Introduction to Algorithmic Trading Strategies
Lecture 2
Hidden Markov Trading Model
Haksun Li haksun.li@numericalmethod.com
www.numericalmethod.com

 Carry trade
 Momentum
 Valuation
 CAPM
 Markov chain
 Hidden Markov model
2

 Algorithmic Trading: Hidden Markov Models on
Foreign Exchange Data. Patrik Idvall, Conny Jonsson.
University essay from Linköpings universitet/Matematiska institutionen; Linköpings universitet/Matematiska institutionen. 2008.
 A tutorial on hidden Markov models and selected applications in speech recognition. Rabiner, L.R.
Proceedings of the IEEE, vol 77 Issue 2, Feb 1989.
3

 FX is the largest and most liquid of all financial markets – multiple trillions a day.
 FX is an OTC market, no central exchanges.






The major players are:
Central banks
Investment and commercial banks
Non-bank financial institutions
Commercial companies
Retails
4

 Reuters
 EBS (Electronic Broking Service)
 Currenex
 FXCM
 FXall
 Hotspot
 Lava FX
5

 Brokerage
 Transaction, e.g., bid-ask
6

 Carry trade
 Momentum
 Valuation
7

 Capture the difference between the rates of two currencies.
 Borrow a currency with low interest rate.
 Buy another currency with higher interest rate.
 Take leverage, e.g., 10:1.
 Risk: FX exchange rate goes against the trade.
 Popular trades: JPY vs. USD, USD vs. AUD
 Worked until 2008.
8




FX tends to trend.
Long when it goes up.
Short when it goes down.
 Irrational traders
 Slow digestion of information among disparate participants
9

 McDonald’s hamburger as a currency.
 The price of a burger in the USA = the price of a burger in Europe


E.g., USD1.25/burger = EUR1/burger
EURUSD = 1.25
10

 Deutsche Bank Currency Return (DBCR) Index

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A combination of
Carry trade
Momentum
Valuation
11

 Individual expected excess return is proportional to the market expected excess return.




𝐸 𝑟 𝑖
− 𝑟 𝑓
= 𝛽 𝑓
𝐸 𝑟
𝑀
− 𝑟 𝑓 𝑟 𝑖
, 𝑟
𝑀 𝑟 𝑓 are geometric returns is an arithmetic return

Sensitivity 𝛽 𝑓
=
𝐶𝑜𝑣 𝑟 𝑖
,𝑟
𝑀
𝑉𝑎𝑟 𝑟
𝑀
12

 Alpha is the excess return above the expected excess return.

 𝛼 = 𝑟 − 𝐸 𝑟 𝑖
For FX, we usually assume 𝑟 𝑓
= 0 .
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




Bayes theorem computes the posterior probability of a hypothesis H after evidence E is observed in terms of the prior probability, 𝑃 𝐻 the prior probability of E, 𝑃 𝐸 the conditional probability of 𝑃 𝐸|𝐻
𝑃 𝐻|𝐸 =
𝑃 𝐸|𝐻 𝑃 𝐻
𝑃 𝐸
14

15 a11 = 0.4
a12 = 0.2
s1: UP a21 = 0.3
a22 = 0.2
s2:
MEAN-
REVERTIN
G a32 = 0.25
a23 = 0.5
a13 = 0.4
a31 = 0.25
s3:
DOWN a33 = 0.5






What is the probability of observing
 Ω = 𝑠
3
, 𝑠
1
, 𝑠
1
, 𝑠
1
𝑃 Ω|Model = P 𝑠
3
, 𝑠
1
, 𝑠
1
, 𝑠
1
|Model
= P 𝑠
P 𝑠
1
3
|𝑠
1
|Model × P 𝑠
1
,Model
|𝑠
3
,Model × P 𝑠
1
|𝑠
1
,Model ×
= 1 × 0.25
× 0.4
× 0.4
= 0.04
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


Given the current information available at time 𝑡 − 1 , the history, e.g., path, is irrelevant.
𝑃 𝑞 𝑡
|𝑞 𝑡−1
, ⋯ , 𝑞
1
= 𝑃 𝑞 𝑡
|𝑞 𝑡−1
Consistent with the weak form of the efficient market hypothesis.
17

 Only observations are observable (duh).


