Post on 01-Mar-2021
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Hidden Markov Models for Financial MarketPredictions
Nguyet Nguyen
Department of Mathematics and StatisticsYoungstown State University
Central Spring Sectional Meeting, Michigan State University,March 15
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
1 Introduction of HMMs
2 HMMs for economics regimes
3 HMMs for stock prices
4 HMM for stock sections
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
History of HMMs
Introduced in 1966 by Baum and Petrie
Baum and his colleagues published HMM training for a singleobservation, 1970
Levonson, Rabiner, and Sondhi presented HMM training formultiple independent observations, 1983
Li, Parizeau, and Plamondo introduced HMM traning formultiple observations, 2000
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
What is a Hidden Markov Model?
Hidden Markov Model (HMM): stochastic signal model with threeassumptions:
The observation at time t, Ot , was generated by some processwhose state, St , is hidden.The hidden process satisfies the first-order Markov property:given St−1, St is independent of Si for any i < t − 1.The hidden state variable is discrete.
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Some applications of HMMs
Figure : 1. Speech recognition 2. Bioinformatics 3. Finance
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Elements of HMM
Observation data, O = (Ot), t = 1, ..,T
Hidden states, S = (Si ), i = 1, 2, ...,N
Hidden state sequence: Q = (qt), t = 1, ...,T
Transition matrix A = (aij)
aij = P(qt = Sj |qt−1 = Si ), i , j = 1, 2, ...,N
Observation symbols per state, V = (vk), k = 1, 2, ...,M
The observation probability B = (bik)
bik = P(Ot = vk |qt = Si ), i = 1, 2, ...,N; k = 1, 2, ...,M
Initial probabilities, vector p, of being in state Si at t = 1
pi = P(q1 = Si ), i = 1, 2, ...,N
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Hidden Markov Model
S2
S1 S1
S2
t t+1
a11
a12
a21
a22
Ot Ot+1
b1(Ot)
b2(Ot)
Parameters of HMM: λ = {A,B, p}
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
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
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
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
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
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
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
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
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
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
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
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
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Forward Algorithm
Define the joint probability αt(i) = P(O1,O2, ...,Ot , qt = Si |λ)
Si
t-1 t
t(i)
forwardNguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Forward algorithm
Initialization, α1(i) = pibi (O1) for i = 1, ...,N
For t = 2, 3, ...,T , for j = 1, ...,N
αt(j) =
[N∑i=1
αt−1(i)aij
]bj(Ot),
P(O|λ) =∑N
i=1 αT (i)
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Backward Algorithm
Define the conditional probabilityβt(j) = P(Ot+1,Ot+2, ..,OT |qt = Sj , λ), for j = 1, ...,N
Sj
t+1 t+2
t+1(j)
backwardNguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Backward Algorithm
Algorithm
Initialization, βT (i) = 1 for i = 1, ...,N
For t = T − 1,T − 2, ..., 1, for i = 1, ...,N
βt(i) =N∑j=1
aijbj(Ot+1)βt+1(j)
P(O|λ) =∑N
i=1 pibi (O1)β1(i)
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
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 anormal distribution.
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Training and Predicting Process
Using the variables above:
Use HMM for single and multiple observation data withnormal distributions.
Calibrate Markov-switching model parameters usingBaum-Welch algorithm
Define state or regime 2 with lower mean/variance
Use the obtained parameters to predict the correspondingstates (regimes), predict the upcoming regime.
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Results
HMM Bear Market (monthly 5/2006−5/2013)
Time
Nor
mal
ized
dat
a
2007 2008 2009 2010 2011 2012 2013
−3
−2
−1
01
2
DJIANDR Bear MarketHMM Bear Market
Figure : Dow Jones observations vs probabilities of being in the bearmarket
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Results
Figure : Forecast bear market using CPI indicator
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Results
HMM Forecast Bear Market (monthly 10/2006−5/2013)
Time
Nor
mal
ized
dat
a
2007 2008 2009 2010 2011 2012 2013
−3
−2
−1
01
2
DJIACredit IndexYield CurveCommodityHMM Bear Market
Figure : Forecast bear market using multiple observations
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Forecast stock price using HMM
S&P 500, a stock market index based on the marketcapitalizations of 500 large companies having common stocklisted on the NYSE or NASDAQ. Monthly percentage changesfrom February 1947 through June 2013.
