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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 Talk at Imperial College () 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 Talk at Imperial College () 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 8 / 46 Part I: Classical physics and …nance (cont’d) Immediate problems and challenges Talk at Imperial College () 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! Talk at Imperial College () ∂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 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 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) 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 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” Talk at Imperial College () May 21 - 2014 10 / 46 Part I: Classical physics and …nance (cont’d) Example 2: Momentum conservation in …nance.... Talk at Imperial College () May 21 - 2014 11 / 46 Part I: Classical physics and …nance (cont’d) Example 2: Momentum conservation in …nance.... consider L =f (V (q1 Talk at Imperial College () q2 )) 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 ) Talk at Imperial College () 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!!! Talk at Imperial College () May 21 - 2014 11 / 46 Part I: Classical physics and …nance (cont’d) Example 3: The microeconomic potential Talk at Imperial College () 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. Talk at Imperial College () 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 Talk at Imperial College () 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 26 / 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? 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? 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 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 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 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 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 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 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 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 32 / 46 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 33 / 46 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 33 / 46 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 33 / 46 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 33 / 46 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 33 / 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.) 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. 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 34 / 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 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 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 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 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 )) 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 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 f = A, with A 2 R0+ and with dom (f ) = R+ 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 )) 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 36 / 46 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 37 / 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?’ 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 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 38 / 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? 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