Nonlocal Diffusions and The Quantum Black-Scholes Equation: Modelling the Market Fear Factor
NNonlocal Diffusions and The QuantumBlack-Scholes Equation: Modelling the MarketFear Factor.
Will HICKS ∗∗ June 28, 2018
Abstract
In this paper, we establish a link between quantum stochastic pro-cesses, and nonlocal diffusions. We demonstrate how the non-commutativeBlack-Scholes equation of Accardi & Boukas (cf [1]) can be written inintegral form. This enables the application of the Monte-Carlo meth-ods adapted to McKean stochastic differential equations (cf [16]) for thesimulation of solutions. We show how unitary transformations can beapplied to classical Black-Scholes systems to introduce novel quantumeffects. These have a simple economic interpretation as a market ‘fearfactor’, whereby recent market turbulence causes an increase in volatil-ity going forward, that is not linked to either the local volatility functionor an additional stochastic variable. Lastly, we extend this system to 2variables, and consider Quantum models for bid-offer spread dynamics.
Keywords—
Quantum Black-Scholes, Hudson-Parthasarathy Quantum Stochas-tic Calculus, Nonlocal Diffusions, McKean Stochastic Differential Equations,Particle Method
The link between the classical Black-Scholes equation and quantum mechanicsand the application of quantum formalism to Mathematical Finance has beeninvestigated by several authors. For example: [1]-[5], [9]-[12], [15], and [17]-[21].In particular, the approach of modelling derivative prices using self-adjoint op-erators on a Hilbert space was suggested by Segal & Segal in [21]. In thispaper the authors noted that, in the real world, the market operates with im-perfect information and that different observables, such as underlying price andoption delta, are usually not simultaneously observable. This fact makes thenon-commutative extension of the Black-Scholes framework a natural step. The ∗∗
42 Cranes Park Avenue, Surbiton, KT5 8BP, United KingdomEmail: [email protected] a r X i v : . [ q -f i n . M F ] J un uthors point out that this approach addresses some of the limitations of theclassical Black-Scholes model, such as the underestimation of the probability ofextreme events - so called “fat tails”. In this sense, non-commutative Quan-tum models present an alternative means to capture complex market dynamics,without the addition of new stochastic variables.In [1], Accardi & Boukas derive a general form for the Quantum Black-Scholesequation based on the Hudson-Parthasarathy calculus (cf [13]) and show thata commutative unitary time development operator acting on the market state,leads to a classical Black-Scholes system. Further they give the quantum stochas-tic differential equation governing the time development operator, and demon-strate how unitary transformations can lead to non-commutativity. An exam-ple of a non-commutative Quantum Black-Scholes partial differential equationis derived, although the authors work in an abstract setting and do not discussspecific unitary transformations and Hilbert space representations of financialmarkets.Therefore, one objective of this work is to use the Accardi-Boukas framework tolook at how different unitary transformations can be used to transform the clas-sical Black-Scholes equation, and to understand how quantum effects becomeapparent. We then go on to explore an example application of the approach inthe modelling of bid-offer spread dynamics. The final objective of the currentwork is to identify suitable Monte-Carlo methods, which can be used for thesimulation of solutions.In section 2, we give an overview of the Accardi-Boukas derivation of the generalform for the Quantum Black-Scholes equation, from [1]. With the objective oflooking at “near classical” Black-Scholes worlds, we then derive specific formsfor the resulting partial differential equations that result from small transla-tions, and rotations. This in turn involves the extension of the Accardi-Boukasequation to systems with more than one underlying variable. We go on to dis-cuss how this approach can be applied to the modelling of bid-offer dynamics.In section 3, we show how this can be linked to the nonlocal diffusion pro-cesses discussed by Luczka, H¨anggi and Gadomski in [14]. Here the impact ofthe diffusion differential operator is spread out through the convolution witha “blurring” function. The Kramers-Moyal expansion of the nonlocal Fokker-Planck equations allows us to derive the moments of the blurring function forthe “near classical” quantum system.This approach allows a natural route to the visualisation of the quantum effectson the system using McKean SDEs (cf [16]). The Monte-Carlo methods, devel-oped by Guyon, and Henry-Labord`ere in [8], can then be adapted to the sim-ulation of solutions. This is discussed in section 4, where we present numericalresults and show how, by introducing small transformations to the system, thestochastic process now reacts to a market downturn by returning higher volatil-ity. This effect is observed even where there is a single static Black-Scholes typevolatility. 2 Quantum Black-Scholes equation