SPY
GOOG
FORD
AAPL
GE
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Training and Predicting Process
Using the variables above:
Use HMM for single and multiple observation data withnormal distributions.
Calibrate Markov-switching model parameters usingBaum-Welch algorithm
Use the obtained parameters to predict stock prices for thenext trading period.
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
●
●●
●
●●
●●●●●●●
●●●
●●
●
●●●●
●
●●●
●
●
●●●
●
●●●●●●
●
●
●
●
●●●
●
●●●
●
●●●
●
●
●●
●●
●●●●●
●
●●
●
●
●●●
●
●●
●
●●●
●●
●
●
●●
●●●●●●
●●
●
●
●
●
●
●
●●
●●
●
●
●●
●●●●
●●●●●
●●
●●●
●●
●
●●
●
●
●●●
●●●●●●
●
●
●
●
●
●
●●●
●
●●●
●●●●
●●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●●
●
●
●●
●
●
●
●●
●
●
●
●●●
●●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●●
●
●
●●
●
●●
●●
●●●●
●●●
●●●
0 50 100 150 200 250
1300
1400
1500
1600
1700
S&P500 Using Close Prices 7/30/2012−7/31/2013
Times
S&
P50
0 P
rices
● True priceEstimated price
Figure : Forecast S&P500 close prices using single observation
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
●
●●
●
●●
●●●●●●●
●●●
●●
●
●●●●
●
●●●
●
●
●●●
●
●●●●●●
●
●
●
●
●●●
●
●●●
●
●●●
●
●
●●
●●
●●●●●
●
●●
●
●
●●●
●
●●
●
●●●
●●
●
●
●●
●●●●●●
●●
●
●
●
●
●
●
●●
●●
●
●
●●
●●●●
●●●●●
●●
●●●
●●
●
●●
●
●
●●●
●●●●●●
●
●
●
●
●
●
●●●
●
●●●
●●●●
●●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●●
●
●
●●
●
●
●
●●
●
●
●
●●●
●●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●●
●
●
●●
●
●●
●●
●●●●
●●●
●●●
0 50 100 150 200 250
1300
1400
1500
1600
1700
S&P500 Using Close−Open−High−Low 7/30/2012−7/31/2013
Times
S&
P50
0 P
rices
● True priceEstimated price
Figure : Forecast S&P500 closing prices using multiple observations(open-close-high-low)
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
●
●
● ●
●
● ●
● ●
● ● ● ● ●
● ● ●
●
● ● ●
● ●
●
●
● ●
●
● ● ●
●
●
● ● ●
●
● ● ●
● ● ● ●
●
●
● ● ●
● ● ●
● ● ●
●
● ● ● ● ●
● ● ● ● ● ● ●
●
●
● ●
● ●
● ● ●
● ● ● ● ● ● ● ●
● ● ● ●
●
● ● ● ● ● ● ● ● ● ●
● ● ● ● ●
● ●
● ● ●
0 20 40 60 80 100
126.
912
7.0
127.
112
7.2
127.
3
SPY 10:51:52 to 10:53:41 on 1/7/2011
Times
S&
P50
0 P
rices
● True priceEstimated price
Figure : Forecast SPY bid price in tick by tick
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Can we use HMMs to make money?
