Introduction to Algorithmic Trading Strategies - PowerPoint Presentation

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Introduction to Algorithmic Trading StrategiesLecture 5

Pairs Trading by Stochastic Spread Methods

Haksun Li

haksun.li@numericalmethod.com

www.numericalmethod.comSlide2

OutlineFirst passage time

Kalman filterMaximum likelihood estimateEM algorithm

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References

As the emphasis of the basic co-integration methods of most papers are on the construction of a synthetic mean-reverting asset, the stochastic spread methods focuses on the dynamic of the price of the synthetic asset.Most referenced academic paper: Elliot, van der Hoek, and Malcolm, 2005, Pairs Trading

Model the spread process as a state-space version of Ornstein-Uhlenbeck processJonathan Chiu, Daniel Wijaya Lukman, Kourosh Modarresi

,

Avinayan

Senthi

Velayutham. High-frequency Trading. Stanford University. 2011The idea has been conceived by a lot of popular pairs trading booksTechnical analysis and charting for the spread, Ehrman, 2005, The Handbook of Pairs TradingARMA model, HMM ARMA model, some non-parametric approach, and a Kalman filter model, Vidyamurthy, 2004, Pairs Trading: Quantitative Methods and Analysis

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Spread as a Mean-Reverting Process

The long term mean =

.

The rate of mean reversion =

.

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Sum of Power Series

We note that

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Unconditional Mean

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Long Term Mean

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Unconditional Variance

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Long Term Variance

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Observations and Hidden State Process

The hidden state process is:

The observations:

We want to compute the

expected

state from observations.

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First Passage Time

Standardized Ornstein-Uhlenbeck process

First passage time

The

pdf

of

has a maximum value at

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A Sample Trading Strategy

,

Buy when

unwind after time

Sell

when

unwind after time

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Kalman Filter

The Kalman filter is an efficient recursive filter that estimates the state of a dynamic system from a series of incomplete and noisy

measurements.13Slide14

Conceptual Diagram

prediction at time

t

Update at time

t+1

as new measurements come in

correct for better estimation

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A Linear Discrete System

: the state transition model applied to the previous state

: the control-input model applied to control vectors

: the noise process drawn from multivariate Normal distribution

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Observations and Noises

: the observation model mapping the true states to observations

: the observation noise

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Discrete System Diagram

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Prediction

predicted a prior state estimate

predicted a prior

estimate covariance

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Update

measurement residual

residual covariance

optimal

Kalman

gain

updated a posteriori state estimate

updated

a posteriori

estimate covariance

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Computing the ‘Best’ State Estimate

Given ,

, , , we define the conditional variance

Start with

,

.

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Predicted (a Priori) State Estimation

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Predicted (a Priori) Variance

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Minimize Posteriori Variance

Let the Kalman updating formula be

We want to solve for K such that the conditional variance is minimized.

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Solve for K

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First Order Condition for k

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Optimal Kalman Filter

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Updated (a Posteriori) State Estimation

So, we have the “optimal” Kalman

updating rule.

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Updated (a Posteriori) Variance

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Parameter Estimation

We need to estimate the parameters

from the observable data before we can use the

Kalman

filter model.

We need to write down the likelihood function in terms of

, and then maximize w.r.t.

.

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Likelihood Function

A likelihood function (often simply the likelihood) is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values.

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Maximum Likelihood Estimate

We find such that

is maximized given the observation.

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Example Using the Normal Distribution

We want to estimate the mean of a sample of size

drawn from a Normal distribution.

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Log-Likelihood

Maximizing the log-likelihood is equivalent to maximizing the following.

First order condition w.r.t.,

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Nelder-Mead

After we write down the likelihood function for the Kalman model in terms of

, we can run any multivariate optimization algorithm, e.g.,

Nelder

-Mead, to search for

.

The disadvantage is that it may not converge well, hence not landing close to the optimal solution.

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Marginal Likelihood

For the set of hidden states,

, we write

Assume we know the conditional distribution of

, we could instead maximize the following.

, or

The expectation is a weighted sum of the (log-) likelihoods weighted by the probability of the hidden states.

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The Q-Function

Where do we get the conditional distribution of

from?Suppose we somehow have an (initial) estimation of the parameters,

. Then the model has no unknowns. We can compute the distribution of

.

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EM Intuition

Suppose we know , we know completely about the mode; we can find

Suppose we know , we can estimate

, by, e.g., maximum likelihood.

What do we do if we don’t know both

and

?

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Expectation-Maximization Algorithm

Expectation step (E-step): compute the expected value of the log-likelihood function, w.r.t., the conditional distribution of

under and .

Maximization step (M-step): find the parameters,

, that maximize the Q-value.

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EM Algorithms for Kalman Filter

Offline: Shumway and Stoffer smoother approach, 1982Online:

Elliott and Krishnamurthy filter approach, 199939Slide40

A Trading Algorithm

From

, , …,

, we estimate

.

Decide whether to make a trade at

, unwind at

, or some time later, e.g.,

.

As

arrives, estimate

.Repeat. 40Slide41

Results (1)

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Results (2)

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Results (3)

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