In this section we follow the notation given, by Accardi & Boukas, in [1]. Thecurrent market is represented by a vector in a Hilbert space: H , which containsall relevant information about the state of the market at an instant in time.The tradeable price for a security is represented by an self-adjoint operator on H : X , and the the spectrum of X represents possible prices.Let L [ R + ; H ] represent functions from the positive real axis (time) to theHilbert space H . Then the random behaviour of tradeable securities can bemodelled using the tensor product of H with the bosonic Fock space: H ⊗ Γ( L [ R + ; H ]). We term this the “market space”. The operator that returns thecurrent price becomes X ⊗ I , where I represents the identity operator. The timedevelopment of X ⊗ I into the future is modelled by: j t ( X ) = U ∗ t X ⊗ I U t H carries the initial state of the market and U t acts by introducing randomfluctuations that fill up the empty states in: Γ( L [ R + ; H ]). The functional formfor U t is derived by Hudson & Parthasarathy in [13], and is given by: dU t = − (cid:32)(cid:18) iH + L ∗ L (cid:19) dt + L ∗ SdA t − LdA † t + (cid:18) − S (cid:19) d Λ t (cid:33) U t dA † t , dA t , d Λ t represent the standard creation, annihilation, and Poisson op-erators of quantum stochastic calculus. H, S and L also operate on the mar-ket space, with S unitary, and H self-adjoint. The multiplication rules of theHudson-Parthasarathy calculus are given below (cf [13]):- dA † t d Λ t dA t dtdA † t d Λ t dA † t d Λ t dA t dt dA t dt S (cid:54) = 1, there is a non-zero Poisson termand the time development operator is non-commutative.The next thing to note is that, where S = 1, the Poisson term disappears.The model can be written using the Ito calculus in place of the more generalHudson-Parthasarathy framework. The Wiener process dW t can be modelledusing: dA t + dA † t .Let V T = j T ( X − K ) + , represent the option price process as at final ex-piry T , and K the operator given by multiplying by the strike. Further, for V t = j t ( X − K ) + the following expansion is assumed:3 t = F ( t, x ) = (cid:80) n,k a n,k ( t − t ) n ( x − x ) k The Hudson-Parthasarathy multiplication rules can be applied to this expansionto give a quantum stochastic differential equation for V t , that corresponds tothe usual Ito expansion used in the derivation of the classical Black-Scholes. Byassuming one can construct a hedge portfolio by holding the underlying and arisk free numeraire asset, Accardi & Boukas are able to derive the general formthe Quantum Black Scholes equation using the assumption that any portfoliomust be self financing. Proposition 1, from [1] gives the full Quantum Black-Scholes equation: a , ( t, j t ( X ))+ a , ( t, j t ( X )) j t ( θ )+ ∞ (cid:88) k =2 a ,k ( t, j t ( X )) j t ( αλ k − α † ) = a t j t ( θ )+ V t r − a t j t ( X ) r (1)Here, a t represents the holding in the underlying asset and is given by theboundary conditions: (cid:80) ∞ k =1 a ,k ( t, j t ( X )) j t ( λ k − α † ) = a t j t ( α † ) (cid:80) ∞ k =1 a ,k ( t, j t ( X )) j t ( αλ k − ) = a t j t ( α ) (cid:80) ∞ k =1 a ,k ( t, j t ( X )) j t ( λ k ) = a t j t ( λ )Further, θ, α and λ are given by: α = [ L ∗ , X ] S , λ = S ∗ XS − X , θ = i [ H, X ] − { L ∗ LX + XL ∗ L + 2 L ∗ XL } .In this case the boundary conditions arise because when the Poisson term: d Λis non-zero, unlike Ito calculus where expansion terms with order above 2 canbe ignored, higher order terms still contain non-vanishing contribution.
The natural Hilbert space for an equity price (say the FTSE price) is: H = L [ R ].In this case, the only unitary transactions we can use are the translations: T (cid:15) : f ( x ) → f ( x − ε )Here we have, for a translation invariant Lebesgue measure µ : (cid:104) T ε f | T ε g (cid:105) = (cid:82) R f ( x − ε ) g ( x − ε ) dµ = (cid:82) R f ( x ) g ( x ) dµ = (cid:104) f | g (cid:105) So S is unitary in this case. Therefore, translating by ε we get: λ = T − ε XT ε f ( x ) − Xf ( x ) = T − ε xf ( x − ε ) − xf ( x ) = ( x + ε ) f ( x ) − xf ( x ) = εf ( x )4o we have λ = ε , and it is clear the example given in [1] relates to a translationby ε = 1. Following the key steps from [1] Proposition 3, and inserting this backinto equation 1, we get the following Quantum Black-Scholes partial differentialequation for this system: Lemma 2.1.
Let u ( t, x ) represent the price at time t, of a derivative contractin the system described above under small translation ε , and with interest rate r . Then the quantum Black-Scholes equation becomes: ∂u ( t, x ) ∂t = rx ∂u ( t, x ) ∂x − u ( t, x ) r + ∞ (cid:88) k =2 ε k − k ! ∂ k u ( t, x ) ∂x k g ( x ) (2) Proof.
