Fractional Black-Scholes Model and Technical Analysis of Stock Price
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 , under which the stock returns follow a kind of long-range dependent process. We also obtain the rate of convergence.
Liu et al. discussed in  the Bollinger bands for the Black-Scholes model. They introduced the corresponding statistics calculated according to the formulation of the Bollinger bands, which is a stationary process, and then they gave the law of large numbers since are mutually independent for each fixed . Thus the Bollinger bands property which seems unthinkable at first glance was proved. The related results have been extended to stochastic volatility model in  and AR-ARCH model in .
It has been noted in  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 ). 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.  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., ): where are constants, is a positive real number, is a fBm with Hurst parameter , and . The fractional Brownian motion is a continuous-time Gaussian process on , which starts at zero with expectation zero for all , and has the following covariance function (see, e.g., [12, 13]): where 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 . The fBm is self-similar, that is, , and the increments are stationary, that is, , and the increments exhibit long-range dependence if , that is, , , where denotes that and have the same distribution. Note that the fBm is in fact a regular Brownian motion if .
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 , 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 be current stock price and the stock price periods ago, where 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 The curve is called the middle Bollinger band, the curve is called the upper Bollinger band, and is called the lower Bollinger band, where is the number of selected periods. The Bollinger bands of S&P500 are shown in Figure 1. Usually we take or 20, . According to Bollinger  and Liu et al. , 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.
(2) Relative Strength Index (RSI). The RSI was developed by Wilder , and it is classified as a momentum oscillator, measuring the velocity and magnitude of directional price movements. If we denote the -period RSI is defined as
The RSI of S&P500 is shown in Figure 2. Usually we take , . 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  discussed the statistical property and the forecasting ability of RSI.
(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 -period ROC is defined as The ROC of S&P500 is shown in Figure 3. Usually we take , . 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  discussed the empirical evidence of ROC. Like all technical indicators, the ROC oscillator should be used in conjunction with other aspects of technical analysis.
3. Some Facts from the Stock Market
Liu et al.  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 . 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.
4. Stationary Property
Let denote the observed stock price under the model (1). And let where is a measurable function: . Then we have the following results:
The process is stationary.
Remark 1. Let be the stock price generated by the model (1), . Then the process is stationary.
Remark 2. Let be the stock price generated by the model (1). Then the process is stationary.
Remark 3. Let be the stock price generated by the model (1). Then the process is stationary.
5. Law of Large Numbers
Let , , where is a subset of . And let which is the observed frequency of the events ().
It is natural to assume ; that is, the length between two observation spots is less than one year. From the above discussion, we can let , . We denote by , , in the following discussion for convenience. Denote by the set of real matrices, and set for each fixed and ; we give the following lemma.
Lemma 4. For all , there exist and a constant ; when is large enough and , one has
Proof. Let , , . And set , , , where is the transpose of ; then , . We denote by the covariance matrix of , denote by the covariance matrix of , and denote by the covariance matrix of and . Let be the covariance matrix of ; then , . Since the fBm has stationary increments, we can get , and
Similar to the conclusion given by Deng and Barkai , we can get from simple calculation
where means . So we have .
When , we can easily derive that the conclusion of Lemma 4 holds. We assume in the following proof. Let be the probability density function of and the distribution function of , and let the marginal distributions for be , , where , . Take the notation , and . Furthermore, we put
First we will consider the integral of on . Referring to Bernstein , we have , where , We take the notation . Then we obtain where . Assume ; then , so we get , , , Choose satisfying , where ; then if and is large enough, there exist , and has relation with , such that Therefore, by (16), (17), and (18), we can derive that there exist satisfying
Then we will consider the integral of on the complementary set of in the following. Let be a random variable satisfying , . Let be the set that contains all elements of the following form: where occurs times and occurs times in , . So we can see that is composed of mutually disjoint elements. Therefore, the complementary set of should be and , ; that is, the complementary set of is the union of mutually disjoint sets.
Since where occurs times and occurs times within , the second inequality holds because , and the last inequality holds because .
Take . We conclude from (19) and (21) that Take ; then we have from which the proof immediately follows.
Then we obtain the law of large numbers.
Theorem 5. There exist and a constant such that
Proof. To simplify notation, we put and set for each fixed and , Since the process is stationary, we have holds for all . In addition, by C-r inequality and Lemma 4, it follows that Let and ; then .
Remark 6. From Theorem 5, it is reasonable to use the stationary distribution of to calculate , which is the relative frequency of the technical indicators falling in the corresponding set.
Corollary 7. Let , ; then . Let which is the observed frequency of the events (), that is, the frequency of stock falling out of the Bollinger bands. Then there exist and a constant such that
Corollary 8. Let , , where . Then . Let which is the observed frequency of the events (). Then there exist and a constant such that
Corollary 9. Let , , where , and and are the indefinite ground and antenna of ROC. Then . Let which is the observed frequency of the events (). Then there exist and a constant such that
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.
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).
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