Introduction to Algorithmic Trading StrategiesLecture 4

Optimal Pairs Trading by Stochastic Control

Haksun Lihaksun.li@numericalmethod.com

www.numericalmethod.com

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Outline Problem formulation Ito’s lemma Dynamic programming Hamilton-Jacobi-Bellman equation Riccati equation Integrating factor

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Reference Optimal Pairs Trading: A Stochastic Control

Approach. Mudchanatongsuk, S., Primbs, J.A., Wong, W. Dept. of Manage. Sci. & Eng., Stanford Univ., Stanford, CA.

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Basket Creation vs. Trading In lecture 3, we discussed a few ways to

construct a mean-reverting basket. In this and the next lectures, we discuss how

to trade a mean-reverting asset, if such exists.

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Stochastic Control We model the difference between the log-

returns of two assets as an Ornstein-Uhlenbeck process.

We compute the optimal position to take as a function of the deviation from the equilibrium.

This is done by solving the corresponding the Hamilton-Jacobi-Bellman equation.

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Formulation Assume a risk free asset, which satisfies

Assume two assets, and . Assumefollows a geometric Brownian motion.

is the spread between the two assets.

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Ornstein-Uhlenbeck Process We assume the spread, the basket that we

want to trade, follows a mean-reverting process.

is the long term equilibrium to which the spread reverts.

is the rate of reversion. It must be positive to ensure stability around the equilibrium value.

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Instantaneous Correlation Let denote the instantaneous correlation

coefficient between and .

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Univariate Ito’s Lemma

Assume is twice differentiable of two real variables

We have

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Log example For G.B.M., ,

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Multivariate Ito’s Lemma

Assume is a vector Ito process is twice differentiable

We have

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Multivariate Example

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What is the Dynamic of Asset At?

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Dynamic of Asset At

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Notations : the value of a self-financing pairs trading

portfolio :the portfolio weight for stock A :the portfolio weight for stock B

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Self-Financing Portfolio Dynamic

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Power Utility Investor preference:

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Problem Formulation , s.t., ,

Note that we simplify GBM to BM of , and remove some constants.

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Dynamic Programming

Consider a stage problem to minimize (or maximize) the accumulated costs over a system path.

s0

s11

s12

timet = 0 t = 1

S21

S22

s23

t = 2

3

2

3

5

Cost =

s3

s12

5

4

3

1

5

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Dynamic Programming Formulation State change:

: time : state : control decision selected at time : a random noise

Cost: Objective: minimize (maximize) the expected

cost. We need to take expectation to account for the

noise, .

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Principle of Optimality Let be an optimal policy for the basic

problem, and assume that when using , a give state occurs at time with positive probability. Consider the sub-problem whereby we are at at time and wish to minimize the “cost-to-go” from time to time .

Then the truncated policy is optimal for this sub-problem.

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Dynamic Programming Algorithm For every initial state , the optimal cost of

the basic problem is equal to , given by the last step of the following algorithm, which proceeds backward in time from period to period

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Value function Terminal condition:

DP equation:

By Ito’s lemma:

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Hamilton-Jacobi-Bellman Equation Cancel on both LHS and RHS. Divide by time discretization, . Take limit as , hence Ito.

The optimal portfolio position is .

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HJB for Our Portfolio Value

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Taking Expectation All disapper because of the expectation

operator. Only the deterministic terms remain. Divide LHR and RHS by .

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Dynamic Programming Solution Solve for the cost-to-go function, . Assume that the optimal policy is .

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First Order Condition Differentiate w.r.t. .

In order to determine the optimal position, , we need to solve for to get , , and .

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The Partial Differential Equation (1)𝐺𝑡−

𝐺𝑉 (𝐺𝑉 (𝑘 (𝜃−𝑥𝑡 ) )+𝐺𝑉𝑥𝜂2

𝐺𝑉𝑉 𝜂2 𝑘 (𝜃− 𝑥𝑡 ))+¿

𝐺𝑥 (𝑘 (𝜃−𝑥𝑡 ))+¿ 12𝐺𝑉𝑉 𝜂

2(𝐺𝑉 (𝑘 (𝜃−𝑥𝑡 ))+𝐺𝑉𝑥𝜂2

𝐺𝑉𝑉 𝜂2 )

