Hidden Markov Model for High Frequency Data

Transcription

1 Hidden Markov Model for High Frequency Data Department of Mathematics, Florida State University Joint Math Meeting, Baltimore, MD, January 15

2 What are HMMs? A Hidden Markov model (HMM) is a stochastic signal model which has 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.

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

5 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 = 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 i (k) = 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

6 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}

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

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

9 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

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 Viterbi algorithm

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 3 Given O, calibrate HMM parameters, λ

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 Baum-Welch algorithm

13 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

14 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)

15 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

16 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)

17 Choose economics indicators 1 Inflation (CPI) 2 Credit Index 3 Yield Curve 4 Commodity 5 Dow Jones Industrial Average

18 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.

19 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

20 Results Figure : Forecast bear market using CPI indicator

21 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

22 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

23 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.

24 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

25 Results 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)

26 Results 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

27 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

28 Thank you! : nnguyen@math.fsu.edu Department of Mathematics, Florida State University