World states may not be known (hidden).
We want to model the hidden states as a Markov Chain.



Two assumptions:
Markov property
𝑃 𝜔 𝑡
|𝑞 𝑡−1
, ⋯ , 𝑞
1
, 𝜔 𝑡−1
, ⋯ , 𝜔
1
= 𝑃 𝜔 𝑡
|𝑞 𝑡
18

19 a11 = ?
a12 = ?
s1: UP a21 = ?
a22 = ?
s2:
MEAN-
REVERTIN
G a32 = ?
a23 = ?
a13 = ?
a31 = ?
s3:
DOWN a33 = ?



Likelihood
Given the parameters, λ, and an observation sequence, Ω, compute 𝑃 Ω|𝜆 .


Decoding
Given the parameters, λ, and an observation sequence, Ω, determine the best hidden sequence Q.


Learning
Given an observation sequence, Ω, and HMM structure, learn λ.
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21






𝑃 Ω|𝜆 = 𝑞 ′ 𝑠
𝑃 Ω, 𝑄|𝜆
= 𝑞 ′ 𝑠
𝑃 Ω|𝑄, 𝜆 × 𝑃 𝑄|𝜆
𝑃 Ω|𝑄, 𝜆 =
𝑇 𝑡=1
𝑃 𝜔 𝑡
|𝑞 𝑡
, 𝜆
𝑃 𝑄|𝜆 = 𝜋 𝑞
1
× 𝑎 𝑞
1 𝑞
2
× 𝑎 𝑞
2 𝑞
3
× ⋯ × 𝑎 𝑞
𝑇−1
But… this is not computationally feasible.
𝑞
𝑇
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



 𝛼 𝑡 𝑖 = 𝑃 𝜔
1
, 𝜔
2
, ⋯ , 𝜔 𝑡
, 𝑞 𝑡
= 𝑠 𝑖
|𝜆 the probability of the partial observation sequence until time t and the system in state 𝑠 𝑖 at time t.

Initialization 𝛼
1
 𝑖 = 𝜋 𝑖 𝑏 𝑖 𝜔
1 𝑏 𝑖
: the conditional distribution of 𝜔

Induction 𝛼 𝑡+1 𝑗 =
𝑁 𝑖=1 𝛼 𝑡 𝑖 𝑎 𝑖𝑗 𝑏 𝑗 𝜔 𝑡+1

Termination
𝑃 Ω|𝜆 =
𝑁 𝑖=1 𝛼
𝑇 𝑖 , the likelihood
23


 𝛽 𝑡 𝑖 = 𝑃 𝜔 𝑡+1
, 𝜔 𝑡+2
, ⋯ , 𝜔
𝑇
|𝑞 𝑡
= 𝑠 𝑖
, 𝜆 the probability of the system in state 𝑠 𝑖 at time t, and the partial observations from then onward till time t


Initialization
 𝛽
𝑇 𝑖 = 1

Induction 𝛽 𝑡 𝑖 =
𝑁 𝑗=1 𝑎 𝑖𝑗 𝑏 𝑗 𝜔 𝑡+1 𝛽 𝑡+1 𝑗
24

25






Given the observations and model, the probability of the system in state 𝑠 𝑖 is: 𝛾
= 𝑡 𝑖 = 𝑃 𝑞 𝑡
𝑃 𝑞 𝑡
=𝑠 𝑖
,Ω|𝜆
𝑃 Ω|𝜆
= 𝑠 𝑖
|Ω, 𝜆 𝑖
= 𝛼 𝑡 𝑖 𝛽 𝑡
𝑃 Ω|𝜆
= 𝛼 𝑡
𝑁 𝑖=1 𝑖 𝛽 𝛼 𝑡 𝑡 𝑖 𝑖 𝛽 𝑡 𝑖
26


 𝑞 𝑡
= argmax
1≤𝑖≤𝑁 𝛾 𝑡 𝑖
This determines the most likely state at every instant, t, without regard to the probability of occurrence of sequences of states.
27

 The maximal probability of the system travelling these states and generating these observations:
 𝛿 𝑡
= max 𝑃 𝑞
1
, 𝑞
2
, ⋯ , 𝑞 𝑡
= 𝑠 𝑖
, 𝜔
0
, ⋯ , 𝜔 𝑡
|𝜆
28



