝛿∈Γ}}, 𝑖≥𝑛, where Γ=[−∞,𝜉]∪[𝜂,∞], and 𝜉 and 𝜂 are the indefinite ground and antenna of ROC. Then 𝐸𝐻_𝑖^(𝑛)=𝑃(𝑅𝑂𝐶^(𝑛)_𝑖𝛿∈Γ). Let 𝐽_𝑁^(𝑛)=1𝑁+1^𝑁_𝑖=0𝐻^(𝑛)_𝑛+𝑖, (31) which is the observed frequency of the events [𝑅𝑂𝐶^(𝑛)_{(𝑛+𝑖)𝛿}∈Γ] (𝑖=0,1,…,𝑁). Then there exist 𝛽∈(0,1) and a constant 𝐶>0 such that 𝐸𝐽_𝑁^(𝑛)−𝑃(𝑅𝑂𝐶^(𝑛)_𝑛𝛿∈Γ)²≤𝐶𝑁^𝛽. (32)

6. Conclusion

In the above discussion, we considered a class of long-range dependent processes, of which the rate of decay is slower than the exponential one, typically a power-like decay. We derived the rate of convergence of the ergodic theorem for several stationary processes associated with the technical analysis in the security market and extended the previous results (see [1–3, 16, 17]). Thus, we established the theoretical foundation of technical analysis for fractional Black-Scholes model.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported in part by the Natural Science Foundation of the Anhui High Education Institutions of China (KJ2013B261), the Natural Science Foundation of the Anhui High Education Institutions of China (KJ2012A257).