Download Applying elementary principles from quantum physics

Applying elementary principles from quantum physics to
…nance: do we know more?
Emmanuel Haven - SoM - University of Leicester
May 21 - 2014
Talk at Imperial College ()
May 21 - 2014
1 / 46
What are we NOT after?
From the outset: looking at the title of this talk...
Talk at Imperial College ()
May 21 - 2014
2 / 46
What are we NOT after?
From the outset: looking at the title of this talk...
It seems we want to make social science or …nance to become
quantum mechanical....
Talk at Imperial College ()
May 21 - 2014
2 / 46
What are we NOT after?
From the outset: looking at the title of this talk...
It seems we want to make social science or …nance to become
quantum mechanical....
This is NOT what we are after
Talk at Imperial College ()
May 21 - 2014
2 / 46
What are we NOT after?
From the outset: looking at the title of this talk...
It seems we want to make social science or …nance to become
quantum mechanical....
This is NOT what we are after
In our book (E. Haven and A. Khrennikov (2013). Quantum Social
Science. Cambridge University Press), we try to make the case we
can use techniques from quantum mechanics but surely without
claiming the macroscopic world is quantum mechanical!!
Talk at Imperial College ()
May 21 - 2014
2 / 46
What are we NOT after?
From the outset: looking at the title of this talk...
It seems we want to make social science or …nance to become
quantum mechanical....
This is NOT what we are after
In our book (E. Haven and A. Khrennikov (2013). Quantum Social
Science. Cambridge University Press), we try to make the case we
can use techniques from quantum mechanics but surely without
claiming the macroscopic world is quantum mechanical!!
Please see our article in New Scientist: Khrennikov, A. and Haven, E.
(2013). Our quantum society. NewScientist July 6 Issue; 26-27
Talk at Imperial College ()
May 21 - 2014
2 / 46
What are we NOT after?
From the outset: looking at the title of this talk...
It seems we want to make social science or …nance to become
quantum mechanical....
This is NOT what we are after
In our book (E. Haven and A. Khrennikov (2013). Quantum Social
Science. Cambridge University Press), we try to make the case we
can use techniques from quantum mechanics but surely without
claiming the macroscopic world is quantum mechanical!!
Please see our article in New Scientist: Khrennikov, A. and Haven, E.
(2013). Our quantum society. NewScientist July 6 Issue; 26-27
All what we say is: “can we use techniques from quantum mechanics
or other areas of physics to aid us in modelling phenomena in …nance
or economics?”
Talk at Imperial College ()
May 21 - 2014
2 / 46
So what are we after then?
We attempt to use techniques from physics to applications outside
physics...
Talk at Imperial College ()
May 21 - 2014
3 / 46
So what are we after then?
We attempt to use techniques from physics to applications outside
physics...
How are we really doing this...?
Talk at Imperial College ()
May 21 - 2014
3 / 46
So what are we after then?
We attempt to use techniques from physics to applications outside
physics...
How are we really doing this...?
Since my training is in economics and not at all in physics or maths
Talk at Imperial College ()
May 21 - 2014
3 / 46
So what are we after then?
We attempt to use techniques from physics to applications outside
physics...
How are we really doing this...?
Since my training is in economics and not at all in physics or maths
....a quote from Nobel prize winner in economics - P. Samuelson may therefore not be out of place....
Talk at Imperial College ()
May 21 - 2014
3 / 46
So what are we after then?
We attempt to use techniques from physics to applications outside
physics...
How are we really doing this...?
Since my training is in economics and not at all in physics or maths
....a quote from Nobel prize winner in economics - P. Samuelson may therefore not be out of place....
Says Samuelson: “There is nothing more pathetic than to have an
economist or a retired engineer try to force analogies between the
concepts of physics and the concepts of engineers. . . ”
Talk at Imperial College ()
May 21 - 2014
3 / 46
So what are we after then?
We attempt to use techniques from physics to applications outside
physics...
How are we really doing this...?
Since my training is in economics and not at all in physics or maths
....a quote from Nobel prize winner in economics - P. Samuelson may therefore not be out of place....
Says Samuelson: “There is nothing more pathetic than to have an
economist or a retired engineer try to force analogies between the
concepts of physics and the concepts of engineers. . . ”
But please note one of the references of Samuelson’s work: .....??
Talk at Imperial College ()
May 21 - 2014
3 / 46
So what are we after then?
We attempt to use techniques from physics to applications outside
physics...
How are we really doing this...?
Since my training is in economics and not at all in physics or maths
....a quote from Nobel prize winner in economics - P. Samuelson may therefore not be out of place....
Says Samuelson: “There is nothing more pathetic than to have an
economist or a retired engineer try to force analogies between the
concepts of physics and the concepts of engineers. . . ”
But please note one of the references of Samuelson’s work: .....??
Samuelson, P. (1977). A quantum theory model of economics. In:
Collected Scienti…c Papers (Vol. 4); H. Nagatani and K. Crowley.
M.I.T. Press, Cambridge, Mass.
Talk at Imperial College ()
May 21 - 2014
3 / 46
Samuelson and metaphors
Did he make such a statement because he believed physics based
models when they are applied to economics are just merely
metaphors?
Talk at Imperial College ()
May 21 - 2014
4 / 46
Samuelson and metaphors
Did he make such a statement because he believed physics based
models when they are applied to economics are just merely
metaphors?
Is ‘metaphorical use’- non-Kosher?
Talk at Imperial College ()
May 21 - 2014
4 / 46
Samuelson and metaphors
Did he make such a statement because he believed physics based
models when they are applied to economics are just merely
metaphors?
Is ‘metaphorical use’- non-Kosher?
After all....??? Model = Function (metaphors + quantitative and/or
qualitative bits. . . ) (F. Verhulst (1998). The Validation of
metaphors. Mathematisch Instituut (Utrecht, The Netherlands))
Talk at Imperial College ()
May 21 - 2014
4 / 46
Samuelson and metaphors
Did he make such a statement because he believed physics based
models when they are applied to economics are just merely
metaphors?
Is ‘metaphorical use’- non-Kosher?
After all....??? Model = Function (metaphors + quantitative and/or
qualitative bits. . . ) (F. Verhulst (1998). The Validation of
metaphors. Mathematisch Instituut (Utrecht, The Netherlands))
But.....aie.... in the ‘formalism approach’to mathematics for instance:
“What is real in mathematics is ‘notation’– not imagined
denotation” (E. Nelson - See:
http://www.math.princeton.edu/~nelson/papers.html)
Talk at Imperial College ()
May 21 - 2014
4 / 46
Samuelson and metaphors
Did he make such a statement because he believed physics based
models when they are applied to economics are just merely
metaphors?
Is ‘metaphorical use’- non-Kosher?
After all....??? Model = Function (metaphors + quantitative and/or
qualitative bits. . . ) (F. Verhulst (1998). The Validation of
metaphors. Mathematisch Instituut (Utrecht, The Netherlands))
But.....aie.... in the ‘formalism approach’to mathematics for instance:
“What is real in mathematics is ‘notation’– not imagined
denotation” (E. Nelson - See:
http://www.math.princeton.edu/~nelson/papers.html)
If metaphors are the imagined denotations then in Nelson’s formalism
approach to mathematics we may just not have models....
Talk at Imperial College ()
May 21 - 2014
4 / 46
OK! Enough of all that...
Let us set aside ‘philosophy’and let me give you an overview of the
topics I would like to discuss in this talk
Talk at Imperial College ()
May 21 - 2014
5 / 46
Brief outline on what follows
Part I: classical physics and …nance
Talk at Imperial College ()
May 21 - 2014
6 / 46
Brief outline on what follows
Part I: classical physics and …nance
Part II: quantum physics and …nance
Talk at Imperial College ()
May 21 - 2014
6 / 46
Brief outline on what follows
Part I: classical physics and …nance
Part II: quantum physics and …nance
And in Part II we will focus on two main topics:
Talk at Imperial College ()
May 21 - 2014
6 / 46
Brief outline on what follows
Part I: classical physics and …nance
Part II: quantum physics and …nance
And in Part II we will focus on two main topics:
i) using basics of quantum physics in modelling information
Talk at Imperial College ()
May 21 - 2014
6 / 46
Brief outline on what follows
Part I: classical physics and …nance
Part II: quantum physics and …nance
And in Part II we will focus on two main topics:
i) using basics of quantum physics in modelling information
ii) using basics of quantum physics in decision making modelling
Talk at Imperial College ()
May 21 - 2014
6 / 46
Part I: Classical physics and …nance
Example 1: Phase space of prices
Talk at Imperial College ()
May 21 - 2014
7 / 46
Part I: Classical physics and …nance
Example 1: Phase space of prices
assume a con…guration space Q = Rn of price vectors !
q = ( q1 ,
q2 ...qn ) where qj is f.i. the price of the share of the jth corporation
(for one and the same asset for instance) (or the jth trader for one and
the same asset)
Talk at Imperial College ()
May 21 - 2014
7 / 46
Part I: Classical physics and …nance
Example 1: Phase space of prices
assume a con…guration space Q = Rn of price vectors !
q = ( q1 ,
q2 ...qn ) where qj is f.i. the price of the share of the jth corporation
(for one and the same asset for instance) (or the jth trader for one and
the same asset)
dynamics of prices can be described by a trajectory !
q ( t ) = ( q1 ( t ) ,
q2 (t )...qn (t )) in Q
Talk at Imperial College ()
May 21 - 2014
7 / 46
Part I: Classical physics and …nance
Example 1: Phase space of prices
assume a con…guration space Q = Rn of price vectors !
q = ( q1 ,
q2 ...qn ) where qj is f.i. the price of the share of the jth corporation
(for one and the same asset for instance) (or the jth trader for one and
the same asset)
dynamics of prices can be described by a trajectory !