2

+¿ 12𝐺𝑥𝑥𝜂

2−

𝐺𝑉𝑥𝜂2𝐺𝑉 (𝑘 (𝜃− 𝑥𝑡 ))+𝐺𝑉𝑥𝜂2

𝐺𝑉𝑉 𝜂2

=0

¿

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The Partial Differential Equation (2)𝐺𝑡−

𝐺𝑉 𝑘 (𝜃−𝑥𝑡 )𝐺𝑉 (𝑘 (𝜃−𝑥𝑡 ))+𝐺𝑉𝑥𝜂 2

𝐺𝑉𝑉 𝜂2 +¿

𝐺𝑥 (𝑘 (𝜃−𝑥𝑡 ))+¿12

[𝐺𝑉 (𝑘 (𝜃−𝑥𝑡 ))+𝐺𝑉𝑥𝜂2 ]2

𝐺𝑉𝑉 𝜂2+¿12𝐺𝑥𝑥𝜂2−

𝐺𝑉𝑥𝐺𝑉 (𝑘 (𝜃−𝑥𝑡 ) )+𝐺𝑉𝑥𝜂

2

𝐺𝑉𝑉

=0

¿

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Dis-equilibrium Let . Rewrite:

Multiply by .

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Simplification Note that The PDE becomes

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The Partial Differential Equation (3)

𝐺𝑡𝐺𝑉𝑉 𝜂2+𝐺𝑥𝑏𝐺𝑉𝑉 𝜂2+12 𝐺𝑥𝑥𝐺𝑉𝑉 𝜂 4−

12 (𝐺𝑉 𝑏+𝐺𝑉𝑥𝜂2 )

2=0

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Ansatz for G

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Another PDE (1)

Divide by .

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Ansatz for

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Boundary Conditions

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Yet Another PDE (1)

Divide by .

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Yet Another PDE (2)

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Expansion in

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Grouping in

(𝑔𝑡+𝑘𝑔 𝛽𝜃+12 𝜂

2𝑔𝛽2+𝜂2𝑔𝛼− 𝜆𝑔𝜂2

𝑘2𝜃2− 𝜆𝑔𝜂2 𝛽2−2 𝜆𝑔𝑘 𝛽𝜃 )+(𝑔𝛽𝑡−𝑘𝑔 𝛽+2𝛼𝑘𝑔 𝜃+2𝜂2𝑔𝛼𝛽+2 𝜆𝑔𝜂2

𝑘2𝜃+2 𝜆𝑔𝑘 𝛽−4𝜆𝑔𝑘 𝜃𝛼−4 𝜆𝑔𝜂2𝛼𝛽)𝑥+(𝑔𝛼 𝑡−2𝛼𝑘𝑔+2𝜂 2𝑔𝛼2− 𝜆𝑔𝜂2

𝑘2−4 𝜆𝑔𝜂2𝛼2+4 𝜆𝑔𝑘𝛼)𝑥2=0

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The Three PDE’s (1)

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PDE in

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PDE in ,

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PDE in , ,

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Riccati Equation A Riccati equation is any ordinary differential

equation that is quadratic in the unknown function.

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Solving a Riccati Equation by Integration Suppose a particular solution, , can be found. is the general solution, subject to some

boundary condition.

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Particular Solution Either or is a particular solution to the ODE.

This can be verified by mere substitution.

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Substitution Suppose .

goes to 0 by the definition of

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Solving

1st order linear ODE

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Solving for

boundary condition:

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Solution (1)

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Solution (2)

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Solution (3)

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Solving

Let

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First Order Non-Constant Coefficients

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Integrating Factor (1)

We try to find an integrating factor s.t.

Divide LHS and RHS by .

By comparison,

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Integrating Factor (2)

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Solution

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,

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Solution

𝛽 (𝑡 )=𝑘𝜃𝜂2

(1+√1−𝛾 )exp ( 2𝑘

√1−𝛾(𝑇 −𝑡 ))−1

1+[1− 21−√1−𝛾 ]exp( 2𝑘

√1−𝛾(𝑇 −𝑡 ))

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Solving

With and solved, we are now ready to solve .

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Computing the Optimal Position

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The Optimal Position

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P&L for Simulated Data

The portfolio increases from $1000 to $4625 in one year.

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Parameter Estimation Can be done using Maximum Likelihood. Evaluation of parameter sensitivity can be

done by Monte Carlo simulation. In real trading, it is better to be conservative

about the parameters. Better underestimate the mean-reverting speed Better overestimate the noise

To account for parameter regime changes, we can use: a hidden Markov chain model moving calibration window