The proof follows the same steps Accardi & Boukas outline in [1] propo-sition 3, with small modifications.For ε = 0, the last term drops out, and the equation reverts to the classicalBlack-Scholes. We investigate the impact of non-zero ε in section 4. For the one dimensional market space: L [ R ], the Lebesgue invariant transla-tions, are the only unitary transformations available. However, the true currentstate of the financial market contains a much richer variety of information thanjust a single price, and by increasing the dimensionality of the Market spaceaccordingly we introduce a wider variety of unitary transformations, that canintroduce non-commutativity. For example, let x represent the FTSE mid-price,and (cid:15) half of the bid-offer spread so that ( x + (cid:15) ) represents the best offer-priceand ( x − (cid:15) ) the best bid-price. Now the market is represented by the Hilbertspace: H = L [ R ], and we can apply rotations, in addition to translations.We make the simplifying assumption that market participants can trade themid-price: x (for example during the end of day auction process) and that themarket has sufficient liquidity to enable participants to alternatively act as mar-ket makers (receiving bid-offer spread) or as hedgers (crossing bid-offer spread)and therefore trade the bid-offer spread: (cid:15) . Therefore we make the followingassumption: Assumption 2.2.
For any derivative payout V ( x T , (cid:15) T ) , we can construct ahedged portfolio, and can proceed with the derivation of the Quantum BlackScholes equation following the basic methodology from [1]. We now have separate creation, annihilation and Poisson operators, for x and (cid:15) ; dA x , dA (cid:15) etc. These can be combined using the multiplication table ([13],Theorem 4.5), by making the assumption that the bid-offer is uncorrelated withthe equity price. This corresponds to assumption 2.3: Assumption 2.3. dA x d Λ (cid:15) = dA (cid:15) d Λ x = d Λ x d Λ (cid:15) = d Λ (cid:15) d Λ x = dA x dA † (cid:15) = dA (cid:15) dA † x = d Λ x dA † (cid:15) = d Λ (cid:15) dA † x = 0 . Assumption 2.4. V t = F ( t, x, (cid:15) ) = (cid:80) n,k,l a n,l,k ( t − t ) n ( x − x ) k ( (cid:15) − (cid:15) ) l We can now derive the relevant Quantum Black-Scholes equation:
Proposition 2.5.
Let H = L [ R ] , and let X ⊗ and (cid:15) ⊗ operate on themarket space: H ⊗ Γ( L [ R + ; H ]) , to return the mid-price, and bid-offer spreadfor a tradeable security respectively. Further, let the notation from [1], and theabove assumptions apply.Then the Quantum Black-Scholes equation in this case is given by: a , , ( t, j t ( X ) , j t ( (cid:15) )) + a , , ( t, j t ( X ) , j t ( (cid:15) )) j t ( θ x ) + a , , ( t, j t ( X ) , j t ( (cid:15) )) j t ( θ (cid:15) )+ ∞ (cid:88) k =2 a ,k, ( t, j t ( X ) , j t ( (cid:15) )) j t ( α x λ k − x α † x ) + ∞ (cid:88) l =2 a , ,l ( t, j t ( X ) , j t ( (cid:15) )) j t ( α (cid:15) λ l − (cid:15) α † (cid:15) )= a x,t j t ( θ x ) + a (cid:15),t j t ( θ (cid:15) ) + V t r − a x,t j t ( X ) r − a (cid:15),t j t ( (cid:15) ) r (3) Where for j t ( X ) : ∞ (cid:88) k =1 a ,k, ( t, j t ( X ) , j t ( (cid:15) )) j t ( λ k − x α † x ) = a x,t j t ( α † x ) ∞ (cid:88) k =1 a ,k, ( t, j t ( X ) , j t ( (cid:15) )) j t ( α x λ k − x ) = a x,t j t ( α x ) ∞ (cid:88) k =1 a ,k, ( t, j t ( X ) , j t ( (cid:15) )) j t ( λ kx ) = a x,t j t ( λ x ) (4) and for j t ( (cid:15) ) : ∞ (cid:88) l =1 a , ,l ( t, j t ( X ) , j t ( (cid:15) )) j t ( λ l − (cid:15) α † (cid:15) ) = a (cid:15),t j t ( α † (cid:15) ) ∞ (cid:88) l =1 a , ,l ( t, j t ( X ) , j t ( (cid:15) )) j t ( α (cid:15) λ l − (cid:15) ) = a (cid:15),t j t ( α (cid:15) ) ∞ (cid:88) l =1 a , ,l ( t, j t ( X ) , j t ( (cid:15) )) j t ( λ l(cid:15) ) = a (cid:15),t j t ( λ (cid:15) ) (5) Proof.