q ( t ) = ( q1 ( t ) ,
q2 (t )...qn (t )) in Q
one can de…ne δqj (t ) = qj (t + ∆t ) qj (t ) and a continuous price
.
change: vj (t ) = qj (t ) = lim∆t !0
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q j (t + ∆t ) q j (t )
∆t
May 21 - 2014
7 / 46
Part I: Classical physics and …nance
Example 1: Phase space of prices
assume a con…guration space Q = Rn of price vectors !
q = ( q1 ,
q2 ...qn ) where qj is f.i. the price of the share of the jth corporation
(for one and the same asset for instance) (or the jth trader for one and
the same asset)
dynamics of prices can be described by a trajectory !
q ( t ) = ( q1 ( t ) ,
q2 (t )...qn (t )) in Q
one can de…ne δqj (t ) = qj (t + ∆t ) qj (t ) and a continuous price
.
q (t + ∆t ) q (t )
j
change: vj (t ) = qj (t ) = lim∆t !0 j
∆t
there exists a phase space: Q V where V
Rn and !
v = ( v1 ,
v2 , ...vn ) 2 V . A state (q, v ) is called a classical state
Talk at Imperial College ()
May 21 - 2014
7 / 46
Part I: Classical physics and …nance (cont’d)
Example 1: Phase space of prices (cont’d)
Talk at Imperial College ()
May 21 - 2014
8 / 46
Part I: Classical physics and …nance (cont’d)
Example 1: Phase space of prices (cont’d)
the analogue of physical mass can also be introduced: the number of
shares of stock j: mj
Talk at Imperial College ()
May 21 - 2014
8 / 46
Part I: Classical physics and …nance (cont’d)
Example 1: Phase space of prices (cont’d)
the analogue of physical mass can also be introduced: the number of
shares of stock j: mj
the market capitalization of …rm j is then: Tj (t ) = mj qj (t )
Talk at Imperial College ()
May 21 - 2014
8 / 46
Part I: Classical physics and …nance (cont’d)
Example 1: Phase space of prices (cont’d)
the analogue of physical mass can also be introduced: the number of
shares of stock j: mj
the market capitalization of …rm j is then: Tj (t ) = mj qj (t )
kinetic energy: 12 ∑nj=1 mj vj2 ; where mj is the number of shares of stock
.
j and vj (t ) = qj (t ) = lim∆t !0
the price of asset j
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q j (t + ∆t ) q j (t )
;
∆t
where t is time and qj is
May 21 - 2014
8 / 46
Part I: Classical physics and …nance (cont’d)
Example 1: Phase space of prices (cont’d)
the analogue of physical mass can also be introduced: the number of
shares of stock j: mj
the market capitalization of …rm j is then: Tj (t ) = mj qj (t )
kinetic energy: 12 ∑nj=1 mj vj2 ; where mj is the number of shares of stock
.
q (t + ∆t ) q (t )
j
j and vj (t ) = qj (t ) = lim∆t !0 j
; where t is time and qj is
∆t
the price of asset j
V (q1 , ...qn ) : could describe interactions between traders as well as
interactions from other factors such as macro-economic factors
Talk at Imperial College ()
May 21 - 2014
8 / 46
Part I: Classical physics and …nance (cont’d)
Example 1: Phase space of prices (cont’d)
the analogue of physical mass can also be introduced: the number of
shares of stock j: mj
the market capitalization of …rm j is then: Tj (t ) = mj qj (t )
kinetic energy: 12 ∑nj=1 mj vj2 ; where mj is the number of shares of stock
.
q (t + ∆t ) q (t )
j
j and vj (t ) = qj (t ) = lim∆t !0 j
; where t is time and qj is
∆t
the price of asset j
V (q1 , ...qn ) : could describe interactions between traders as well as
interactions from other factors such as macro-economic factors
a simple …nancial potential could be: (qi qj )2 : price di¤erences on
the same asset amongst two traders i and j
Talk at Imperial College ()
May 21 - 2014
8 / 46
Part I: Classical physics and …nance (cont’d)
Example 1: Phase space of prices (cont’d)
the analogue of physical mass can also be introduced: the number of
shares of stock j: mj
the market capitalization of …rm j is then: Tj (t ) = mj qj (t )
kinetic energy: 12 ∑nj=1 mj vj2 ; where mj is the number of shares of stock
.
q (t + ∆t ) q (t )
j
j and vj (t ) = qj (t ) = lim∆t !0 j
; where t is time and qj is
∆t
the price of asset j
V (q1 , ...qn ) : could describe interactions between traders as well as
interactions from other factors such as macro-economic factors
a simple …nancial potential could be: (qi qj )2 : price di¤erences on
the same asset amongst two traders i and j
classical price dynamics are then de…ned by a price momentum:
v (t + ∆t ) v j (t )
= ∂∂qV
pj = mj vj . We then have: mj lim∆t !0 j
∆t
j
Talk at Imperial College ()
May 21 - 2014
8 / 46
Part I: Classical physics and …nance (cont’d)
Example 1: Phase space of prices (cont’d)
the analogue of physical mass can also be introduced: the number of
shares of stock j: mj
the market capitalization of …rm j is then: Tj (t ) = mj qj (t )
kinetic energy: 12 ∑nj=1 mj vj2 ; where mj is the number of shares of stock
.
q (t + ∆t ) q (t )
j
j and vj (t ) = qj (t ) = lim∆t !0 j
; where t is time and qj is
∆t
the price of asset j
V (q1 , ...qn ) : could describe interactions between traders as well as
interactions from other factors such as macro-economic factors
a simple …nancial potential could be: (qi qj )2 : price di¤erences on
the same asset amongst two traders i and j
classical price dynamics are then de…ned by a price momentum:
v (t + ∆t ) v j (t )
= ∂∂qVj
pj = mj vj . We then have: mj lim∆t !0 j
∆t
See also: E. Haven and A. Khrennikov (2013). Quantum Social
Science. Cambridge University Press and A. Khrennikov (2010).
Ubiquitous quantum structure. Springer
Talk at Imperial College ()
May 21 - 2014
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Part I: Classical physics and …nance (cont’d)
Immediate problems and challenges
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May 21 - 2014
9 / 46
Part I: Classical physics and …nance (cont’d)
Immediate problems and challenges
Query 1:
Talk at Imperial College ()
May 21 - 2014
9 / 46
Part I: Classical physics and …nance (cont’d)
Immediate problems and challenges
Query 1:
what about conservation of total energy: is
hold: no second law of Newton!
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∂E
∂t
= 0? If this does not
May 21 - 2014
9 / 46
Part I: Classical physics and …nance (cont’d)
Immediate problems and challenges
Query 1:
what about conservation of total energy: is ∂E
∂t = 0? If this does not
hold: no second law of Newton!
time translation invariance: occurs when L is time independent.
Unlikely in …nance! Example: the potential could be made time
.
dependent: (qi qj )2t , with t time (this implies a move from: L(q, q )
.
to L(q, q, t ) and Hamiltonian conservation does not obtain)
Talk at Imperial College ()
May 21 - 2014
9 / 46
Part I: Classical physics and …nance (cont’d)
Immediate problems and challenges
Query 1:
what about conservation of total energy: is ∂E
∂t = 0? If this does not
hold: no second law of Newton!
time translation invariance: occurs when L is time independent.
Unlikely in …nance! Example: the potential could be made time
.
dependent: (qi qj )2t , with t time (this implies a move from: L(q, q )
.
to L(q, q, t ) and Hamiltonian conservation does not obtain)
Rt
.
Query 2: is action stationary: δA = δ to1 L(q, q )dt = 0?
Talk at Imperial College ()
May 21 - 2014
9 / 46
Part I: Classical physics and …nance (cont’d)
Immediate problems and challenges
Query 1:
what about conservation of total energy: is ∂E
∂t = 0? If this does not
hold: no second law of Newton!
time translation invariance: occurs when L is time independent.
Unlikely in …nance! Example: the potential could be made time
.
dependent: (qi qj )2t , with t time (this implies a move from: L(q, q )
.
to L(q, q, t ) and Hamiltonian conservation does not obtain)
Rt
.
Query 2: is action stationary: δA = δ to1 L(q, q )dt = 0?
See Ilinski, K. (2001). Physics of …nance: gauge modelling in
non-equilibrium pricing. J. Wiley.
Talk at Imperial College ()
May 21 - 2014
9 / 46
Part I: Classical physics and …nance (cont’d)
Immediate problems and challenges
Query 1:
what about conservation of total energy: is ∂E
∂t = 0? If this does not
hold: no second law of Newton!
time translation invariance: occurs when L is time independent.
Unlikely in …nance! Example: the potential could be made time
.
dependent: (qi qj )2t , with t time (this implies a move from: L(q, q )
.
to L(q, q, t ) and Hamiltonian conservation does not obtain)
Rt
.
Query 2: is action stationary: δA = δ to1 L(q, q )dt = 0?
See Ilinski, K. (2001). Physics of …nance: gauge modelling in
non-equilibrium pricing. J. Wiley.
exp (r ∆)
we have cash at ti : we want to buy shares at time ti +1 : Si +01 is the
number of shares you can buy at time ti +1 at price Si +1 .
Talk at Imperial College ()
May 21 - 2014
9 / 46
Part I: Classical physics and …nance (cont’d)
Immediate problems and challenges
Query 1:
what about conservation of total energy: is ∂E
∂t = 0? If this does not
hold: no second law of Newton!
time translation invariance: occurs when L is time independent.
Unlikely in …nance! Example: the potential could be made time
.
dependent: (qi qj )2t , with t time (this implies a move from: L(q, q )
.
to L(q, q, t ) and Hamiltonian conservation does not obtain)
Rt
.
Query 2: is action stationary: δA = δ to1 L(q, q )dt = 0?
