Vladimir R. Evstigneev
Slide 1
Support and Resistance Levels as Parameters
of a Flexible Boundaries Problem in the Calculus of Variations
Slide 2
A couple of two interchangeable problems appears:
1. Direct problem - once we've got an instantaneous drift function and
an instantaneous diffusion function, construct a probability density
function with required properties.
2. Inverse problem - having constructed a probability dinsity function
with required properties, find the drift term and the instantaneous
diffusion term in order to estimate the spectrum.
Solution to the latter problem permits to build a vehicle for the PDF
to evolve in time, either in terms of a series decomposition as
a solution to the corresponding Fokker - Planck equation, or in terms
of (a Mercer theorem-based approximation to) the kernel of an integral
transform, once we might prefer a sourcewise representation as our
basic tool to make the PDF evolve in time.
A solution with flexible boundaries is an alternative approach to the extreme values problem, and
therefore allows to determine certain critical values of "abnormal prices".
Slide 3
We shall follow the two strings of inference, respectively.
Let's embark on solving the direct problem first.
Recall that the "twin functions", i.e. the instantaneous drift term μ (x) and
the instantaneous diffusion term σ (x) define a Fokker - Planck equation
d
p ( xt)
dt
d
2
2
( ( x)  p ( xt) ) 
dx
d
(  ( x)  p ( xt) )
dx
for the PDF p(x,t) and an equation of motion
d
x( t)
dt
 ( x)
for the mean value.
Slide 4
Consider the simplest mean-reverting Uhlenbeck - Ornstein process.
 ( x)  m  x
( x)  s
2
The stationary solution to the corresponding Fokker - Planck equation
is the famous normal distribution.
Ne
N






d
 ( x)   ( x)
dx
 ( x)
dx

N e
( m  x)
2 s
2
2
1
s  2 
The arbitrary constant is normalized with regard to
the entire real axis.
Slide 5
Consider now a more complicated family of equations of motion. Each of them is related to
a problem in the calculus of variations.
 ( x)
1





 (   1)  
 1 
x
2
2



2 2   6   x 

    3   x  2   x  
    3   x  2   x    1  (   1)
 1
x  (   1)  
x  (   1) 

3
The above-given drift term, e.g., is related to the following variational problem.
F( P( x) p ( x) u( x)  ( x) x)
2
3
2
p ( x)  ln( p ( x) )   x p ( x)    x  p ( x)    x p ( x)   ( x)  ( x  ( x)   u( x) )    u( x)  p ( x) 
Here F( P(x) p (x) u( x)  ( x) x) is a functional the definite integral of which is to be minimized,
U(x) is the utility function meeting the Stiglitz condition, i.e. the Epstein - Zin utility,
u ( x)
 ( x)
dU( x)
dx
d
2
2
U( x)
dx
1
1
  u ( x)
an auxiliary function  ( x) is a functional Lagrange multiplier, 1   c
pertains to a stochastic
pricing kernel that guarantees fair (equilibtium) pricing, ρ is the zero-coupon rate and c is the known
marginal utility of the present value of the investment portfolio; x is the portfolio value bounded within
the interval [0, 2]. The Greek scalars are the parameters to be estimamted via maximum likelihood.
1 1
  u( x)  x p ( x)
1  c
Slide 6
We are especially interested in solving a variational problem with the following functional,
2
F( P( x) p ( x) m( x)  ( x) x)
3
p ( x)  ln( p ( x) )   x p ( x)    x  p ( x)    x  p ( x)    ( x  m( x) )  p ( x)     ( x)  p ( x)
where m(x) is the "local center" function with flexible left and right boundaries, ξ and ζ .
This problem is solved by a PDF p(x) and m(x) and a third function, f(x), as shown below; n=ln(N).

p ( x)
N  exp   (   1)  x 
3



2
m( x)
    3          n

f ( x)

    n   3      
 
2
2
3   (   1)  x 



3   (   1)  [ 2   (   )   ] 2      n

      n
x 
 3      3    
     3      3    

2 
 2



2



3   (   1)  [ 2     (   ) ]

2
 x  3    
2
   2  3
x 
x
2 



   n



 x      3     
2
2
The function f(x) defines the instantaneous drift and diffusion terms as follows.
 ( x)
d
( x)  ( x)  f ( x)
dx
( x)
f ( x)
2
2
  3 (  )  3   2  n  3 (  )




2

x
 
 
Slide 7
Recall that the idea of a "local center" stems from behavioral finance and is dated back to
Herbert Simon's famous writings on suboptimal decision making.
The boundary functions that determine the transversality conditions are assumed to be as simple
as possible, so they are set linear with ι as "leap back" parameter to be determined via maximum
likelihood as well.
At the left boundary...
 ( x)
x 
At the right boundary...
 ( x)
x
Transversality conditions imply, as x approaches ξ ...