Symbol Initial Investment ($) Earning ($) Earning %
SPY 9,000.00 2050.66 22.79
GOOG 30,000.00 29,036.4 96.79
FORD 250.00 10.10 4.04
AAPL 950.00 19.06 2.01
GE 1,700.00 490.00 28.82
TOTAL 41,900.00 31,606.22 75.43
Table : One year daily stock trading portfolio from December 2012 toDecember 2013
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
HMM for stock selections
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Stock Factors
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
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} andcheck factor performances and then rank factor performances(factor did well for that regime will have higher rank andhigher weight)
3 Add all factor’s ranks to find a composite score (from 0 to100) for each stock
4 Pick top 50 stocks
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Economic Growth (GDP)
Growth (Quarterly GDP Growth Rate) - 2 Regimes Monthly Data 1990-01-31 to 2014-12-31 (Log Scale)
Regime Parameters (2014-12-31)
Mu Sigma
Regime 1 (Unshaded)
Regime 2 (Shaded)
Regime 1 (Unshaded)
Regime 2 (Shaded)
0.79 0.37 0.55 1.26
Data Statistics
Mean Variance
Regime 1 (Unshaded)
Regime 2 (Shaded)
Regime 1 (Unshaded)
Regime 2 (Shaded)
0.82 -0.00 0.16 0.60
Growth (Quarterly GDP Growth Rate) - 2 Regimes Monthly Data 1990-01-31 to 2014-12-31 (Log Scale)
Regime Parameters (2014-12-31)
Mu Sigma
Regime 1 (Unshaded)
Regime 2 (Shaded)
Regime 1 (Unshaded)
Regime 2 (Shaded)
0.79 0.37 0.55 1.26
Data Statistics
Mean Variance
Regime 1 (Unshaded)
Regime 2 (Shaded)
Regime 1 (Unshaded)
Regime 2 (Shaded)
0.82 -0.00 0.16 0.60
1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 2012 20148,913
9,441
10,000
10,593
11,220
11,885
12,589
13,335
14,125
14,962
15,849
8,913
9,441
10,000
10,593
11,220
11,885
12,589
13,335
14,125
14,962
15,849
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Top Decile of Cash/Enterprise Value vs. S&P 500 Monthly Data 1999-12-31 to 2014-12-31 (Log Scale)
TitleGain/
AnnumStandard Deviation
Downside Deviation
Batting Average
Sharpe Ratio
Info Ratio
Tracking Error
Max Drawdown
Top Decile of Cash/Enterprise Value 14.2% 24.5% 19.2% 59.4% 0.50 0.93 12.8% -65.7% (2007-05-31..2009-03-01)
S&P 500 Index 2.3% 15.7% 12.3% 0.03 -52.5% (2007-09-30..2009-03-01)
Top Decile of Cash/Enterprise Value vs. S&P 500 Monthly Data 1999-12-31 to 2014-12-31 (Log Scale)
TitleGain/
AnnumStandard Deviation
Downside Deviation
Batting Average
Sharpe Ratio
Info Ratio
Tracking Error
Max Drawdown
Top Decile of Cash/Enterprise Value 14.2% 24.5% 19.2% 59.4% 0.50 0.93 12.8% -65.7% (2007-05-31..2009-03-01)
S&P 500 Index 2.3% 15.7% 12.3% 0.03 -52.5% (2007-09-30..2009-03-01)
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
40
50
63
79
100
126
158
200
251
316
398
501
631
794
1,000
40
50
63
79
100
126
158
200
251
316
398
501
631
794
1,000Top Decile of Cash/Enterprise Value (2014-12-31) = 729.75Rescaled S&P 500 Index (2014-12-31) = 140.13
*Not Including Transaction Costs. *Equity Lines Start at 100 on 1999-12-31.
-10-505
1015202530354045505560
-10-505
1015202530354045505560Excess Return
Cumulative Excess Return (1/10 Scale)
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Top Decile of 1-Month Momentum vs. S&P 500 Monthly Data 1999-12-31 to 2014-12-31 (Log Scale)
TitleGain/
AnnumStandard Deviation
Downside Deviation
Batting Average
Sharpe Ratio
Info Ratio
Tracking Error
Max Drawdown
Top Decile of 1-Month Momentum 5.0% 22.1% 16.3% 51.7% 0.14 0.21 12.5% -60.1% (2007-09-30..2009-03-01)
S&P 500 Index 2.3% 15.7% 12.3% 0.03 -52.5% (2007-09-30..2009-03-01)
Top Decile of 1-Month Momentum vs. S&P 500 Monthly Data 1999-12-31 to 2014-12-31 (Log Scale)
TitleGain/
AnnumStandard Deviation
Downside Deviation
Batting Average
Sharpe Ratio
Info Ratio
Tracking Error
Max Drawdown
Top Decile of 1-Month Momentum 5.0% 22.1% 16.3% 51.7% 0.14 0.21 12.5% -60.1% (2007-09-30..2009-03-01)
S&P 500 Index 2.3% 15.7% 12.3% 0.03 -52.5% (2007-09-30..2009-03-01)
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
45
50
56
63
71
79
89
100
112
126
141
158
178
200
224
251
45
50
56
63
71
79
89
100
112
126
141
158
178
200
224
251Top Decile of 1-Month Momentum (2014-12-31) = 206.60Rescaled S&P 500 Index (2014-12-31) = 140.13
*Not Including Transaction Costs. *Equity Lines Start at 100 on 1999-12-31.
