Presentation about Statistical Arbitrage (Stat-Arb), using

45 Pages·2005·0.53 MB·English

Presentation about Statistical Arbitrage (Stat-Arb), using
Presentation about Statistical Arbitrage (Stat-Arb), using
Cointegration on the Equity Market
Cointegration on the Equity Market
By Yoann BOURGEOIS and Marc MINKO
By Yoann BOURGEOIS and Marc MINKO
Derivative Models Review Group (DMRG-Paris)
Derivative Models Review Group (DMRG-Paris)
HSBC-CCF
HSBC-CCF
DMRG-Paris - 17/06/05
PLAN
PLAN
(cid:1)
Introduction
(cid:1)
Part I: Mathematical Framework
(cid:1)
Part II: Description of the proposed
strategy and first results
(cid:1)
Conclusion
DMRG-Paris - 17/06/05
Introduction
Introduction
(cid:1)
Single stocks in the Equity Market generally are not stationary.
(cid:1)
But, their yields, in many cases are.
(cid:1)
From the econometrical point of view, they are generally told to be
Integrated of order 1.
(cid:1)
Cointegration is a mathematical theory that helps to handle the problem
generated by non-stationary data.
(cid:1)
With the help of this theory, we propose to build linear combinations of
these single stocks that are stationary.
(cid:1)
Such combinations can be traded and are called synthetic assets.
(cid:1)
Eventually, these stationary assets have the mean reversion property and
we will use this property in order to set up arbitrage strategies.
DMRG-Paris - 17/06/05
Part I: Mathematical Framework
Part I: Mathematical Framework
(cid:1)
Description of the
framework of our strategy
(cid:1)
Statistical Analysis of
models that are Integrated of
order 1 (ie I(1))
DMRG-Paris - 17/06/05
Description of the framework of our strategy (1)
Description of the framework of our strategy (1)
(cid:2)
Reminder about Vector AutoRegressive models (VAR)
(cid:1)
In what follows, we consider a VAR process X (p 1), which can be
t
written:
k
(cid:1)
(cid:2) (cid:6) (cid:4) (cid:5) (cid:4) (cid:1)
X X D
(cid:3)
t i t i t t
(cid:2)
i 1
(cid:3) (cid:3)
with 1 t T
D (cid:2) (1,t,t2)'
t
X ,..., X known
(cid:1)
-k 1 0
(cid:1) (cid:1)
i.i.d with law N (0, )
t p
Remark: here, we suppose that the errors are i.i.d with a gaussian law, but
it can easily be generalised with errors i.i.d with finite moments of order 2.
DMRG-Paris - 17/06/05
DescriptCioand roef dthaen sfr laemqueewl oornk s oef p oluacre s t(r2a)tegy (2)
DescriptCioand roef dthaen sfr laemqueewl oornk s oef p oluacre s t(r2a)tegy (2)
Definition: Let’s introduce the characteristic polynomial: A ( z ) with
k
(cid:1)
(cid:3) (cid:2) i(cid:1)
A(z) I z
i
(cid:1)
i 1
Inversibility THEOREM for a VAR process
The VAR process X can be written as a function of its initial values and of the errors:
t
k t(cid:5)1
(cid:1) (cid:1)
(cid:1) (cid:4) (cid:8) (cid:8) (cid:4) (cid:8) (cid:2) (cid:8) (cid:7)
X C ( X ... X ) C ( D )
t t(cid:5)s s 0 k (cid:5)k(cid:6)s j t(cid:5)j t(cid:5)j
s(cid:4)1 j(cid:4)0
k(cid:3)n (cid:2)
(cid:1) (cid:1)
with C (cid:1) I et (cid:6)n (cid:5)1: C (cid:1) C (cid:4) . Let C(z)(cid:1) znC .
0 n n(cid:5)j j n
j(cid:4)1 n(cid:1)0
(cid:3)(cid:1) (cid:2) (cid:1)
Then, 0/ : this serie converges and inside the disc of radius :
C(z)A(z) (cid:1) I ie C(z) (cid:1) A(z)-1
DMRG-Paris - 17/06/05
Description of the framework of our strategy (3)
Description of the framework of our strategy (3)
Remark: the solution given by this theorem is valid whatever the
parameters are. On the contrary, it is reminded in what follows that the
parameters have to be constrained in order to define a stationary VAR
process.
X
Definition: a process is told strongly stationary iif
t
(cid:3) (cid:2) (cid:1)
h 1: Law(X ,..., X ) Law(X ,..., X )
(cid:1) (cid:1)
t t t h t h
1 m 1 m
It is told weakly stationary of order 2 iif
(cid:1) (cid:1)
EX constant et VarX constant
t t
Remark: in part I, strong stationarity is used since the errors are gaussian.
DMRG-Paris - 17/06/05
Description of the framework of our strategy (4)
Description of the framework of our strategy (4)
FUNDAMENTAL HYPOTHESIS :
(cid:1) (cid:3) (cid:2) (cid:1)
A(z) 0 z 1 or z 1
Remark: this fundamental hypothesis excludes explosive roots with |z|<1
as well as seasonal roots (|z|=1 and z different from 1). If z=1 is a root, then the
process is told to have a unit root.
DMRG-Paris - 17/06/05
Description of the framework of our strategy (5)
Description of the framework of our strategy (5)
THEOREM defining the necessary and sufficient condition for the
stationarity of a VAR process
Under the fundamental hypothesis, a necessary and sufficient condition
(cid:8) (cid:7)
for X EX to be stationary is A(1) 0. In such a case,
t t
the MA representation of the VAR process is obtained :
(cid:1) (cid:1)
X (cid:3) (C (cid:2) (cid:1) (cid:6)D ) where (cid:5)(cid:1) (cid:4) 0/ : C(z) (cid:3) znC (cid:3) A(z)(cid:1)1
(cid:1) (cid:1)
t n t n t n n
(cid:2) (cid:2)
n 0 n 0
(cid:2) (cid:1)(cid:1)
is convergent for z 1
(cid:2)
Remark: (i) when =0, one recognizes the WOLD theorem.
(ii) it is checked that with gaussian errors, the strong stationarity is
recovered, whereas in the general case, the weak stationarity is obtained.
DMRG-Paris - 17/06/05
Description of the framework of our strategy (6)
Description of the framework of our strategy (6)
(cid:1)
Basic definitions for cointegration
Preliminary remark:
Many economic variables are non stationary and the kind of non-
stationarity that is considered here can be removed by one or several
differentiations. In what follows, we will suppose that:
(cid:1) (cid:1)
is i.i.d with law N (0, )
t p
Definition:
(cid:1)
(cid:4) (cid:2) (cid:1)
a process Y / : Y EY C is integrated
(cid:3)
t t t i t i
(cid:1)
i 0
(cid:2)
(cid:1)
(cid:3) (cid:2) (cid:1)
of order 0 C C 0
i
(cid:1)
i 0
DMRG-Paris - 17/06/05