2
3
p ( x)   ln( p ( x) )    x    x   
d

 ( x)   ( x)      x   x
dx

0
Transversality conditions imply, as x approaches ζ ...


2
3
p ( x)   ln( p ( x) )    x    x   
d

 ( x)   ( x)      x   x
dx

0
Meeting the transversality conditions requires that the two parameters, α and β , be
expressed in terms of the flexible boundaries regarded as parameters, too.
Slide 8
Consider a time series of monthly S&P500 quotes since January, 1990, up to March, 2014.
Regard the flexible boundaries as natural baundaries of the domain for the PDF. These
boundaries may as well be regarded as "natural" support and resistance levels.
The trading rule may, thus, read as follows:
long if the last price was below the lower boundary (therefore, disobeying the "core"
distribution) plus two times ι ;
short if the last price was above the upper boundary plus two times ι .
The performance of a back testing portfolio is presented below. M is the naive buy-and-hold
portfolio and G is a time track of the support-resistence boundaries-based portfolio. The length
of the moving period ("window") equals 25 observation.
10
10
8
8
6
6
G
M
4
4
2
2
0
2
0
100
200

0
300
Slide 9
Consider now hourly quotes of the EUR/USD exchange rates. Select randomly a sample
set, say, since 19 October to 3 December 2012. The trading rule is almost the same safe that the position sign is multiplied by the sign of the difference betwen the last quote
and the first momet of the PDF estimated over the corresponding moving period.
Here again, M is the porfolio mimicking the market and G is the rule-based portfolio.
A few words must be said as concerns the second string of inference, i.e. the inverse
problem. As one starts with a variational problem, the drift term, μ (x), can be easily
expressed in terms of σ (x) and f(x). It is nor an easy task to find σ (x), however.
2
0
1
1
 2M 
G
3
0
4
2.710
4
4
2.7210
2.7410

4
4
2.7610
Slide 10
Define the three parameterizing functions and a Sturm - Liouville equation, as usual.
P0( x)
( x)
P1( x)
2
d
( x)   ( x)
dx
d
P2( x)
2
2
dx
P0( x) 
d
2
2
dx
( x) 
d
 ( x)
dx
u ( x)  P1( x) 
d
u ( x)  P2( x)  u ( x)
dx
0
Now turn to the inhomogeneous form of the latter equation.
Slide 11
2d
3 f ( x) 
Let the right-hand part look like
( x)
f ( x)
f ( x)
dx
. Then...
2
2
d
f ( x)    k (   x)   2 k f ( x)  (   x)
Let the right-hand side be mean reverting, dx
. Now...
( x)
2
k (   x)  
Once we've got a σ (x), we are welcome to proceed directly to μ (x), i.e. the right-hand part
of an equation of motion.
d
x( t)
dt
 ( x)
Speaking of the two examples given in the above, in the first instance the right-hand part of
the equation of motion would look like this...
d
x( t)
dt
2 f ( x) 
3
d
f ( x)  f ( x)
dx
...while in the second instance - like this...
d
x( t)
dt
2 k ( x   )  f ( x)    k (   x)
2

It is terribly tempting to solve these non-linear equations in order to find their equilibrium points.
Slide 12
Or alternatively, one may consider a normal form of the Sturm - Liouville equation, with ω (x)
instead of u(x) as its solutions (the mapping from u to ω is homeomorphic and easily attainable).
d
2
2
( x)  h ( x) ( x)
0
dx
h ( x)
P2( x)
P0( x)

 
P1( x) 
2
1 d P1( x) 
  


4  P0( x) 
2 dx P0( x) 
1
Then the specific form of σ (x) depends upon the appropriate form of the normal equivalent
equation. One may consider various specifications of h(x), the simplest one shown below.
h ( x)
d
2
2
dx
( x) 
2
 d ( x)   f ( x)  d ( x)   1  f ( x) 2  d f ( x)   ( x)

2

2 ( x)  dx
dx
dx



1

1

0
The Form of σ (x) is determined by the considerations of convineience as concerns
solving the Sturm - Liouville equation.
We obtain, eventually, a family of equations of motion corresponding to one and
the same PDF (which is a challengong problem itself).