First, the equations for time-development operators for X ⊗
1, and (cid:15) ⊗ dU x,t = − (cid:32)(cid:18) iH + L ∗ x L x (cid:19) dt + L ∗ x SdA x − L x dA † x + (cid:18) − S (cid:19) d Λ x U (cid:15),t = − (cid:32)(cid:18) iH + L ∗ (cid:15) L (cid:15) (cid:19) dt + L ∗ (cid:15) SdA (cid:15) − L (cid:15) dA † (cid:15) + (cid:18) − S (cid:19) d Λ (cid:15) Then, applying the Hudson-Parthasarathy multiplication rules to the expan-sion given in assumption 2.4 gives: dV t = (cid:18) a , , ( t, j t ( x ) , j t ( (cid:15) )) + a , , ( t, j t ( x ) , j t ( (cid:15) )) j t ( θ x ) + a , , ( t, j t ( x ) , j t ( (cid:15) )) j t ( θ (cid:15) )+ ∞ (cid:88) k =2 a ,k, ( t, j t ( X ) , j t ( (cid:15) )) j t ( α x λ k − x α † x ) + ∞ (cid:88) l =2 a , ,l ( t, j t ( X ) , j t ( (cid:15) )) j t ( α (cid:15) λ l − (cid:15) α † (cid:15) ) (cid:19) dt + (cid:18) a , , ( t, j t ( X ) , j t ( (cid:15) )) j t ( α x ) + ∞ (cid:88) k =2 a ,k, ( t, j t ( X ) , j t ( (cid:15) )) j t ( α x λ k − x ) (cid:19) dA x + (cid:18) a , , ( t, j t ( X ) , j t ( (cid:15) )) j t ( α (cid:15) ) + ∞ (cid:88) l =2 a , ,l ( t, j t ( X ) , j t ( (cid:15) )) j t ( α (cid:15) λ k − (cid:15) ) (cid:19) dA (cid:15) + (cid:18) a , , ( t, j t ( X ) , j t ( (cid:15) )) j t ( α † x ) + ∞ (cid:88) k =2 a ,k, ( t, j t ( X ) , j t ( (cid:15) )) j t ( λ k − x α † x ) (cid:19) dA † x + (cid:18) a , , ( t, j t ( X ) , j t ( (cid:15) )) j t ( α † (cid:15) ) + ∞ (cid:88) l =2 a , ,l ( t, j t ( X ) , j t ( (cid:15) )) j t ( λ k − (cid:15) α † (cid:15) ) (cid:19) dA † (cid:15) (6)Where θ x , θ (cid:15) are given by: θ x = i [ H, X ] − (cid:18) L ∗ x L x X + XL ∗ x L x − L ∗ x XL x (cid:19) θ (cid:15) = i [ H, (cid:15) ] − (cid:18) L ∗ (cid:15) L (cid:15) (cid:15) + (cid:15)L ∗ (cid:15) L (cid:15) − L ∗ (cid:15) (cid:15)L (cid:15) (cid:19) α x , α (cid:15) are given by: α x = [ L ∗ x , X ] Sα (cid:15) = [ L ∗ (cid:15) , (cid:15) ] S and finally λ x , λ (cid:15) are given by: λ x = S ∗ XS − Xλ (cid:15) = S ∗ (cid:15)S − (cid:15) By assumption 2.2 we can form a hedge portfolio which we now use: V t = a x,t j t ( X ) + a (cid:15),t j t ( (cid:15) ) + b t β , for risk free numeraire asset β . dV t = a x,t dj t ( X ) + a (cid:15),t dj t ( (cid:15) ) + b t βrdt (cid:15) and x we have: dV t = a x,t (cid:0) j t ( α † x ) dA † x + j t ( λ x ) d Λ x + j t ( α x ) dA x (cid:1) + a (cid:15),t (cid:0) j t ( α † (cid:15) ) dA † (cid:15) + j t ( λ (cid:15) ) d Λ (cid:15) + j t ( α (cid:15) ) dA (cid:15) (cid:1) + (cid:0) j t ( θ x ) + ( V t − a x,t j t ( X ) − a (cid:15),t j t ( (cid:15) )) r (cid:1) dt (7)Equating the risky terms between equations (6), and (7) leads to the boundaryconditions, (4) and (5) on a x,t and a (cid:15),t . Similarly, equating the dt terms, leadsto the Quantum Black-Scholes equation for this system: equation (3).Now, let f (cid:0) x, (cid:15) (cid:1) represent a vector in H , and apply a rotation matrix: S = (cid:20) cos ( φ ) − sin ( φ ) sin ( φ ) cos ( φ ) (cid:21) We have: Sf (cid:0) x, (cid:15) (cid:1) = f (cid:0) cos ( φ ) x − sin ( φ ) (cid:15), cos ( φ ) (cid:15) + sin ( φ ) x (cid:1) XSf = xf (cid:0) cos ( φ ) x − sin ( φ ) (cid:15), cos ( φ ) (cid:15) + sin ( φ ) x (cid:1) S ∗ XSf = (cid:0) cos ( φ ) x + sin ( φ ) (cid:15) (cid:1) f ( x, (cid:15) )So, we end up with: λ x = (cid:18)(cid:0) cos ( φ ) − (cid:1) x + sin ( φ ) (cid:15) (cid:19) , λ (cid:15) = (cid:18)(cid:0) cos ( φ ) − (cid:1) (cid:15) − sin ( φ ) x (cid:19) .Finally, inserting this back into equation (3), we get the Black-Scholes equa-tion for the system (following notation from [1]): Proposition 2.6.