See Ilinski, K. (2001). Physics of …nance: gauge modelling in
non-equilibrium pricing. J. Wiley.
exp (r ∆)
we have cash at ti : we want to buy shares at time ti +1 : Si +01 is the
number of shares you can buy at time ti +1 at price Si +1 .
the accumulated cash at time ti +1 - with start value 1 at time ti is
given by: exp(r0 ∆)
Talk at Imperial College ()
May 21 - 2014
9 / 46
Part I: Classical physics and …nance (cont’d)
Immediate problems and challenges (cont’d)
Talk at Imperial College ()
May 21 - 2014
10 / 46
Part I: Classical physics and …nance (cont’d)
Immediate problems and challenges (cont’d)
alternatively - one can buy shares now at ti at price Si and at time
exp (r1 ∆)
ti +1 :
Si
Talk at Imperial College ()
May 21 - 2014
10 / 46
Part I: Classical physics and …nance (cont’d)
Immediate problems and challenges (cont’d)
alternatively - one can buy shares now at ti at price Si and at time
exp (r1 ∆)
ti +1 :
Si
consider then the two situations: exp(r0 ∆)Si +11 > exp(r1 ∆)Si 1 and
the opposite: exp(r1 ∆)Si 1 > exp(r0 ∆)Si +11
Talk at Imperial College ()
May 21 - 2014
10 / 46
Part I: Classical physics and …nance (cont’d)
Immediate problems and challenges (cont’d)
alternatively - one can buy shares now at ti at price Si and at time
exp (r1 ∆)
ti +1 :
Si
consider then the two situations: exp(r0 ∆)Si +11 > exp(r1 ∆)Si 1 and
the opposite: exp(r1 ∆)Si 1 > exp(r0 ∆)Si +11
a non-arbitrage condition could be that:
exp(r1 ∆)Si 1 exp( r0 ∆)Si +1 + exp(r0 ∆)Si +11 exp( r1 ∆)Si 2 = 0
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May 21 - 2014
10 / 46
Part I: Classical physics and …nance (cont’d)
Immediate problems and challenges (cont’d)
alternatively - one can buy shares now at ti at price Si and at time
exp (r1 ∆)
ti +1 :
Si
consider then the two situations: exp(r0 ∆)Si +11 > exp(r1 ∆)Si 1 and
the opposite: exp(r1 ∆)Si 1 > exp(r0 ∆)Si +11
a non-arbitrage condition could be that:
exp(r1 ∆)Si 1 exp( r0 ∆)Si +1 + exp(r0 ∆)Si +11 exp( r1 ∆)Si 2 = 0
an action can be formulated as:
A = ∑i∞= ∞ αi (exp(r1 ∆)Si 1 exp( r0 ∆)Si +1 +
exp(r0 ∆)Si +11 exp( r1 ∆)Si 2)
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May 21 - 2014
10 / 46
Part I: Classical physics and …nance (cont’d)
Immediate problems and challenges (cont’d)
alternatively - one can buy shares now at ti at price Si and at time
exp (r1 ∆)
ti +1 :
Si
consider then the two situations: exp(r0 ∆)Si +11 > exp(r1 ∆)Si 1 and
the opposite: exp(r1 ∆)Si 1 > exp(r0 ∆)Si +11
a non-arbitrage condition could be that:
exp(r1 ∆)Si 1 exp( r0 ∆)Si +1 + exp(r0 ∆)Si +11 exp( r1 ∆)Si 2 = 0
an action can be formulated as:
A = ∑i∞= ∞ αi (exp(r1 ∆)Si 1 exp( r0 ∆)Si +1 +
exp(r0 ∆)Si +11 exp( r1 ∆)Si 2)
2
R ∞ 1 ∂S 1
In the limit: ∆ ! 0 : A = 12
µ dt (Ilinski (p. 96)):
∞ σ2
∂t S
“corresponds to geometric R.W. with time dependent volatility and
average rate of share return”
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May 21 - 2014
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Part I: Classical physics and …nance (cont’d)
Example 2: Momentum conservation in …nance....
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May 21 - 2014
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Part I: Classical physics and …nance (cont’d)
Example 2: Momentum conservation in …nance....
consider L =f (V (q1
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q2 ))
May 21 - 2014
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Part I: Classical physics and …nance (cont’d)
Example 2: Momentum conservation in …nance....
consider L =f (V (q1 q2 ))
.
.
momentum conservation if V (q1 q2 ) is considered: p 1 + p 2 = 0
.
.
since p 1 = V 0 (q1 q2 ) and p 2 = V 0 (q1 q2 )
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May 21 - 2014
11 / 46
Part I: Classical physics and …nance (cont’d)
Example 2: Momentum conservation in …nance....
consider L =f (V (q1 q2 ))
.
.
momentum conservation if V (q1 q2 ) is considered: p 1 + p 2 = 0
.
.
since p 1 = V 0 (q1 q2 ) and p 2 = V 0 (q1 q2 )
but this conservation may surely not always happen!!!
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May 21 - 2014
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Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential
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May 21 - 2014
12 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential
Baaquie, Belal (2013). Statistical microeconomics. Physica A 392(19);
4400–4416.
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May 21 - 2014
12 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential
Baaquie, Belal (2013). Statistical microeconomics. Physica A 392(19);
4400–4416.
∂ U [q]
the demand function in economics is obtained via: ∂q = 0 with the
i
N
constraint: ∑ pi qi = m
i =1
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May 21 - 2014
12 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential
Baaquie, Belal (2013). Statistical microeconomics. Physica A 392(19);
4400–4416.
∂ U [q]
the demand function in economics is obtained via: ∂q = 0 with the
i
N
constraint: ∑ pi qi = m
i =1
di
N
the demand function: D[p] = m
2 ∑i =1 p ai ; ai , di > 0 is proposed.
i
ai identi…es the demand for a speci…c commodity; di is determined by
the relative importance of quantity qi in the demand for the total
collection of N commodities.
Talk at Imperial College ()
May 21 - 2014
12 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
Talk at Imperial College ()
May 21 - 2014
13 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
the supply function: F [q] = 12 ∑N
i =1 αi qi ; αi indicates relative
importance of quantity qi in the total supply of N commodities
Talk at Imperial College ()
May 21 - 2014
13 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
the supply function: F [q] = 12 ∑N
i =1 αi qi ; αi indicates relative
importance of quantity qi in the total supply of N commodities
C [q] (C [q] is the cost function),
pro…t is: π [q] = ∑N
i = 1 p i qi
∂π [q]
∂q = 0 =) q = q(p)
i
Talk at Imperial College ()
May 21 - 2014
13 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
the supply function: F [q] = 12 ∑N
i =1 αi qi ; αi indicates relative
importance of quantity qi in the total supply of N commodities
C [q] (C [q] is the cost function),
pro…t is: π [q] = ∑N
i = 1 p i qi
∂π [q]
∂q i = 0 =) q = q(p)
one writes: S (p) = F (q(p)): pro…t maximizing output at each price
Talk at Imperial College ()
May 21 - 2014
13 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
the supply function: F [q] = 12 ∑N
i =1 αi qi ; αi indicates relative
importance of quantity qi in the total supply of N commodities
C [q] (C [q] is the cost function),
pro…t is: π [q] = ∑N
i = 1 p i qi
∂π [q]
∂q i = 0 =) q = q(p)
one writes: S (p) = F (q(p)): pro…t maximizing output at each price
bi
N
for a given cost function, one can obtain: S (p) = m
2 ∑i =1 si pi and
bi ; si > 0
Talk at Imperial College ()
May 21 - 2014
13 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
Talk at Imperial College ()
May 21 - 2014
14 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
in economics: one postulates that the interplay of the supply and
demand functions determines the stationary prices of commodities
Talk at Imperial College ()
May 21 - 2014
14 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
in economics: one postulates that the interplay of the supply and
demand functions determines the stationary prices of commodities
the trade o¤ between supply and demand is encoded in the so called
‘microeconomic potential’: V [p] = D[p] + S[p]
Talk at Imperial College ()
May 21 - 2014
14 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
in economics: one postulates that the interplay of the supply and
demand functions determines the stationary prices of commodities
the trade o¤ between supply and demand is encoded in the so called
‘microeconomic potential’: V [p] = D[p] + S[p]
D[p] ! ∞ ; pi ! 0
V [p] !
S[p] ! ∞ ; pi ! ∞
Talk at Imperial College ()
May 21 - 2014
14 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
in economics: one postulates that the interplay of the supply and
demand functions determines the stationary prices of commodities
the trade o¤ between supply and demand is encoded in the so called
‘microeconomic potential’: V [p] = D[p] + S[p]
D[p] ! ∞ ; pi ! 0
V [p] !
S[p] ! ∞ ; pi ! ∞
minimizing price p0 is given by:
∂ V [p]
∂D[p]
∂S[p]
∂p jp=p0 = 0 =) ∂p jp=p0 = ∂p jp=p0
i
Talk at Imperial College ()
i
i
May 21 - 2014
14 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
in economics: one postulates that the interplay of the supply and
demand functions determines the stationary prices of commodities
the trade o¤ between supply and demand is encoded in the so called
‘microeconomic potential’: V [p] = D[p] + S[p]
D[p] ! ∞ ; pi ! 0
V [p] !
S[p] ! ∞ ; pi ! ∞
minimizing price p0 is given by:
∂ V [p]
∂D[p]
∂S[p]
∂p i jp=p0 = 0 =) ∂p i jp=p0 = ∂p i jp=p0
a minimum value of the potential V [p] is attained at price vector p0
when a small variation of prices yields a change of demand that is
exactly the opposite to change of supply
Talk at Imperial College ()
May 21 - 2014
14 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
in economics: one postulates that the interplay of the supply and
demand functions determines the stationary prices of commodities
the trade o¤ between supply and demand is encoded in the so called
‘microeconomic potential’: V [p] = D[p] + S[p]
D[p] ! ∞ ; pi ! 0
V [p] !
