Hidden Markov Models for Financial Market Predictions

Transcription

1 Hidden Markov Models for Financial Market Predictions Department of Mathematics and Statistics Youngstown State University Central Spring Sectional Meeting, Michigan State University, March 15

2 1 Introduction of HMMs 2 HMMs for economics regimes 3 HMMs for stock prices 4 HMM for stock sections

3 History of HMMs Introduced in 1966 by Baum and Petrie Baum and his colleagues published HMM training for a single observation, 1970 Levonson, Rabiner, and Sondhi presented HMM training for multiple independent observations, 1983 Li, Parizeau, and Plamondo introduced HMM traning for multiple observations, 2000

4 What is a Hidden Markov Model? Hidden Markov Model (HMM): stochastic signal model with three assumptions: The observation at time t, O t, was generated by some process whose state, S t, is hidden. The hidden process satisfies the first-order Markov property: given S t 1, S t is independent of S i for any i < t 1. The hidden state variable is discrete.

5 Some applications of HMMs Figure : 1. Speech recognition 2. Bioinformatics 3. Finance

6 Elements of HMM Observation data, O = (O t ), t = 1,.., T Hidden states, S = (S i ), i = 1, 2,..., N Hidden state sequence: Q = (q t ), t = 1,..., T Transition matrix A = (a ij ) a ij = P(q t = S j q t 1 = S i ), i, j = 1, 2,..., N Observation symbols per state, V = (v k ), k = 1, 2,..., M The observation probability B = (b ik ) b ik = P(O t = v k q t = S i ), i = 1, 2,..., N; k = 1, 2,..., M Initial probabilities, vector p, of being in state S i at t = 1 p i = P(q 1 = S i ), i = 1, 2,..., N

7 Hidden Markov Model S1 a11 S1 a12 b1(ot) a21 S2 a22 S2 Ot b2(ot) Ot+1 t t+1 Parameters of HMM: λ = {A, B, p}

8 Three problems and corresponding solutions for HMMs 1 Given (O, λ), compute the probability of observations, P(O λ)

9 Three problems and corresponding solutions for HMMs 1 Given (O, λ), compute the probability of observations, P(O λ) Forward, backward algorithm

10 Three problems and corresponding solutions for HMMs 1 Given (O, λ), compute the probability of observations, P(O λ) Forward, backward algorithm 2 Given (O, λ), simulate the most likely hidden states, Q

11 Three problems and corresponding solutions for HMMs 1 Given (O, λ), compute the probability of observations, P(O λ) Forward, backward algorithm 2 Given (O, λ), simulate the most likely hidden states, Q Viterbi algorithm

12 Three problems and corresponding solutions for HMMs 1 Given (O, λ), compute the probability of observations, P(O λ) Forward, backward algorithm 2 Given (O, λ), simulate the most likely hidden states, Q Viterbi algorithm 3 Given O, calibrate HMM parameters, λ

13 Three problems and corresponding solutions for HMMs 1 Given (O, λ), compute the probability of observations, P(O λ) Forward, backward algorithm 2 Given (O, λ), simulate the most likely hidden states, Q Viterbi algorithm 3 Given O, calibrate HMM parameters, λ Baum-Welch algorithm

14 Forward Algorithm Define the joint probability α t (i) = P(O 1, O 2,..., O t, q t = S i λ) S i t (i) t-1 t

15 Forward algorithm Initialization, α 1 (i) = p i b i (O 1 ) for i = 1,..., N For t = 2, 3,..., T, for j = 1,..., N [ N α t (j) = i=1 α t 1 (i)a ij ]b j (O t ), P(O λ) = N i=1 α T (i)

16 Backward Algorithm Define the conditional probability β t (j) = P(O t+1, O t+2,.., O T q t = S j, λ), for j = 1,..., N S j t+1 (j) t+1 t+2

17 Backward Algorithm Algorithm Initialization, β T (i) = 1 for i = 1,..., N For t = T 1, T 2,..., 1, for i = 1,..., N β t (i) = N a ij b j (O t+1 )β t+1 (j) j=1 P(O λ) = N i=1 p ib i (O 1 )β 1 (i)

18 Forecast economics regimes using HMM 1 Inflation (CPI) 2 Credit Index 3 Yield Curve 4 Commodity 5 Dow Jones Industrial Average HMM assumptions: There are two states represent Bull and Bear market. The observation corresponding with each state follows a normal distribution.

19 Training and Predicting Process Using the variables above: Use HMM for single and multiple observation data with normal distributions. Calibrate Markov-switching model parameters using Baum-Welch algorithm Define state or regime 2 with lower mean/variance Use the obtained parameters to predict the corresponding states (regimes), predict the upcoming regime.