Let u ( t, x, (cid:15) ) represent the price at time t, of a derivativecontract in the system described above under rotation φ , and with interest rate r . Then the quantum Black-Scholes equation becomes: ∂u ( t, x, (cid:15) ) ∂t = rx ∂u ( t, x, (cid:15) ) ∂x + r(cid:15) ∂u ( t, x, (cid:15) ) ∂(cid:15) − u ( t, x, (cid:15) ) r + ∞ (cid:88) k =2 (( cos ( φ ) − x + sin ( φ ) (cid:15) ) k − k ! ∂ k u ( t, x, (cid:15) ) ∂x k g ( x, (cid:15) )+ ∞ (cid:88) l =2 (( cos ( φ ) − (cid:15) − sin ( φ ) x ) l − l ! ∂ l u ( t, x, (cid:15) ) ∂(cid:15) l g ( x, (cid:15) ) (8) Proof.
We assume that the operators L x , L ∗ x , L (cid:15) , L ∗ (cid:15) involve multiplication bya polynomial in x, (cid:15) , and therefore commute with λ x , λ (cid:15) . Therefore, from theboundary conditions we have: 8 ∞ k =1 a ,k, ( t, j t ( X ) , j t ( (cid:15) )) j t ( λ k − x ) = a x,t (cid:80) ∞ l =1 a , ,l ( t, j t ( X ) , j t ( (cid:15) )) j t ( λ l − (cid:15) ) = a (cid:15),t Inserting this into 3 gives: a , , ( t, j t ( X ) , j t ( (cid:15) )) + a , , ( t, j t ( X ) , j t ( (cid:15) )) j t ( X ) r + a , , ( t, j t ( X ) , j t ( (cid:15) )) j t ( (cid:15) ) r + ∞ (cid:88) k =2 a ,k, ( t, j t ( X ) , j t ( (cid:15) )) j t ( λ k − x ( α x α ∗ x − λ x ( θ x − xr )))+ ∞ (cid:88) l =2 a , ,l ( t, j t ( X ) , j t ( (cid:15) )) j t ( λ l − (cid:15) ( α (cid:15) α ∗ (cid:15) − λ (cid:15) ( θ (cid:15) − (cid:15)r )))= V t r (9)Now writing g ( x, (cid:15) ) = j t ( α x α ∗ x − λ x ( θ x − xr )), g ( x, (cid:15) ) = j t ( α (cid:15) α ∗ (cid:15) − λ (cid:15) ( θ (cid:15) − (cid:15)r )),and a ,k, ( t, j t ( X ) , j t ( (cid:15) )) = k ! ∂ k u∂x k , a , ,l ( t, j t ( X ) , j t ( (cid:15) )) = l ! ∂ l u∂(cid:15) l , we have theresult given.For small rotations, we have cos ( φ ) = 1 − ε + o ( ε ), and sin ( φ ) = ε + o ( ε ).Inserting this into equation (8), we have a new partial differential equation,where the coefficient of the k th partial derivative, for k ≥
3, with respect to x, (cid:15) , is correct to o ( ε k − ). This form for small rotations is more amenable tothe methods we apply in section 3. ∂u ( t, x, (cid:15) ) ∂t = rx ∂u ( t, x, (cid:15) ) ∂x + r(cid:15) ∂u ( t, x, (cid:15) ) ∂(cid:15) − u ( t, x, (cid:15) ) r + ∞ (cid:88) k =2 ( ε(cid:15) − ( ε / x ) k − k ! ∂ k u ( t, x, (cid:15) ) ∂x k g ( x, (cid:15) )+ ∞ (cid:88) l =2 ( − εx − ( ε / (cid:15) ) l − l ! ∂ l u ( t, x, (cid:15) ) ∂(cid:15) l g ( x, (cid:15) ) (10)As is the case for equation (2), this reduces to the classical Black-Scholes for2 uncorrelated random variables (in this case price: x , and bid-offer spread: (cid:15) )when ε = 0.For the classical case, the addition of the bid-offer spread is in some ways un-necessary when using the model for derivative