# The Relative Strength Index

First introduced by J. Welles Wilder Jr., the RSI is one of the most popular and versatile technical indicators. Mainly used as a contrarian indicator where extreme values signal a reaction that can be exploited. Typically, we use the following steps to calculate the default RSI:

• Calculate the change in the closing prices from the previous ones.
• Separate the positive net changes from the negative net changes.
• Calculate a smoothed moving average on the positive net changes and on the absolute values of the negative net changes.
• Divide the smoothed positive changes by the smoothed negative changes. We will refer to this calculation as the Relative Strength — RS.
• Apply the normalization formula shown below for every time step to get the RSI. GBPUSD in the first panel with the 13-period RSI in the second panel.

The above chart shows the hourly values of the GBPUSD in black with the 13-period RSI. We can generally note that the RSI tends to bounce close to 25 while it tends to pause around 75. To code the RSI in Python, we need an OHLC array composed of four columns that cover open, high, low, and close prices.

`def adder(data, times):  for i in range(1, times + 1):  new = np.zeros((len(data), 1), dtype = float)  data = np.append(data, new, axis = 1) return datadef deleter(data, index, times):  for i in range(1, times + 1):  data = np.delete(data, index, axis = 1) return data def jump(data, jump):  data = data[jump:, ]  return datadef ma(data, lookback, close, where):   data = adder(data, 1)  for i in range(len(data)):  try:  data[i, where] = (data[i - lookback + 1:i + 1, close].mean())  except IndexError:  pass  data = jump(data, lookback)  return datadef ema(data, alpha, lookback, what, where):  alpha = alpha / (lookback + 1.0)  beta = 1 - alpha  data = ma(data, lookback, what, where)data[lookback + 1, where] = (data[lookback + 1, what] * alpha) + (data[lookback, where] * beta) for i in range(lookback + 2, len(data)):  try:  data[i, where] = (data[i, what] * alpha) + (data[i - 1, where] * beta)  except IndexError:  pass  return datadef rsi(data, lookback, close, where):  data = adder(data, 5)  for i in range(len(data)):  data[i, where] = data[i, close] - data[i - 1, close]  for i in range(len(data)):  if data[i, where] > 0:  data[i, where + 1] = data[i, where]  elif data[i, where] < 0:  data[i, where + 2] = abs(data[i, where])  lookback = (lookback * 2) - 1 # From exponential to smoothed data = ema(data, 2, lookback, where + 1, where + 3) data = ema(data, 2, lookback, where + 2, where + 4) data[:, where + 5] = data[:, where + 3] / data[:, where + 4]  data[:, where + 6] = (100 - (100 / (1 + data[:, where + 5]))) data = deleter(data, where, 6) data = jump(data, lookback) return data`

# The Relative Volatility Index

Our first step into calculating the dynamic relative strength index is to calculate the relative volatility index. Volatility is the magnitude of fluctuations of a variable around its mean value. In time series, a volatile data is one that moves swiftly from one level to another and is generally away from its mean while a stable or low-volatility data is one that looks closer to its moving average (mean). USDTRY pair showing low and high volatility environments.

There are many ways to measure volatility such as the average true range, the mean absolute deviation, and the standard deviation. We will use standard deviation to create the relative volatility index.

The most basic type of volatility is the standard deviation. It is one of the pillars of descriptive statistics and an important element in some technical indicators such as the famous Bollinger bands. But first let us define what variance is before we find the standard deviation.

Variance is the squared deviations from the mean (a dispersion measure), we take the square deviations so as to force the distance from the mean to be non-negative, finally we take the square root to make the measure have the same units as the mean, in a way we are comparing apples to apples (mean to standard deviation standard deviation). Variance is calculated through this formula:

Following what we have said, standard deviation is therefore:

`def volatility(Data, lookback, what, where):  for i in range(len(Data)): try: Data[i, where] = (Data[i - lookback + 1:i + 1, what].std()) except IndexError: pass  return Data`

The exciting job now is to turn the historical volatility otherwise known as the historical standard deviation into an understandable indicator that resembles the relative strength index. The below plot shows the relative volatility index which is simply the RSI applied to 10-period standard deviation of the closing prices. EURUSD in the first panel with the 10-period Relative Volatility Index in the second panel.