S[p] ! ∞ ; pi ! ∞
minimizing price p0 is given by:
∂ V [p]
∂D[p]
∂S[p]
∂p i jp=p0 = 0 =) ∂p i jp=p0 = ∂p i jp=p0
a minimum value of the potential V [p] is attained at price vector p0
when a small variation of prices yields a change of demand that is
exactly the opposite to change of supply
bi
di
N
N
given V [p] = D[p] + S[p] = m
2 ∑ i = 1 p a i + ∑ i = 1 s i pi
i
0 ; a, b > 0
Talk at Imperial College ()
; di , si >
May 21 - 2014
14 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
Talk at Imperial College ()
May 21 - 2014
15 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
the global minimum obtained with this functional form is:
∂ V [p]
∂p i jp=p0
Talk at Imperial College ()
= 0 =) p0i =
ai di
b i si
1
a i +b i
May 21 - 2014
15 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
the global minimum obtained with this functional form is:
1
a +b
∂ V [p]
∂p i jp=p0
= 0 =) p0i = abii dsii i i
in classical microeconomics, using the functional forms for supply and
demand, as mentioned above:
D[p ] = S[p ] )
Talk at Imperial College ()
di
(p i )a i
= si ( pi ) b i
) pi =
di
si
1 / (a i +b i )
May 21 - 2014
15 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
the global minimum obtained with this functional form is:
∂ V [p]
∂p i jp=p0
1
a +b
= 0 =) p0i = abii dsii i i
in classical microeconomics, using the functional forms for supply and
demand, as mentioned above:
1 / (a i +b i )
D[p ] = S[p ] ) (pdi)ai = si (pi )bi ) pi = dsii
i
for the case of a = b the two approaches yield the same answer (a and
b are shape parameters in resp. demand and supply functions)
Talk at Imperial College ()
May 21 - 2014
15 / 46
Part I: Classical physics and …nance (cont’d)
Example 3: The microeconomic potential....(cont’d)
the global minimum obtained with this functional form is:
∂ V [p]
∂p i jp=p0
1
a +b
= 0 =) p0i = abii dsii i i
in classical microeconomics, using the functional forms for supply and
demand, as mentioned above:
1 / (a i +b i )
D[p ] = S[p ] ) (pdi)ai = si (pi )bi ) pi = dsii
i
for the case of a = b the two approaches yield the same answer (a and
b are shape parameters in resp. demand and supply functions)
the ‘potential’based equilibrium price is a ‘most likely’price, whilst the
D[p ] = S[p ] does not have that interpretation
Talk at Imperial College ()
May 21 - 2014
15 / 46
Part I: Classical physics and …nance (cont’d)
Talk at Imperial College ()
May 21 - 2014
16 / 46
Part II: Quantum physics and …nance: introduction
Before we start, in order to remove the exotic-ity....from the subject....
Talk at Imperial College ()
May 21 - 2014
17 / 46
Part II: Quantum physics and …nance: introduction
Before we start, in order to remove the exotic-ity....from the subject....
Some pubs with applications in the social science area:
Talk at Imperial College ()
May 21 - 2014
17 / 46
Part II: Quantum physics and …nance: introduction
Before we start, in order to remove the exotic-ity....from the subject....
Some pubs with applications in the social science area:
Khrennikov, A. (1999). Classical and quantum mechanics on
information spaces with applications to cognitive, psychological, social
and anomalous phenomena. Foundations of Physics 29; 1065-1098
Talk at Imperial College ()
May 21 - 2014
17 / 46
Part II: Quantum physics and …nance: introduction
Before we start, in order to remove the exotic-ity....from the subject....
Some pubs with applications in the social science area:
Khrennikov, A. (1999). Classical and quantum mechanics on
information spaces with applications to cognitive, psychological, social
and anomalous phenomena. Foundations of Physics 29; 1065-1098
Busemeyer and Bruza (2012)(Quantum models on cognition and
decision; CUP); Baaquie (2007) (Quantum …nance; CUP); Khrennikov
(2010) (Ubiquitous quantum structure; Springer); Haven and
Khrennikov (2013) (Quantum social science; CUP); Bagarello (2012)
(Quantum dynamics for classical systems; J. Wiley)
Talk at Imperial College ()
May 21 - 2014
17 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Before we start (continued)
Talk at Imperial College ()
May 21 - 2014
18 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Before we start (continued)
Segal, W. ; Segal I. E. (1998). The Black-Scholes pricing formula in
the quantum context. Proceedings of the National Academy of
Sciences of the USA 95; 4072-4075
Talk at Imperial College ()
May 21 - 2014
18 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Before we start (continued)
Segal, W. ; Segal I. E. (1998). The Black-Scholes pricing formula in
the quantum context. Proceedings of the National Academy of
Sciences of the USA 95; 4072-4075
Shubik M. (1999). Quantum economics, uncertainty and the optimal
grid size. Economics Letters 64 (3); 277-278
Talk at Imperial College ()
May 21 - 2014
18 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Before we start (continued)
Segal, W. ; Segal I. E. (1998). The Black-Scholes pricing formula in
the quantum context. Proceedings of the National Academy of
Sciences of the USA 95; 4072-4075
Shubik M. (1999). Quantum economics, uncertainty and the optimal
grid size. Economics Letters 64 (3); 277-278
Aerts, D.; Broekaert, J.; Gabora, L.; Sozzo, S. (2013). Quantum
structure and human thought. Behavioral and Brain Sciences 36;
274-276
Talk at Imperial College ()
May 21 - 2014
18 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Before we start (continued)
Segal, W. ; Segal I. E. (1998). The Black-Scholes pricing formula in
the quantum context. Proceedings of the National Academy of
Sciences of the USA 95; 4072-4075
Shubik M. (1999). Quantum economics, uncertainty and the optimal
grid size. Economics Letters 64 (3); 277-278
Aerts, D.; Broekaert, J.; Gabora, L.; Sozzo, S. (2013). Quantum
structure and human thought. Behavioral and Brain Sciences 36;
274-276
Khrennikova, P.; Haven, E..; Khrennikov, A. (2014). An application of
the theory of open quantum systems to model the dynamics of party
governance in the US political system. International Journal of
Theoretical Physics 53(4); 1346-1360
Talk at Imperial College ()
May 21 - 2014
18 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Before we start (continued)
Segal, W. ; Segal I. E. (1998). The Black-Scholes pricing formula in
the quantum context. Proceedings of the National Academy of
Sciences of the USA 95; 4072-4075
Shubik M. (1999). Quantum economics, uncertainty and the optimal
grid size. Economics Letters 64 (3); 277-278
Aerts, D.; Broekaert, J.; Gabora, L.; Sozzo, S. (2013). Quantum
structure and human thought. Behavioral and Brain Sciences 36;
274-276
Khrennikova, P.; Haven, E..; Khrennikov, A. (2014). An application of
the theory of open quantum systems to model the dynamics of party
governance in the US political system. International Journal of
Theoretical Physics 53(4); 1346-1360
Hawkins, R. J.; Aoki, M.; Frieden, B. J. (2010). Asymmetric
information and macroeconomic dynamics. Physica A 389; 3565-3571
Talk at Imperial College ()
May 21 - 2014
18 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Some relevant background pubs - purely in physics
Talk at Imperial College ()
May 21 - 2014
19 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Some relevant background pubs - purely in physics
Nelson, E. (1966). Derivation of the Schrödinger equation from
Newtonian mechanics. Physical Review 150; 1079-
Talk at Imperial College ()
May 21 - 2014
19 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Some relevant background pubs - purely in physics
Nelson, E. (1966). Derivation of the Schrödinger equation from
Newtonian mechanics. Physical Review 150; 1079Bohm, D and Hiley, B. (1993). The undivided universe: an ontological
interpretation of quantum theory. Routledge - London.
Talk at Imperial College ()
May 21 - 2014
19 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Some relevant background pubs - purely in physics
Nelson, E. (1966). Derivation of the Schrödinger equation from
Newtonian mechanics. Physical Review 150; 1079Bohm, D and Hiley, B. (1993). The undivided universe: an ontological
interpretation of quantum theory. Routledge - London.
Brandenburger A.; Yanofsky N. (2008). A classi…cation of
hidden-variable properties. Journal of Physics A 41; 425302
Talk at Imperial College ()
May 21 - 2014
19 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Some relevant background pubs - purely in physics
Nelson, E. (1966). Derivation of the Schrödinger equation from
Newtonian mechanics. Physical Review 150; 1079Bohm, D and Hiley, B. (1993). The undivided universe: an ontological
interpretation of quantum theory. Routledge - London.
Brandenburger A.; Yanofsky N. (2008). A classi…cation of
hidden-variable properties. Journal of Physics A 41; 425302
Madelung, E. (1926). Quantenttheorie in Hydrodynamischer Form.
Zeitschrift fur Physik 40; 322-
Talk at Imperial College ()
May 21 - 2014
19 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Some relevant background pubs - purely in physics
Nelson, E. (1966). Derivation of the Schrödinger equation from
Newtonian mechanics. Physical Review 150; 1079Bohm, D and Hiley, B. (1993). The undivided universe: an ontological
interpretation of quantum theory. Routledge - London.
Brandenburger A.; Yanofsky N. (2008). A classi…cation of
hidden-variable properties. Journal of Physics A 41; 425302
Madelung, E. (1926). Quantenttheorie in Hydrodynamischer Form.
Zeitschrift fur Physik 40; 322Reginatto, M. (1998). Derivation of the equations of nonrelativistic
quantum mechanics using the principle of minimum Fisher information.
Physical Review A 58(3); 1775-1778
Talk at Imperial College ()
May 21 - 2014
19 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Some relevant background pubs - purely in physics (cont’d)
Talk at Imperial College ()
May 21 - 2014
20 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Some relevant background pubs - purely in physics (cont’d)
Bohm D. (1952a). A suggested interpretation of the quantum theory in
terms of hidden variables. Physical Review ; 85; 166-179.