pricing. For derivative contractsdepending on the close price, one can usually hedge daily at the closing priceduring the end of day auction process. For many trading desks this may be suf-ficient in practice, and terms involving the bid-offer spread will drop out of themodel. In the quantum case, examination of equations (8) and (10) shows thatwe expect interference between the bid-offer spread dynamics and the price dy-namics. For small rotations, these equations are singular PDEs, and we expectthe behaviour in most regions to approximate classical behaviour. However,9hen the higher derivative terms are larger, quantum interference may be sig-nificant. We discuss this more in sections 3 and 4. In this section, we derive the Fokker-Planck equations associated to the Quan-tum Black-Scholes equations: (2), and (10). We show how these can be writtenin integral form, by using the Kramers-Moyal expansion (see for example [7]).This enables us to link the Quantum Black-Scholes models of the previous sec-tion to nonlocal diffusions (see for example the paper by Luczka, H¨anggi andGadomski: [14]). We assume zero interest rates in this section to help clarifythe notation without changing the key dynamics. The integral form for theFokker-Planck equations is given by: ∂p ( t, x, (cid:15) ) ∂t = 12 ∂ ∂x (cid:18) (cid:90) ∞−∞ (cid:90) ∞−∞ (cid:0) H ( y x , y (cid:15) | x, (cid:15) ) g ( x, (cid:15) ) p ( x − y x , (cid:15) − y (cid:15) , t ) (cid:1) dy x dy (cid:15) (cid:19) + 12 ∂ ∂(cid:15) (cid:18) (cid:90) ∞−∞ (cid:90) ∞−∞ (cid:0) H ( y x , y (cid:15) | x, (cid:15) ) g ( x, (cid:15) ) p ( x − y x , (cid:15) − y (cid:15) , t ) (cid:1) dy x dy (cid:15) (cid:19) (11)The function H ( y x , y (cid:15) | x, (cid:15) ) has the effect of ”blurring” the impact of the dif-fusion operator. In the case that H ( y x , y (cid:15) | x, (cid:15) ) is a Dirac delta function, thediffusion operator is localised as usual, and the associated Fokker-Planck equa-tion reduces to the standard Kolmogorov forward equation associated with theclassical Black-Scholes.We start with the following general form for equations (2) and (10): ∂u ( t, x, (cid:15) ) ∂t = g ( x, (cid:15) ) ∞ (cid:88) k =2 f ( x, (cid:15), ε ) k − k ! ∂ k u ( t, x, (cid:15) ) ∂x k + g ( x, (cid:15) ) ∞ (cid:88) l =2 f ( x, (cid:15), ε ) l − l ! ∂ l u ( t, x, (cid:15) ) ∂(cid:15) l (12) Proposition 3.1.
The Fokker-Planck equation associated to equation (12), with r = 0 is given by: ∂p ( t, x, (cid:15) ) ∂t = ∞ (cid:88) k =2 ( − k k ! ∂ k (cid:0) g ( x, (cid:15) ) f ( x, (cid:15), ε ) k − p ( t, x, (cid:15) ) (cid:1) ∂x k + ∞ (cid:88) l =2 ( − l l ! ∂ l (cid:0) g ( x, (cid:15) ) f ( x, (cid:15), ε ) l − p ( t, x, (cid:15) ) (cid:1) ∂(cid:15) l (13) Proof.