Initialization
 𝛿
1 𝑖 = 𝜋 𝑖 𝑏 𝑖
Recursion 𝜔
1

 𝛿 𝑡
 𝑗 = max 𝛿 𝑡−1 𝑖 𝑎 𝑖𝑗 𝑏 𝑗 𝜔 𝑡 the probability of the most probable state sequence for the first t observations, ending in state j 𝜓 𝑡
 𝑗 = argmax 𝛿 𝑡−1 the state chosen at t 𝑖 𝑎 𝑖𝑗



Termination
𝑃 ∗ = max 𝛿
𝑇 𝑞 ∗ = argmax 𝛿
𝑇 𝑖 𝑖
29

30

 Our objective is to find λ that maximizes 𝑃 Ω|𝜆 .
 For any given λ, we can compute 𝑃 Ω|𝜆 .
 Then solve a maximization problem.
 Algorithm: Nelder-Mead.
31

 the probability of being in state 𝑠 𝑖 𝑠 𝑗 at time 𝑡 , and state at time 𝑡 + 1 , given the model and the observation sequence
 𝜉 𝑡 𝑖, 𝑗 = 𝑃 𝑞 𝑡
= 𝑠 𝑖
, 𝑞 𝑡+1
= 𝑠 𝑗
|Ω, 𝜆
32

 𝜉 𝑡 𝑖, 𝑗 = 𝑃 𝑞 𝑡
= 𝑠 𝑖
, 𝑞 𝑡+1
= 𝑠 𝑗
|Ω, 𝜆




=
𝑃 𝑞 𝑡
=𝑠 𝑖
,𝑞 𝑡+1
=𝑠 𝑗
,Ω|𝜆
𝑃 Ω|𝜆
= 𝛼 𝑡 𝑖 𝑎 𝑖𝑗 𝑏 𝑗 𝜔 𝑡+1
𝑃 Ω|𝜆 𝛽 𝑡+1 𝑗 𝛾 𝑡 𝑖 = 𝑃 𝑞 𝑡
=
𝑁 𝑗=1 𝜉 𝑡 𝑖, 𝑗
= 𝑠 𝑖
|Ω, 𝜆
33

 By summing up over time,

 𝛾 𝑡 𝑖 ~ the number of times 𝑠 𝑖 is visited 𝜉 𝑡 𝑖, 𝑗 state 𝑠 𝑖
~ the number of times the system goes from to state 𝑠 𝑗
 Thus, the parameters λ are:


 𝜋 𝑖 𝑎 𝑖𝑗 𝑏 𝑗
= 𝛾
1 𝑣
= 𝑘 𝑖 , initial state probabilities
𝑇−1 𝑡=1
𝑇−1 𝑡=1 𝜉 𝑡 𝛾 𝑡 𝑖,𝑗 𝑖
=
, transition probabilities
𝑇−1 𝑡=1,𝜔𝑡=𝑣𝑘
𝑇−1 𝑡=1 𝛾 𝑡 𝛾 𝑡 𝑗 𝑗
, conditional probabilities
34

 Guess what λ is.


Compute λ 𝑖 using the estimation equations.
Practically, we can estimate the initial λ by Nelder-
Mead to get “closer” to the solution.
35

 Our formulation so far assumes discrete conditional probabilities.


The formulations that take other probability density functions are similar.
But the computations are more complicated, and the solutions may not even be analytical, e.g., t-distribution.
36

 t-distribution


Gaussian Mixture Model a weighted sum of Normal distributions
37

 Compute the next state.
 Compute the expected return.
 Long (short) when expected return > (<) 0.


Long (short) when expected return > (<) c.
c = the transaction costs
 Any other ideas?
38

 EURUSD daily prices from 2003 to 2006.
 6 unknown factors.
 Λ is estimated on a rolling basis.






Evaluations:
Hypothesis testing
Sharpe ratio
VaR
Max drawdown alpha
39

40

41

 More data (the 6 factors) do not always help (esp. for the discrete case).
 Parameters unstable.
42

 How can we improve the HMM model(s)? Ideas?
43