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# Creating the Dynamic Relative Strength Index

The idea of the dynamic relative strength index is to select the lookback period based on the recent volatility following this intuition:

• Whenever the volatility is high according to the current RVI, we need to bias the lookback period on the RSI downwards to account more for the recent values.
• Whenever the volatility is low according to the current RVI, we need to bias the lookback period on the RSI upwards to account less for the recent values.

Therefore, the two main components of the dynamic relative strength index are:

• The relative volatility index.
• The relative strength index.

How are we going to calculate this and how does it use volatility? It is a very simple concept, let us take a look at the below code:

`lookback = 10def dynamic_relative_strength_index(data, lookback, close, where):  # Calculating the Relative Volatility Index data = relative_volatility_index(data, lookback, close, where)  # Adding a column data = adder(data, 1)  # Calculating the Lookback Periods on the Dynamic Relative Strength Index for i in range(len(data)):  if data[i, where] >= 0 and data[i, where] <= 10 : data[i, where + 1] = 90  if data[i, where] > 10 and data[i, where] <= 20 : data[i, where + 1] = 80  if data[i, where] > 20 and data[i, where] <= 30 : data[i, where + 1] = 70  if data[i, where] > 30 and data[i, where] <= 40 : data[i, where + 1] = 60  if data[i, where] > 40 and data[i, where] <= 50 : data[i, where + 1] = 50  if data[i, where] > 50 and data[i, where] <= 60 : data[i, where + 1] = 40  if data[i, where] > 60 and data[i, where] <= 70 : data[i, where + 1] = 30  if data[i, where] > 70 and data[i, where] <= 80 : data[i, where + 1] = 20  if data[i, where] > 80 and data[i, where] <= 90 : data[i, where + 1] = 10  if data[i, where] > 90 and data[i, where] <= 100 : data[i, where + 1] = 5  # Calculating the Relative Strength Index data = rsi(data, 5, close, where + 2) data = rsi(data, 10, close, where + 3) data = rsi(data, 20, close, where + 4) data = rsi(data, 30, close, where + 5) data = rsi(data, 40, close, where + 6) data = rsi(data, 50, close, where + 7) data = rsi(data, 60, close, where + 8) data = rsi(data, 70, close, where + 9) data = rsi(data, 80, close, where + 10) data = rsi(data, 90, close, where + 11)  # Adding a column data = adder(data, 1)  # Dynamic Relative Strength Index for i in range(len(data)):  if data[i, where + 1] == 5: data[i, where + 12] = data[i, where + 2]  if data[i, where + 1] == 10: data[i, where + 12] = data[i, where + 3]  if data[i, where + 1] == 20: data[i, where + 12] = data[i, where + 4]  if data[i, where + 1] == 30: data[i, where + 12] = data[i, where + 5]  if data[i, where + 1] == 40: data[i, where + 12] = data[i, where + 6]  if data[i, where + 1] == 50: data[i, where + 12] = data[i, where + 7]  if data[i, where + 1] == 60: data[i, where + 12] = data[i, where + 8]  if data[i, where + 1] == 70: data[i, where + 12] = data[i, where + 9]  if data[i, where + 1] == 80: data[i, where + 12] = data[i, where + 10]  if data[i, where + 1] == 90: data[i, where + 12] = data[i, where + 11]  # Cleaning data = deleter(data, where, 12)  return data`

What the above code simply does is calculate the 10-period relative volatility index, then calculates an array of RSI’s on the market price using the following lookback periods {5, 10, 20, 30, 40, 50, 60, 70, 80, 90}. Then the code will loop around the data and segment the relative volatility index into 10 parts with each part assigned a period. For example:

• If the RVI is showing a reading of 8, then it falls into the first part. With such low volatility, the lookback calculated on the RSI is 90. This is to account for the low current volatility.
• If the RVI is showing a reading of 85, then it falls into the ninth part. With such high volatility, the lookback calculated on the RSI is 10. This is to account for the high current volatility. EURGBP in the first panel with the Dynamic Relative Strength Index in the second panel.

The dynamic relative strength index witnesses from time to time spikes and this due to the change in volatility. It takes into account the current market properties and is therefore better at detecting short-term and long-term opportunities and signals. Unlike the normal RSI, the dynamic RSI, when around 50, does not mean we are far from a signal due to the fact that it can spike at any time. EURUSD in the first panel with the Dynamic Relative Strength Index in the second panel.

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