Talk at Imperial College ()
May 21 - 2014
20 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Some relevant background pubs - purely in physics (cont’d)
Bohm D. (1952a). A suggested interpretation of the quantum theory in
terms of hidden variables. Physical Review ; 85; 166-179.
Bohm D. (1952b). A suggested interpretation of the quantum theory in
terms of hidden variables. Physical Review ; 85; 180-193.
Talk at Imperial College ()
May 21 - 2014
20 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Some relevant background pubs - purely in physics (cont’d)
Bohm D. (1952a). A suggested interpretation of the quantum theory in
terms of hidden variables. Physical Review ; 85; 166-179.
Bohm D. (1952b). A suggested interpretation of the quantum theory in
terms of hidden variables. Physical Review ; 85; 180-193.
Some research funding and conferences
Talk at Imperial College ()
May 21 - 2014
20 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Some relevant background pubs - purely in physics (cont’d)
Bohm D. (1952a). A suggested interpretation of the quantum theory in
terms of hidden variables. Physical Review ; 85; 166-179.
Bohm D. (1952b). A suggested interpretation of the quantum theory in
terms of hidden variables. Physical Review ; 85; 180-193.
Some research funding and conferences
NSF (Busemeyer et al); Belgian Fund for Scienti…c Research
(D’Hooghe, Aerts and Haven); Leverhulme (Pothos et al)
Talk at Imperial College ()
May 21 - 2014
20 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Some relevant background pubs - purely in physics (cont’d)
Bohm D. (1952a). A suggested interpretation of the quantum theory in
terms of hidden variables. Physical Review ; 85; 166-179.
Bohm D. (1952b). A suggested interpretation of the quantum theory in
terms of hidden variables. Physical Review ; 85; 180-193.
Some research funding and conferences
NSF (Busemeyer et al); Belgian Fund for Scienti…c Research
(D’Hooghe, Aerts and Haven); Leverhulme (Pothos et al)
Conferences: longest conference series in the world on quantum
foundations: Andrei Khrennikov’s Linnaeus Universities series
Talk at Imperial College ()
May 21 - 2014
20 / 46
Part II: Quantum physics and …nance: introduction
(cont’d)
Some relevant background pubs - purely in physics (cont’d)
Bohm D. (1952a). A suggested interpretation of the quantum theory in
terms of hidden variables. Physical Review ; 85; 166-179.
Bohm D. (1952b). A suggested interpretation of the quantum theory in
terms of hidden variables. Physical Review ; 85; 180-193.
Some research funding and conferences
NSF (Busemeyer et al); Belgian Fund for Scienti…c Research
(D’Hooghe, Aerts and Haven); Leverhulme (Pothos et al)
Conferences: longest conference series in the world on quantum
foundations: Andrei Khrennikov’s Linnaeus Universities series
UCI; 7th Quantum Interaction conference (Un. of Leicester); 8th
Quantum Interaction conference (ETH Zürich); IQSA (2014)
Talk at Imperial College ()
May 21 - 2014
20 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information
Estimate a quantity ‘x0 ’when noise ‘x’is present: xobs : xobs = x0 + x
Talk at Imperial College ()
May 21 - 2014
21 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information
Estimate a quantity ‘x0 ’when noise ‘x’is present: xobs : xobs = x0 + x
Brody et al. (2006) argue that amongst the three possible sources
which trigger asset price changes, one considers the information ‡ow
around the position of the asset
Talk at Imperial College ()
May 21 - 2014
21 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information
Estimate a quantity ‘x0 ’when noise ‘x’is present: xobs : xobs = x0 + x
Brody et al. (2006) argue that amongst the three possible sources
which trigger asset price changes, one considers the information ‡ow
around the position of the asset
R
Fisher information I can be de…ned as: P1 dP
dx dx, where P (.) is
the pdf on noise ‘x’
Talk at Imperial College ()
May 21 - 2014
21 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information
Estimate a quantity ‘x0 ’when noise ‘x’is present: xobs : xobs = x0 + x
Brody et al. (2006) argue that amongst the three possible sources
which trigger asset price changes, one considers the information ‡ow
around the position of the asset
R
Fisher information I can be de…ned as: P1 dP
dx dx, where P (.) is
the pdf on noise ‘x’
When P is peaked around ‘x’, it means there are little ‡uctuations
and therefore the level of information in xobs is high
Talk at Imperial College ()
May 21 - 2014
21 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information
Estimate a quantity ‘x0 ’when noise ‘x’is present: xobs : xobs = x0 + x
Brody et al. (2006) argue that amongst the three possible sources
which trigger asset price changes, one considers the information ‡ow
around the position of the asset
R
Fisher information I can be de…ned as: P1 dP
dx dx, where P (.) is
the pdf on noise ‘x’
When P is peaked around ‘x’, it means there are little ‡uctuations
and therefore the level of information in xobs is high
In econometrics we are taught that the mean squared errors in an
unbiased estimate of x0 must exceed 1/I . 1/I is the Cramer-Rao
bound
Talk at Imperial College ()
May 21 - 2014
21 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Using Jaynes (1957); Hawkins and Frieden (2014) show that one can
optimize I with Lagrangian multipliers
Talk at Imperial College ()
May 21 - 2014
22 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Using Jaynes (1957); Hawkins and Frieden (2014) show that one can
optimize I with Lagrangian multipliers
Hawkins and Frieden (2014) show that if P is obtained from the very
basic quantum mechanical premise: the probability amplitude ψ, then
this ψ follows
h a Schrödinger-likei di¤erential equation:
d 2 ψ (x )
dx 2
=
1
4
λ0 + ∑N
n = 1 λ n fn ( x ) ψ ( x )
Talk at Imperial College ()
May 21 - 2014
22 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Using Jaynes (1957); Hawkins and Frieden (2014) show that one can
optimize I with Lagrangian multipliers
Hawkins and Frieden (2014) show that if P is obtained from the very
basic quantum mechanical premise: the probability amplitude ψ, then
this ψ follows
h a Schrödinger-likei di¤erential equation:
d 2 ψ (x )
dx 2
=
1
4
λ0 + ∑N
n = 1 λ n fn ( x ) ψ ( x )
λn fn (x ) as a potential generalizes the potentials we considered in part
I of the talk
Talk at Imperial College ()
May 21 - 2014
22 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Using Jaynes (1957); Hawkins and Frieden (2014) show that one can
optimize I with Lagrangian multipliers
Hawkins and Frieden (2014) show that if P is obtained from the very
basic quantum mechanical premise: the probability amplitude ψ, then
this ψ follows
h a Schrödinger-likei di¤erential equation:
d 2 ψ (x )
dx 2
=
1
4
λ0 + ∑N
n = 1 λ n fn ( x ) ψ ( x )
λn fn (x ) as a potential generalizes the potentials we considered in part
I of the talk
the wave function within its format of a probability amplitude is now
acquiring a macroscopic identity as a device which can be used to
formalize information
Talk at Imperial College ()
May 21 - 2014
22 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Jaynes, E.T. (1957). Information Theory and Statistical Mechanics.
Physical Review 106, 120-130
Talk at Imperial College ()
May 21 - 2014
23 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Jaynes, E.T. (1957). Information Theory and Statistical Mechanics.
Physical Review 106, 120-130
Hawkins, R.J., Frieden, B. R. (2014). Fisher information and
quantization in …nancial economics. ESRC Seminar Series: Financial
Modelling Post 2008: Where Next? (University of Leicester, UK)
Talk at Imperial College ()
May 21 - 2014
23 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Jaynes, E.T. (1957). Information Theory and Statistical Mechanics.
Physical Review 106, 120-130
Hawkins, R.J., Frieden, B. R. (2014). Fisher information and
quantization in …nancial economics. ESRC Seminar Series: Financial
Modelling Post 2008: Where Next? (University of Leicester, UK)
Brody, D., Hughston, L.P., Macrina, A. (2006). Information based
asset pricing. Working paper (Department of Mathematics - King’s
College (London))
Talk at Imperial College ()
May 21 - 2014
23 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
The immediate questions now become: ‘what type of probability’?
and ‘what information’?
Talk at Imperial College ()
May 21 - 2014
24 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
The immediate questions now become: ‘what type of probability’?
and ‘what information’?
We have pronounced the word ‘probability’in relation to ψ
Talk at Imperial College ()
May 21 - 2014
24 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
The immediate questions now become: ‘what type of probability’?
and ‘what information’?
We have pronounced the word ‘probability’in relation to ψ
But frankly: what is its interpretation?
Talk at Imperial College ()
May 21 - 2014
24 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
The immediate questions now become: ‘what type of probability’?
and ‘what information’?
We have pronounced the word ‘probability’in relation to ψ
But frankly: what is its interpretation?
Consider the non-arbitrage theorem, and assume the risk free rate is
zero
Talk at Imperial College ()
May 21 - 2014
24 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
The immediate questions now become: ‘what type of probability’?
and ‘what information’?
We have pronounced the word ‘probability’in relation to ψ
But frankly: what is its interpretation?
Consider the non-arbitrage theorem, and assume the risk free rate is
zero
Denote the state prices in that theorem with si
Talk at Imperial College ()
May 21 - 2014
24 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
The immediate questions now become: ‘what type of probability’?
and ‘what information’?
We have pronounced the word ‘probability’in relation to ψ
But frankly: what is its interpretation?
Consider the non-arbitrage theorem, and assume the risk free rate is
zero
Denote the state prices in that theorem with si
We could formulate, kets:
p
p
p
p
s1 jstate1i + s2 jstate2i + s3 jstate3i + s4 jstate4i + ....
