## Hidden Markov Models & Regime Change: S&P500

In this post, we will employ a **statistical time series approach** using** Hidden Markov Models **(HMM), to firstly obtain visual evidence of regime change in the **S&P500.**

Detecting significant, unforeseen changes in underlying market conditions (termed “**market regimes**“) is one of the greatest challenges faced by algorithmic traders today. It is therefore critical that traders account for shifts in these market regimes during trading strategy development.

## Why use Hidden Markov Models?

Hidden Markov Models infer “**hidden states**” in data by using observations (in our case, **returns**) correlated to these states (in our case, **bullish, bearish, or unknown**).

They are hence a suitable technique for detecting regime change, enabling algorithmic traders to optimize entries/exits and risk management accordingly.

We will make use of the **depmixS4** package in **R** to analyse regime change in the S&P500 Index.

With any state-space modelling effort in quantitative finance, there are usually three main types of problems to address:

**Prediction**– forecasting**future**states of the market**Filtering**– estimating the**present**state of the market**Smoothing**– estimating the**past**states of the market

We will be using the **Filtering** approach.

Additionally, we will assume that since S&P500 returns are *continuous*, the probability of seeing a particular return **R** in time **t, **with market regime** M **being in state **m, **where the model used has parameter-set** P,** is described by a **multivariate normal distribution **with mean **μ **and standard deviation** σ** [1].

Mathematically, this can be expressed as:

\(p(R_t | M_t = m, P) = N(R_t | μ_m, σ_m)\)

As noted earlier, we will utilize the **Dependent Mixture Models package in R** (**depmixS4**) for the purposes of:

- Fitting a Hidden Markov Model to S&P500 returns data.
- Determining posterior probabilities of being in one of three market states (bullish, bearish or unknown), at any given time.

We will then use the **plotly** R graphing library to plot both the S&P500 returns, and the market states the index was estimated to have been in over time.

You may replicate the following R source code to conduct this analysis on the S&P500.

#### Step 1: Load required R libraries

`library(quantmod)`

library(plotly)

library(depmixS4)

#### Step 2: Get S&P500 data from June 2014 to March 2017

`getSymbols("^GSPC", from="2014-06-01", to="2017-03-31")`

#### Step 3: Calculate differenced logarithmic returns using S&P500 EOD Close prices.

`sp500_temp = diff(log(Cl(GSPC)))`

sp500_returns = as.numeric(sp500_temp)

#### Step 4: Plot returns from (3) above on plot_ly scatter plot.

`plot_ly(x = index(GSPC), y = sp500_returns, type="scatter", mode="lines") %>%`

`layout(xaxis = list(title="Date/Time (June 2014 to March 2017)"), yaxis = list(title="S&P500 Differenced Logarithmic Returns"))`

## S&P500 Differenced Logarithmic Returns

(June 2014 to March 2017)

#### Step 5: Fit Hidden Markov Model to S&P500 returns, with three “states”

`hidden_markov_model <- depmix(sp500_returns ~ 1, family = gaussian(), nstates = 3, data = data.frame(sp500_returns=sp500_returns))`

`model_fit <- fit(hidden_markov_model)`

#### Step 6: Calculate posterior probabilities for each of the market states

`posterior_probabilities <- posterior(model_fit)`

#### Step 7: Overlay calculated probabilities on S&P500 cumulative returns

`sp500_gret = 1 + sp500_returns`

`sp500_gret <- sp500_gret[-1]`

`sp500_cret = cumprod(sp500_gret)`

`plot_ly(name="Unknown", x = index(GSPC), y = posterior_probabilities$S1, type="scatter", mode="lines", line=list(color="grey")) %>%`

`add_trace(name="Bullish", y = posterior_probabilities$S2, line=list(color="blue")) %>%`

`add_trace(name="Bearish", y = posterior_probabilities$S3, line=list(color="red")) %>%`

`add_trace(name="S&P500", y = c(rep(NA,1), sp500_cret-1), line=list(color="black"))`

## S&P500 Market Regime Probabilities

(June 2014 to March 2017)

**Interpretation:** In any one “market regime”, the corresponding line/curve will “cluster” towards the top of the y-axis (i.e. near a probability of 100%).

For example, during a brief bullish run starting on 01 June 2014, the **blue** line/curve clustered near y-axis value 1.0. This correlates as you can see, with movement in the S&P500 (**black** line/curve). The same applies to **bearish** and “**unknown**” market states.

*An interesting insight one can draw from this graphic, is how the Hidden Markov Model successfully reveals high volatility in the market between June 2014 and March 2015 (constantly changing states between bullish, bearish and unknown).*

**References:**

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[1] Murphy, K.P. (2012) Machine Learning – A Probabilistic Perspective, MIT Press.

https://www.cs.ubc.ca/~murphyk/MLbook/

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**Influences:**

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The honourable Mr. Michael Halls-Moore. QuantStart.com

http://www.quantstart.com/

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