20 Results HMM Bear Market (monthly 5/2006 5/2013) Normalized data DJIA NDR Bear Market HMM Bear Market Time Figure : Dow Jones observations vs probabilities of being in the bear market

21 Results Figure : Forecast bear market using CPI indicator

22 Results HMM Forecast Bear Market (monthly 10/2006 5/2013) Normalized data DJIA Credit Index Yield Curve Commodity HMM Bear Market Time Figure : Forecast bear market using multiple observations

23 Forecast stock price using HMM S&P 500, a stock market index based on the market capitalizations of 500 large companies having common stock listed on the NYSE or NASDAQ. Monthly percentage changes from February 1947 through June SPY GOOG FORD AAPL GE

24 Training and Predicting Process Using the variables above: Use HMM for single and multiple observation data with normal distributions. Calibrate Markov-switching model parameters using Baum-Welch algorithm Use the obtained parameters to predict stock prices for the next trading period.

25 S&P500 Using Close Prices 7/30/2012 7/31/2013 S&P500 Prices True price Estimated price Times Figure : Forecast S&P500 close prices using single observation

26 S&P500 Using Close Open High Low 7/30/2012 7/31/2013 S&P500 Prices True price Estimated price Times Figure : Forecast S&P500 closing prices using multiple observations (open-close-high-low)

27 SPY 10:51:52 to 10:53:41 on 1/7/2011 S&P500 Prices True price Estimated price Times Figure : Forecast SPY bid price in tick by tick

28 Can we use HMMs to make money? Symbol Initial Investment ($) Earning ($) Earning % SPY 9, GOOG 30, , FORD AAPL GE 1, TOTAL 41, , Table : One year daily stock trading portfolio from December 2012 to December 2013

29 HMM for stock selections

30 Stock Factors

31 HMM for stock selections 1 Each month, look at regimes of the four macro variables, e.g. {CPI, SP500, VIX, GDP} = {2, 1, 1, 2} 2 Look back all months with the same regimes {2, 1, 1, 2} and check factor performances and then rank factor performances (factor did well for that regime will have higher rank and higher weight) 3 Add all factor s ranks to find a composite score (from 0 to 100) for each stock 4 Pick top 50 stocks

32 Economic Growth (GDP) 15,849 14,962 14,125 13,335 12,589 11,885 11,220 10,593 10,000 9,441 Growth (Quarterly GDP Growth Rate) - 2 Regimes Monthly Data to (Log Scale) 15,849 14,962 14,125 13,335 12,589 11,885 11,220 10,593 10,000 9,441 8, ,913 Regime Parameters ( ) Mu Sigma Regime 1 Regime 2 (Unshaded) (Shaded) Regime 1 Regime 2 (Unshaded) (Shaded) Data Statistics Mean Variance Regime 1 Regime 2 (Unshaded) (Shaded) Regime 1 Regime 2 (Unshaded) (Shaded)

33 1, Top Decile of Cash/Enterprise Value vs. S&P 500 Title Gain/ Standard Downside Batting Sharpe Info Tracking Annum Deviation Deviation Average Ratio Ratio Error Monthly Data to (Log Scale) 50 *Not Including Transaction Costs. 40 *Equity Lines Start at 100 on Top Decile of Cash/Enterprise Value ( ) = Rescaled S&P 500 Index ( ) = Excess Return Cumulative Excess Return (1/10 Scale) Max Drawdown Top Decile of Cash/Enterprise Value 14.2% 24.5% 19.2% 59.4% % -65.7% ( ) S&P 500 Index 2.3% 15.7% 12.3% % ( ) 1,

34 Top Decile of 1-Month Momentum vs. S&P Top Decile of 1-Month Momentum ( ) = Rescaled S&P 500 Index ( ) = Title Gain/ Standard Downside Batting Sharpe Info Tracking Annum Deviation Deviation Average Ratio Ratio Error Monthly Data to (Log Scale) Excess Return Cumulative Excess Return (1/10 Scale) *Not Including Transaction Costs. *Equity Lines Start at 100 on Max Drawdown Top Decile of 1-Month Momentum 5.0% 22.1% 16.3% 51.7% % -60.1% ( ) S&P 500 Index 2.3% 15.7% 12.3% % ( )

35 Factor Weight Monthly Data to Earnings/Price Weight = 0.13 Free Cash Flow/Enterprise Value Weight = Sales/Enterprise Value Weight = Month Momentum Weight = Month Momentum Weight =

36 Top Decile of Model Composite Score vs. S&P 500 Monthly Data to (Log Scale) 631 Top Decile of Model Composite Score ( ) = Rescaled S&P 500 Index ( ) = *Not Including Transaction Costs. *Equity Lines Start at 100 on Excess Return 30 Cumulative Excess Return (1/10 Scale) Title Gain/ Standard Downside Batting Sharpe Info Tracking Max Annum Deviation Deviation Average Ratio Ratio Error Drawdown Top Decile of Model Composite Score 11.1% 21.8% 18.1% 58.9% % -61.9% ( ) S&P 500 Index 2.3% 15.7% 12.3% % ( )

37 Thank you! : ntnguyen01@ysu.edu Department of Mathematics & Statistics Youngstown State University