Talk at Imperial College ()
May 21 - 2014
24 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Assume we make a ‘measurement’: state 2 occurs with probability s2
Talk at Imperial College ()
May 21 - 2014
25 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Assume we make a ‘measurement’: state 2 occurs with probability s2
Financially, assuming the risk free rate is zero, we could say one is
willing to pay s2 units of currency!! for say ‘1’unit of currency if state
2 occurs and nothing else if another state occurs
Talk at Imperial College ()
May 21 - 2014
25 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Assume we make a ‘measurement’: state 2 occurs with probability s2
Financially, assuming the risk free rate is zero, we could say one is
willing to pay s2 units of currency!! for say ‘1’unit of currency if state
2 occurs and nothing else if another state occurs
We could make the statement: the more one is willing to pay - the
higher one thinks the probability s2 will be
Talk at Imperial College ()
May 21 - 2014
25 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Assume we make a ‘measurement’: state 2 occurs with probability s2
Financially, assuming the risk free rate is zero, we could say one is
willing to pay s2 units of currency!! for say ‘1’unit of currency if state
2 occurs and nothing else if another state occurs
We could make the statement: the more one is willing to pay - the
higher one thinks the probability s2 will be
What type of probability is this?
Talk at Imperial College ()
May 21 - 2014
25 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Assume we make a ‘measurement’: state 2 occurs with probability s2
Financially, assuming the risk free rate is zero, we could say one is
willing to pay s2 units of currency!! for say ‘1’unit of currency if state
2 occurs and nothing else if another state occurs
We could make the statement: the more one is willing to pay - the
higher one thinks the probability s2 will be
What type of probability is this?
See Ballentine, L. (2007). Objective and subjective probabilities in
quantum mechanics. In: Quantum Theory. G. Adenier; A. Yu.
Khrennikov; P. Lahti; V. I. Man’ko; T. M. Nieuwenhuizen (Eds);
American Institute of Physics Proc. 962, 28-33.
Talk at Imperial College ()
May 21 - 2014
25 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
A propensity based probability? Then this would mean: ‘the
probability the next measurement will yield state 2 is given by ‘s2 ’
Talk at Imperial College ()
May 21 - 2014
26 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
A propensity based probability? Then this would mean: ‘the
probability the next measurement will yield state 2 is given by ‘s2 ’
Is this reasonable? Not within the no-arbitrage theorem: this
probability is a synthetic probability!
Talk at Imperial College ()
May 21 - 2014
26 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
A propensity based probability? Then this would mean: ‘the
probability the next measurement will yield state 2 is given by ‘s2 ’
Is this reasonable? Not within the no-arbitrage theorem: this
probability is a synthetic probability!
A frequency based probability? Then this would mean: ‘in the long
run of similar measurements on this state, the fraction of ‘state 2’
should be close to the level of probability s2
Talk at Imperial College ()
May 21 - 2014
26 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
A propensity based probability? Then this would mean: ‘the
probability the next measurement will yield state 2 is given by ‘s2 ’
Is this reasonable? Not within the no-arbitrage theorem: this
probability is a synthetic probability!
A frequency based probability? Then this would mean: ‘in the long
run of similar measurements on this state, the fraction of ‘state 2’
should be close to the level of probability s2
Is this reasonable? Not within the no-arbitrage theorem: this
probability is a synthetic probability!
Talk at Imperial College ()
May 21 - 2014
26 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
A propensity based probability? Then this would mean: ‘the
probability the next measurement will yield state 2 is given by ‘s2 ’
Is this reasonable? Not within the no-arbitrage theorem: this
probability is a synthetic probability!
A frequency based probability? Then this would mean: ‘in the long
run of similar measurements on this state, the fraction of ‘state 2’
should be close to the level of probability s2
Is this reasonable? Not within the no-arbitrage theorem: this
probability is a synthetic probability!
A subjective probability? Then this would mean: ‘my degree of
belief that the next measurement will yield state 2 will be given by s2 ’
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Back to the ‘wave function’in our non-arbitrage context: what
information does it represent?
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Back to the ‘wave function’in our non-arbitrage context: what
information does it represent?
Within purely the non-arbitrage theorem setting (thus disregarding
the insurance price set up): this wave function would surely NOT
describe an observer independent reality. In fact it does not describe
any reality?
Talk at Imperial College ()
May 21 - 2014
27 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Back to the ‘wave function’in our non-arbitrage context: what
information does it represent?
Within purely the non-arbitrage theorem setting (thus disregarding
the insurance price set up): this wave function would surely NOT
describe an observer independent reality. In fact it does not describe
any reality?
Within purely the non-arbitrage theorem setting (but NOT
disregarding the insurance price set up): this wave function would
describe a subjective probability which depends on the observer’s
knowledge
Talk at Imperial College ()
May 21 - 2014
27 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Let us consider two examples (on Fisher information)
Talk at Imperial College ()
May 21 - 2014
28 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Let us consider two examples (on Fisher information)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Let us consider two examples (on Fisher information)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
Example 2: A sketch of the mechanism of how Fisher
information and payo¤ functions can co-exist
Talk at Imperial College ()
May 21 - 2014
28 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
Talk at Imperial College ()
May 21 - 2014
29 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
Basil Hiley and David Bohm show that with the inputting of the polar
form of the wave function into the SE, one gets for the real part of
2
the complex equation, a HJ equation with term: R1 ∂∂qR2 , with R
amplitude function of the polar form of wave function.
Talk at Imperial College ()
May 21 - 2014
29 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
Basil Hiley and David Bohm show that with the inputting of the polar
form of the wave function into the SE, one gets for the real part of
2
the complex equation, a HJ equation with term: R1 ∂∂qR2 , with R
amplitude function of the polar form of wave function.
See also: Holland P. (2000). The quantum theory of motion: an
account of the de Broglie-Bohm causal interpretation of quantum
mechanics. Cambridge University Press.
Talk at Imperial College ()
May 21 - 2014
29 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
Basil Hiley and David Bohm show that with the inputting of the polar
form of the wave function into the SE, one gets for the real part of
2
the complex equation, a HJ equation with term: R1 ∂∂qR2 , with R
amplitude function of the polar form of wave function.
See also: Holland P. (2000). The quantum theory of motion: an
account of the de Broglie-Bohm causal interpretation of quantum
mechanics. Cambridge University Press.
This quantity is proportional to Fisher information
Talk at Imperial College ()
May 21 - 2014
29 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
Basil Hiley and David Bohm show that with the inputting of the polar
form of the wave function into the SE, one gets for the real part of
2
the complex equation, a HJ equation with term: R1 ∂∂qR2 , with R
amplitude function of the polar form of wave function.
See also: Holland P. (2000). The quantum theory of motion: an
account of the de Broglie-Bohm causal interpretation of quantum
mechanics. Cambridge University Press.
This quantity is proportional to Fisher information
Reginatto, M. (1998). Derivation of the equations of nonrelativistic
quantum mechanics using the principle of minimum Fisher
information. Physical Review A; 58(3); 1775-1778
Talk at Imperial College ()
May 21 - 2014
29 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
Talk at Imperial College ()
May 21 - 2014
30 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
One now obtains: the second law: m
Q the so called quantum potential
Talk at Imperial College ()
d 2 q (t )
dt 2
=
∂V (q,t )
∂q
∂Q (q,t )
∂q ,
May 21 - 2014
with
30 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
One now obtains: the second law: m
Q the so called quantum potential
d 2 q (t )
dt 2
=
∂V (q,t )
∂q
∂Q (q,t )
∂q ,
with
Say an amplitude function: R (q ) = c (q 2 + d ), c, d > 0;
∂Q
= (q 2 +4qd )2
Q (q ) = q 2 +2d and the ‘force’: ∂q
Talk at Imperial College ()
May 21 - 2014
30 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
One now obtains: the second law: m
Q the so called quantum potential
d 2 q (t )
dt 2
=
∂V (q,t )
∂q
∂Q (q,t )
∂q ,
with
Say an amplitude function: R (q ) = c (q 2 + d ), c, d > 0;
∂Q
= (q 2 +4qd )2
Q (q ) = q 2 +2d and the ‘force’: ∂q
With q small:
4q
:
(d )2
with price q going up there is resistance for it to
continue going up; alternatively, with q large:
occurs
Talk at Imperial College ()
4
q3
: the opposite
May 21 - 2014
30 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
One now obtains: the second law: m
Q the so called quantum potential
d 2 q (t )
dt 2
=
∂V (q,t )
∂q
∂Q (q,t )
∂q ,
with
Say an amplitude function: R (q ) = c (q 2 + d ), c, d > 0;
∂Q
= (q 2 +4qd )2
Q (q ) = q 2 +2d and the ‘force’: ∂q
With q small:
4q
:
(d )2
with price q going up there is resistance for it to
continue going up; alternatively, with q large:
occurs
4
q3
: the opposite
The price trajectory q (t ) can be found as the solution of the second
0
law equation with initial condition q (t0 ) = q0 , q 0 (t0 ) = q0
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
The quantum potential is closely related to Fisher information
Talk at Imperial College ()
May 21 - 2014
31 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
The quantum potential is closely related to Fisher information
A pricing rule can be derived using the quantum potential as input
Talk at Imperial College ()
May 21 - 2014
31 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
The quantum potential is closely related to Fisher information
A pricing rule can be derived using the quantum potential as input
The pricing rule forms part of the Second law
Talk at Imperial College ()
May 21 - 2014
31 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
The quantum potential is closely related to Fisher information
A pricing rule can be derived using the quantum potential as input
The pricing rule forms part of the Second law
We can derive trajectories from that second law
Talk at Imperial College ()
May 21 - 2014
31 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
The quantum potential is closely related to Fisher information
A pricing rule can be derived using the quantum potential as input
The pricing rule forms part of the Second law
We can derive trajectories from that second law
Within this mechanism sketched here: Fisher information and price
trajectories co-exist
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
Talk at Imperial College ()
May 21 - 2014
32 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
BUT: for any smooth Bohmian trajectory its quadratic variation is
zero
Talk at Imperial College ()
May 21 - 2014
32 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
BUT: for any smooth Bohmian trajectory its quadratic variation is
zero
This runs counter the usual property of the trajectories of continuous
square integrable martingales where quadratic variation is non-zero
Talk at Imperial College ()
May 21 - 2014
32 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
BUT: for any smooth Bohmian trajectory its quadratic variation is
zero
This runs counter the usual property of the trajectories of continuous
square integrable martingales where quadratic variation is non-zero
It is unfortunately NOT at all easy to obtain non-zero quadratic
variation.