For a derivative payout h ( x, (cid:15) ), with zero interest rates, we have the fol-lowing price in risk neutral measure Q : u ( x t , (cid:15) t , t ) = E Q (cid:2) h ( x T , (cid:15) T ) (cid:3) = (cid:82) R h ( y x , y (cid:15) ) p ( y x , y (cid:15) | x, (cid:15), t ) dy x dy (cid:15) p ( y x , y (cid:15) | x, (cid:15), t ) represents the risk neutral probability density for the vari-ables observed at time T , conditional on the values at time t . h ( x, (cid:15) ) representsa derivative payout at T . We then write the right hand integral as: (cid:82) R g ( y x , y (cid:15) ) p ( y x , y (cid:15) | x, (cid:15), t ) dy x dy (cid:15) = (cid:82) t (cid:82) R Lh ( y x , y (cid:15) ) p ( y x , y (cid:15) | x, (cid:15), s ) dy x dy (cid:15) ds Where L represents the operator: Lh ( x, (cid:15) ) = (cid:18) g ( x, (cid:15) ) (cid:80) ∞ k =2 f ( x,(cid:15),ε ) k − k ! ∂ k ∂x k + g ( x, (cid:15) ) (cid:80) ∞ l =2 f ( x,(cid:15),ε ) l − l ! ∂ l ∂(cid:15) l (cid:19) h ( x, (cid:15) )The Fokker-Planck equation, is given by the adjoint operator L ∗ . Therefore,since: (cid:82) t (cid:82) R Lh ( y x , y (cid:15) ) p ( y x , y (cid:15) | x, (cid:15), s ) dy x dy (cid:15) ds = (cid:82) t (cid:82) R h ( y x , y (cid:15) ) L ∗ p ( y x , y (cid:15) | x, (cid:15), s ) dy x dy (cid:15) ds If we truncate equation (12) at a certain order for the derivative: N , the resultfollows by integrating by parts N times. Proceeding with higher and higher N , we can match the derivative terms of any arbitary order, and the resultfollows.The objective now, is to write equation (13) in the form of (11). To do this wecan follow a Moment Matching algorithm. We use the following expansion: g ( x, (cid:15) ) p ( x − y x , (cid:15) − y (cid:15) , t ) = (cid:80) ∞ i,j =0 ( − ( i + j ) ( i + j )! y ix y j(cid:15) d i + j ( g ( x,(cid:15) ) p ( x,(cid:15) )) dx i d(cid:15) j Inserting this into equation (11) gives: ∂p ( t, x, (cid:15) ) ∂t = 12 ∂ ∂x (cid:18) ∞ (cid:88) i,j =0 ( − ( i + j ) ( i + j )! ∂ i + j ( g ( x, (cid:15) ) p ( x, (cid:15) )) ∂x i ∂(cid:15) j (cid:90) ∞−∞ (cid:90) ∞−∞ H ( y x , y (cid:15) | x, (cid:15) ) y ix y j(cid:15) dy x dy (cid:15) (cid:19) + 12 ∂ ∂(cid:15) (cid:18) ∞ (cid:88) i,j =0 ( − ( i + j ) ( i + j )! ∂ i + j ( g ( x, (cid:15) ) p ( x, (cid:15) )) ∂x i ∂(cid:15) j (cid:90) ∞−∞ (cid:90) ∞−∞ H ( y x , y (cid:15) | x, (cid:15) ) y ix y j(cid:15) dy x dy (cid:15) (cid:19) (14)Now by equating the coefficients of the derivatives with respect to x and (cid:15) ,between equations (14) and (13) one can calculate the moments of the “blurring”function H ( y x , y (cid:15) | x, (cid:15) ). For the translation case, g ( x, (cid:15) ) = 0, and the probabilitydensity is a function of x only. In the translation case, of section 2.1, since the coefficients of each differen-tial term in equation (2) is a constant multiplied by g ( x ), the moments of the“blurring” function H ( y ) will not depend of x . Equation (14) becomes: ∂p ( t, x ) ∂t = 12 (cid:18) ∞ (cid:88) j =0 ( − ( j ) j ! d ( j +2) ( g ( x ) p ( x )) dx ( j +2) (cid:90) ∞−∞ H ( y ) y j dy (cid:19) (15)11imilarly, the Fokker-Planck associated with equation (2), with r = 0, is givenby: ∂p ( t, x ) ∂t = ∞ (cid:88) k =2 ( − k ε k − k ! ∂ k ( g ( x ) p ( t, x )) ∂x k (16)Now the moments of the “blurring” function can be matched by equating di-rectly equations (15) and (16): Proposition 3.2.
Let H i represent the i th moment of H ( y ) , for the Fokker-Planck equation (13), relating to the translation case described in section 2.1.Then, H i is given by: H i = − ε ) i ( i +1)( i +2) Proof. H i follows (for i ≥
0) by equating the coefficients for: ∂ ( i +2) ∂x ( i +2) , betweenequations (15) and (16).We find that, in this case, H ( y ) is a normalised function that tends to a Diracfunction as ε tends to zero, and for ε = 0 we end up with classical 2nd orderFokker-Planck equation. This is discussed further in section 4. In the rotation case of section 2.2, the coefficients of each differential term inequation (13) are functions of x and (cid:15) . Therefore, we require the moments for the“blurring” function also to be functions of x , and (cid:15) : H ( y x , y (cid:15) | e, (cid:15) ). Once we havecalculated the coefficients for the differential terms, we can use these to form aninhomogeneous 2nd order differential equation for the moments of H ( y x , y (cid:15) | e, (cid:15) ).In this case, from equation (13) we have: f ( x, (cid:15) ) = ε(cid:15) − ( ε / x , and f ( x, (cid:15) ) = − εx − ( ε / (cid:15) . Therefore, the Fokker-Planck equation associated with equation(10), with r = 0, is given by: ∂p ( t, x, (cid:15) ) ∂t = ∞ (cid:88) k =2 k ! ∂ k (cid:18)(cid:0) ( ε / x − ε(cid:15) (cid:1) k − g ( x, (cid:15) ) p ( t, x, (cid:15) ) (cid:19) ∂x k + ∞ (cid:88) l =2 l ! ∂ l (cid:18)(cid:0) εx + ( ε / (cid:15) (cid:1) l − g ( x, (cid:15) ) p ( t, x, (cid:15) ) (cid:19) ∂(cid:15) l (17)The moments of the “blurring” function will now follow by equating coefficientsfor the differential terms between equations (14), and (17). Proposition 3.3.