-12.5-10.0
-7.5-5.0-2.50.02.55.07.5
10.012.515.017.520.022.525.027.5
-12.5-10.0
-7.5-5.0-2.50.02.55.07.5
10.012.515.017.520.022.525.027.5Excess Return
Cumulative Excess Return (1/10 Scale)
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Factor Weight Monthly Data 1999-12-31 to 2014-12-31Factor Weight Monthly Data 1999-12-31 to 2014-12-31
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
0.10
0.15
0.20
0.25
0.30
0.10
0.15
0.20
0.25
0.30
Earnings/Price Weight 2014-12-31 = 0.13
0.15
0.20
0.25
0.30
0.15
0.20
0.25
0.30
Free Cash Flow/Enterprise Value Weight 2014-12-31 = 0.33
0.10
0.15
0.20
0.25
0.30
0.10
0.15
0.20
0.25
0.30
Sales/Enterprise Value Weight 2014-12-31 = 0.20
0.10
0.15
0.20
0.25
0.30
0.10
0.15
0.20
0.25
0.30
12-Month Momentum Weight 2014-12-31 = 0.27
0.10
0.15
0.20
0.25
0.30
0.10
0.15
0.20
0.25
0.30
1-Month Momentum Weight 2014-12-31 = 0.07
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Top Decile of Model Composite Score vs. S&P 500 Monthly Data 1999-12-31 to 2014-12-31 (Log Scale)
TitleGain/
AnnumStandard Deviation
Downside Deviation
Batting Average
Sharpe Ratio
Info Ratio
Tracking Error
Max Drawdown
Top Decile of Model Composite Score 11.1% 21.8% 18.1% 58.9% 0.42 0.82 10.8% -61.9% (2007-05-31..2009-03-01)
S&P 500 Index 2.3% 15.7% 12.3% 0.03 -52.5% (2007-09-30..2009-03-01)
Top Decile of Model Composite Score vs. S&P 500 Monthly Data 1999-12-31 to 2014-12-31 (Log Scale)
TitleGain/
AnnumStandard Deviation
Downside Deviation
Batting Average
Sharpe Ratio
Info Ratio
Tracking Error
Max Drawdown
Top Decile of Model Composite Score 11.1% 21.8% 18.1% 58.9% 0.42 0.82 10.8% -61.9% (2007-05-31..2009-03-01)
S&P 500 Index 2.3% 15.7% 12.3% 0.03 -52.5% (2007-09-30..2009-03-01)
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
50
63
79
100
126
158
200
251
316
398
501
631
50
63
79
100
126
158
200
251
316
398
501
631Top Decile of Model Composite Score (2014-12-31) = 484.87Rescaled S&P 500 Index (2014-12-31) = 140.13
*Not Including Transaction Costs. *Equity Lines Start at 100 on 1999-12-31.
-15
-10
-5
0
5
10
15
20
25
30
35
-15
-10
-5
0
5
10
15
20
25
30
35Excess ReturnCumulative Excess Return (1/10 Scale)
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions
Outline Introduction of HMMs HMMs for economics regimes HMMs for stock prices HMM for stock sections
Thank you!
Nguyet Nguyen: ntnguyen01@ysu.edu
Department of Mathematics & Statistics
Youngstown State University
Nguyet Nguyen Hidden Markov Models for Financial Market Predictions