Talk at Imperial College ()
May 21 - 2014
32 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 1: A sketch of the mechanism of how Fisher
information and price trajectories can co-exist
BUT: for any smooth Bohmian trajectory its quadratic variation is
zero
This runs counter the usual property of the trajectories of continuous
square integrable martingales where quadratic variation is non-zero
It is unfortunately NOT at all easy to obtain non-zero quadratic
variation.
See: Choustova O. (2007). Quantum modeling of nonlinear dynamics
of prices of shares: Bohmian approach. Theoretical and Mathematical
Physics; 152(7); 1213-1222.
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 2: A sketch of the mechanism of how Fisher
information and payo¤ functions can co-exist
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 2: A sketch of the mechanism of how Fisher
information and payo¤ functions can co-exist
Consider a …nancial payo¤ function: simply a function de…ned on
R0+ ! R
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 2: A sketch of the mechanism of how Fisher
information and payo¤ functions can co-exist
Consider a …nancial payo¤ function: simply a function de…ned on
R0+ ! R
Assumption 1. There exists a level of public information in the
economy relative to the particular payo¤ function.
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 2: A sketch of the mechanism of how Fisher
information and payo¤ functions can co-exist
Consider a …nancial payo¤ function: simply a function de…ned on
R0+ ! R
Assumption 1. There exists a level of public information in the
economy relative to the particular payo¤ function.
Assumption 2. Private information is information the holder of the
…nancial contract possesses (this contract precisely describes the
payo¤ function) on how he/she thinks the actual market price will be
positioned relative to the price domain of the payo¤ function as
stipulated in the contract.
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Example 2: A sketch of the mechanism of how Fisher
information and payo¤ functions can co-exist
Consider a …nancial payo¤ function: simply a function de…ned on
R0+ ! R
Assumption 1. There exists a level of public information in the
economy relative to the particular payo¤ function.
Assumption 2. Private information is information the holder of the
…nancial contract possesses (this contract precisely describes the
payo¤ function) on how he/she thinks the actual market price will be
positioned relative to the price domain of the payo¤ function as
stipulated in the contract.
Assumption 3. The precise form of private information is given by
the functional form of the wave function ψ(x ),which follows the
Schrödinger-like di¤erential equation.
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Private information is not a ‘new’quantity: Kyle depth (Kyle (1985)):
(ratio of the amount of noise trading to the amount of private
information the informed trader is expected to have.)
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Private information is not a ‘new’quantity: Kyle depth (Kyle (1985)):
(ratio of the amount of noise trading to the amount of private
information the informed trader is expected to have.)
Kyle, A.S. (1985). Continuous auctions and insider trading.
Econometrica 53, 1315-1335.
Talk at Imperial College ()
May 21 - 2014
34 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Private information is not a ‘new’quantity: Kyle depth (Kyle (1985)):
(ratio of the amount of noise trading to the amount of private
information the informed trader is expected to have.)
Kyle, A.S. (1985). Continuous auctions and insider trading.
Econometrica 53, 1315-1335.
We would like to de…ne that for a given level of public information,
the functional form of the wave function may be a¤ected (or not) by
the type of payo¤ function
Talk at Imperial College ()
May 21 - 2014
34 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Private information is not a ‘new’quantity: Kyle depth (Kyle (1985)):
(ratio of the amount of noise trading to the amount of private
information the informed trader is expected to have.)
Kyle, A.S. (1985). Continuous auctions and insider trading.
Econometrica 53, 1315-1335.
We would like to de…ne that for a given level of public information,
the functional form of the wave function may be a¤ected (or not) by
the type of payo¤ function
What does it mean?
Talk at Imperial College ()
May 21 - 2014
34 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Private information is not a ‘new’quantity: Kyle depth (Kyle (1985)):
(ratio of the amount of noise trading to the amount of private
information the informed trader is expected to have.)
Kyle, A.S. (1985). Continuous auctions and insider trading.
Econometrica 53, 1315-1335.
We would like to de…ne that for a given level of public information,
the functional form of the wave function may be a¤ected (or not) by
the type of payo¤ function
What does it mean?
If functional form is a¤ected: this means that public information on
its own is not deemed to be su¢ cient to infer that the price interval
in which the actual price will fall will be very close to the price
interval as stipulated in the contract
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Consider for instance a …nancial payo¤ function which only has
theoretical value: a Dirac δ function
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Consider for instance a …nancial payo¤ function which only has
theoretical value: a Dirac δ function
Payo¤ increases dramatically with the narrowing of the spread
parameter α
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Consider for instance a …nancial payo¤ function which only has
theoretical value: a Dirac δ function
Payo¤ increases dramatically with the narrowing of the spread
parameter α
Measurement of I : very high given the steepness of decay of the
wave function
Talk at Imperial College ()
May 21 - 2014
35 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Consider for instance a …nancial payo¤ function which only has
theoretical value: a Dirac δ function
Payo¤ increases dramatically with the narrowing of the spread
parameter α
Measurement of I : very high given the steepness of decay of the
wave function
Very sharp slope on density function over a very small domain:
If
very high
l (dom (f ))
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Consider for instance a …nancial payo¤ function which only has
theoretical value: a Dirac δ function
Payo¤ increases dramatically with the narrowing of the spread
parameter α
Measurement of I : very high given the steepness of decay of the
wave function
Very sharp slope on density function over a very small domain:
If
very high
l (dom (f ))
From a …nancial point of view, private information is to be seen as
highly relevant: there is a lot of information needed to know what the
payo¤ will be
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Consider for instance a …nancial payo¤ function f = A, with A 2 R0+
and with dom (f ) = R+
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Consider for instance a …nancial payo¤ function f = A, with A 2 R0+
and with dom (f ) = R+
The wave function will have a long tail on a large domain:
If
!0
l (dom (f ))
Talk at Imperial College ()
May 21 - 2014
36 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in modelling information (cont’d)
Consider for instance a …nancial payo¤ function f = A, with A 2 R0+
and with dom (f ) = R+
The wave function will have a long tail on a large domain:
If
!0
l (dom (f ))
From a …nancial point of view, private information is to be seen as
virtually irrelevant, since there is no information needed at all in order
to know what the payo¤ will be: the payo¤ is guaranteed
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: modelling
information: graphs
Example 1
Pay off
Pay off Function
Public Info
Example 2
Pay off
Pay off Function
A
Public Info
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
Assume you are reading a book: ideas are mixed up with other ideas:
‘how is my son doing outside with his little scooter?’
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
Assume you are reading a book: ideas are mixed up with other ideas:
‘how is my son doing outside with his little scooter?’
Superposition of thoughts:
ja >= c1 jidea1 > +c2 jidea2 > +c3 jidea3 > +.....; with of course:
jci j2 =probability of each idea to occur in the superposed thought
Talk at Imperial College ()
May 21 - 2014
38 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
Assume you are reading a book: ideas are mixed up with other ideas:
‘how is my son doing outside with his little scooter?’
Superposition of thoughts:
ja >= c1 jidea1 > +c2 jidea2 > +c3 jidea3 > +.....; with of course:
jci j2 =probability of each idea to occur in the superposed thought
Similarly we could think of values versus price of assets:
jp >= a1 jvalue1 > +a2 jvalue2 > +a3 jvalue3 > +.....; with of course
jai j2 =probability of each value to occur
Talk at Imperial College ()
May 21 - 2014
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Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
Some problems: i) are jidea1 >; jidea2 >; jidea3 > ... linearly
independent? Are they a basis for a space?
Talk at Imperial College ()
May 21 - 2014
39 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
Some problems: i) are jidea1 >; jidea2 >; jidea3 > ... linearly
independent? Are they a basis for a space?
More di¢ cult: do we have an additive inverse? What is its meaning:
jidea1 > +j idea1 >=?0
Talk at Imperial College ()
May 21 - 2014
39 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
Some problems: i) are jidea1 >; jidea2 >; jidea3 > ... linearly
independent? Are they a basis for a space?
More di¢ cult: do we have an additive inverse? What is its meaning:
jidea1 > +j idea1 >=?0
What is the meaning of: < idea1 jidea2 >= 0?
Talk at Imperial College ()
May 21 - 2014
39 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
Some problems: i) are jidea1 >; jidea2 >; jidea3 > ... linearly
independent? Are they a basis for a space?
More di¢ cult: do we have an additive inverse? What is its meaning:
jidea1 > +j idea1 >=?0
What is the meaning of: < idea1 jidea2 >= 0?
What is the meaning of the eigenvalues of the matrix of the
Hamiltonian?
Talk at Imperial College ()
May 21 - 2014
39 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In decision making, the concept of probability interference has now
made some inroads
Talk at Imperial College ()
May 21 - 2014
40 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In decision making, the concept of probability interference has now
made some inroads
The Ellsberg paradox is a well known decision making paradox and it
can be explained with the idea of a two stage gamble
Talk at Imperial College ()
May 21 - 2014
40 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In decision making, the concept of probability interference has now
made some inroads
The Ellsberg paradox is a well known decision making paradox and it
can be explained with the idea of a two stage gamble
You gamble the …rst time and then you decide to gamble a second
time on the basis of you being told: i) you won in the …rst gamble; ii)
you lost in the …rst gamble; iii) you have no information on how you
did in the …rst gamble.
Talk at Imperial College ()
May 21 - 2014
40 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In decision making, the concept of probability interference has now
made some inroads
The Ellsberg paradox is a well known decision making paradox and it
can be explained with the idea of a two stage gamble
You gamble the …rst time and then you decide to gamble a second
time on the basis of you being told: i) you won in the …rst gamble; ii)
you lost in the …rst gamble; iii) you have no information on how you
did in the …rst gamble.
The so called ‘sure-thing’principle in economics says that: if you
prefer to gamble the second time, knowing you won the …rst gamble
and you are preferring to gamble the second time, given you know you
lost in the …rst gamble; then you should be …ne to gamble the second
time even if you have no information whether you lost or won in the
…rst gamble.
Talk at Imperial College ()
May 21 - 2014
40 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
Work by Sha…r and Tversky show many decision makers will violate
this sure-thing principle
Talk at Imperial College ()
May 21 - 2014
41 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
Work by Sha…r and Tversky show many decision makers will violate
this sure-thing principle
Two approaches can be proposed: a Markov approach and a
quantum-like approach
Talk at Imperial College ()
May 21 - 2014
41 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
Work by Sha…r and Tversky show many decision makers will violate
this sure-thing principle
Two approaches can be proposed: a Markov approach and a
quantum-like approach
In the Markov approach: the probability of gambling in the unknown
case should be equal to the average of the probabilities of gambling in
the known cases.
Talk at Imperial College ()
May 21 - 2014
41 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
Work by Sha…r and Tversky show many decision makers will violate
this sure-thing principle
Two approaches can be proposed: a Markov approach and a
quantum-like approach
In the Markov approach: the probability of gambling in the unknown
case should be equal to the average of the probabilities of gambling in
the known cases.
With observed frequencies one gets: 0.36 which should be the
average of 0.59 and 0.69 (in respectively ‘known to lose’and ‘known
to win’) - and clearly it is not
Talk at Imperial College ()
May 21 - 2014
41 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In the quantum-like approach, one can de…ne basis states with the
following kets (we use the same set up as in Busemeyer (for full
references see our book: pp. 152-154):
Talk at Imperial College ()
May 21 - 2014
42 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In the quantum-like approach, one can de…ne basis states with the
following kets (we use the same set up as in Busemeyer (for full
references see our book: pp. 152-154):
jWG > (you simultaneously believe you won in the …rst gamble and
you will undertake a second gamble);
Talk at Imperial College ()
May 21 - 2014
42 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In the quantum-like approach, one can de…ne basis states with the
following kets (we use the same set up as in Busemeyer (for full
references see our book: pp. 152-154):
jWG > (you simultaneously believe you won in the …rst gamble and
you will undertake a second gamble);
jWN >; (you simultaneously believe you won in the …rst gamble and
will not gamble);
Talk at Imperial College ()
May 21 - 2014
42 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In the quantum-like approach, one can de…ne basis states with the
following kets (we use the same set up as in Busemeyer (for full
references see our book: pp. 152-154):
jWG > (you simultaneously believe you won in the …rst gamble and
you will undertake a second gamble);
jWN >; (you simultaneously believe you won in the …rst gamble and
will not gamble);
jLG >; jLN > with corresponding probability amplitudes:
ψWG ; ψWN ; ψLG ; ψLN .
Talk at Imperial College ()
May 21 - 2014
42 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In the quantum-like approach, one can de…ne basis states with the
following kets (we use the same set up as in Busemeyer (for full
references see our book: pp. 152-154):
jWG > (you simultaneously believe you won in the …rst gamble and
you will undertake a second gamble);
jWN >; (you simultaneously believe you won in the …rst gamble and
will not gamble);
jLG >; jLN > with corresponding probability amplitudes:
ψWG ; ψWN ; ψLG ; ψLN .
If there exists an initial state vector ψ and one gets the information
you lost or won, then Busemeyer (references in our book : pp.
152-154) proposes that a unitary operator U is applied: U.ψ.
Talk at Imperial College ()
May 21 - 2014
42 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
If you are informed you lost the gamble: the initial state is
transformed from: [ψWG ; ψWN ; ψLG ; ψLN ] to: [0, 0, ψLG , ψLN ].
Talk at Imperial College ()
May 21 - 2014
43 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
If you are informed you lost the gamble: the initial state is
transformed from: [ψWG ; ψWN ; ψLG ; ψLN ] to: [0, 0, ψLG , ψLN ].
An unknown state will be a superposition of the lost and win states.
Talk at Imperial College ()
May 21 - 2014
43 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
If you are informed you lost the gamble: the initial state is
transformed from: [ψWG ; ψWN ; ψLG ; ψLN ] to: [0, 0, ψLG , ψLN ].
An unknown state will be a superposition of the lost and win states.
As has been shown now by many authors (see our book pp. 152-154
for references), the quantum-like model can accommodate observed
percentages by using the probability interference term.
Talk at Imperial College ()
May 21 - 2014
43 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
This of course also leads us into the di¢ cult debate of what a
‘quantum probability’is as opposed to a non quantum probability
Talk at Imperial College ()
May 21 - 2014
44 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
This of course also leads us into the di¢ cult debate of what a
‘quantum probability’is as opposed to a non quantum probability
Is quantum probability subjective probability? Would it encapsulate
subjective expectations of an experimenter?
Talk at Imperial College ()
May 21 - 2014
44 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
This of course also leads us into the di¢ cult debate of what a
‘quantum probability’is as opposed to a non quantum probability
Is quantum probability subjective probability? Would it encapsulate
subjective expectations of an experimenter?
In the double slit experiment (see our book: pp. 122-123 and Chapter
8) the law of total probability is violated
Talk at Imperial College ()
May 21 - 2014
44 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
This of course also leads us into the di¢ cult debate of what a
‘quantum probability’is as opposed to a non quantum probability
Is quantum probability subjective probability? Would it encapsulate
subjective expectations of an experimenter?
In the double slit experiment (see our book: pp. 122-123 and Chapter
8) the law of total probability is violated
Consider a simple example: let a = +1 : “the democrats will win”;
a = 1 : the negation; a = +1. An example for the b-variable: “you
buy a condominium in midtown Manhattan”, b = 1 : the negation;
b = +1.
Talk at Imperial College ()
May 21 - 2014
44 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
This of course also leads us into the di¢ cult debate of what a
‘quantum probability’is as opposed to a non quantum probability
Is quantum probability subjective probability? Would it encapsulate
subjective expectations of an experimenter?
In the double slit experiment (see our book: pp. 122-123 and Chapter
8) the law of total probability is violated
Consider a simple example: let a = +1 : “the democrats will win”;
a = 1 : the negation; a = +1. An example for the b-variable: “you
buy a condominium in midtown Manhattan”, b = 1 : the negation;
b = +1.
The law of total probability: P(b = j ) = P(a = +1)P(b = j ja =
+1) + P(a = 1)P(b = j ja = 1),where j = +1 or j = 1.
Talk at Imperial College ()
May 21 - 2014
44 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
This of course also leads us into the di¢ cult debate of what a
‘quantum probability’is as opposed to a non quantum probability
Is quantum probability subjective probability? Would it encapsulate
subjective expectations of an experimenter?
In the double slit experiment (see our book: pp. 122-123 and Chapter
8) the law of total probability is violated
Consider a simple example: let a = +1 : “the democrats will win”;
a = 1 : the negation; a = +1. An example for the b-variable: “you
buy a condominium in midtown Manhattan”, b = 1 : the negation;
b = +1.
The law of total probability: P(b = j ) = P(a = +1)P(b = j ja =
+1) + P(a = 1)P(b = j ja = 1),where j = +1 or j = 1.
Decision making paradoxes violate this law of total probability (and so
does the double slit experiment)
Talk at Imperial College ()
May 21 - 2014
44 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In classical systems: representation of the classical world is done via
Abelian variables which are numbers
Talk at Imperial College ()
May 21 - 2014
45 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In classical systems: representation of the classical world is done via
Abelian variables which are numbers
In quantum systems: representation of the world is done via
non-Abelian variables and operators
Talk at Imperial College ()
May 21 - 2014
45 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In classical systems: representation of the classical world is done via
Abelian variables which are numbers
In quantum systems: representation of the world is done via
non-Abelian variables and operators
Data are numbers AND measurements are also numbers: one-to-one
correspondence
Talk at Imperial College ()
May 21 - 2014
45 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In classical systems: representation of the classical world is done via
Abelian variables which are numbers
In quantum systems: representation of the world is done via
non-Abelian variables and operators
Data are numbers AND measurements are also numbers: one-to-one
correspondence
Data and eigenvalues: are they in one-to-one correspondence?
Talk at Imperial College ()
May 21 - 2014
45 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In classical systems: representation of the classical world is done via
Abelian variables which are numbers
In quantum systems: representation of the world is done via
non-Abelian variables and operators
Data are numbers AND measurements are also numbers: one-to-one
correspondence
Data and eigenvalues: are they in one-to-one correspondence?
If not: then information is partial
Talk at Imperial College ()
May 21 - 2014
45 / 46
Part II: Quantum physics and …nance: using basics of
quantum physics in decision making modelling
In classical systems: representation of the classical world is done via
Abelian variables which are numbers
In quantum systems: representation of the world is done via
non-Abelian variables and operators
Data are numbers AND measurements are also numbers: one-to-one
correspondence
Data and eigenvalues: are they in one-to-one correspondence?
If not: then information is partial
See T. Robinson (2014) - in our module taught in the Department of
Physics and Astronomy (University of Leicester) ‘Quantum …nance
and social science’: slides 3 and 4 (module taught by E. Haven and
T. Robinson)
Talk at Imperial College ()
May 21 - 2014
45 / 46
THANK YOU!!!
Talk at Imperial College ()
May 21 - 2014
46 / 46