Where the moments of the “blurring” function: H ( y x , y (cid:15) | x, (cid:15) ) are given by: ix = (cid:82) ∞−∞ (cid:82) ∞−∞ H ( y x , y (cid:15) | x, (cid:15) ) y ix dy x dy (cid:15) a j(cid:15) = (cid:82) ∞−∞ (cid:82) ∞−∞ H ( y x , y (cid:15) | x, (cid:15) ) y j(cid:15) dy x dy (cid:15) and a , a x , a (cid:15) are assumed to be: a = 1 , a x , a (cid:15) = 0 Then for the higher moments we have, for n ≥ : ( − n a n − x + 2 n ∂a n − x ∂x + n ( n − ∂ a nx ∂x n ! = (( ε / x − ε(cid:15) ) n − n ( n − − ( ε / ( n − (18)( − n a n − (cid:15) + 2 n ∂a n − (cid:15) ∂(cid:15) + n ( n − ∂ a n(cid:15) ∂(cid:15) n ! = (( ε / (cid:15) + εx ) n − n ( n − − ( ε / ( n − (19) Proof.
We first calculate the coefficients for ∂ n ( g ( x,(cid:15) ) p ( x,(cid:15) )) ∂x n from equation (17).The 2nd order coefficient is given by: (cid:80) i ≥ i − ε / i − ( i ) i ! = (cid:80) i ≥ ( ε / i = − ( ε / Similarly, the 3rd order coefficient is given by: (cid:80) i ≥ i − ε / ( i − ( i ) (( ε / x − ε(cid:15) ) i ! = (( ε / x − ε(cid:15) )3! (cid:80) i ≥ ( i +1)( ε / i = (( ε / x − ε(cid:15) )3!(1 − ( ε / In general, the nth order coefficient is given by: (cid:80) i ≥ n ( i − ε / ( i − ( in ) (( ε / x − ε(cid:15) ) n − i !( n − = (( ε / x − ε(cid:15) ) ( n − n ! (cid:80) i ≥ ( i + 1)( i + 2) ... ( i + n − ε / i The final summation can be calculated by differentiating ( n −
2) times, theinfinite sum 1 / (1 − v ), where v = ( ε / n ≥ ε / x − ε(cid:15) ) n − n ( n − − ( ε / ( n − ∂ n ( g ( x, (cid:15) ) p ( x, (cid:15) )) ∂x n (20)Following similar logic for (cid:15) we have the coefficient:(( ε / (cid:15) + εx ) n − n ( n − − ( ε / ( n − ∂ n ( g ( x, (cid:15) ) p ( x, (cid:15) )) ∂(cid:15) n (21)These coefficients can now be used to calculate a 2nd order inhomogeneous dif-ferential equation for the moments of H ( y x , y (cid:15) | x, (cid:15) ). We start by expanding the13 /∂x , and ∂ /∂(cid:15) in equation (14).Since, we assume from section 2.2, that x, (cid:15) are uncorrelated, equation (14)can be written: ∂p ( t, x, (cid:15) ) ∂t = 12 ∞ (cid:88) i =0 ( − ( i ) i ! (cid:32) ∂ i ( g ( x, (cid:15) ) p ( x, (cid:15) )) ∂x i ∂ a ix ∂x + ∂ i +2 ( g ( x, (cid:15) ) p ( x, (cid:15) )) ∂x i +2 a ix +2 ∂ i +1 ( g ( x, (cid:15) ) p ( x, (cid:15) )) ∂x i +1 ∂a ix ∂x (cid:33) + 12 ∞ (cid:88) j =0 ( − ( j ) j ! (cid:32) ∂ j ( g ( x, (cid:15) ) p ( x, (cid:15) )) ∂(cid:15) j ∂ a j(cid:15) ∂(cid:15) + ∂ j +2 ( g ( x, (cid:15) ) p ( x, (cid:15) )) ∂(cid:15) j a j(cid:15) +2 ∂ j +1 ( g ( x, (cid:15) ) p ( x, (cid:15) )) ∂(cid:15) j +1 ∂a j(cid:15) ∂(cid:15) (cid:33) (22)The coefficients for ∂ n ( g ( x,(cid:15) ) p ( x,(cid:15) )) ∂x n from equation (22) are now given by: ∂ a ∂x ( g ( x, (cid:15) ) p ( x, (cid:15) ))for n = 0, ( ∂ a x ∂x + 2 ∂a ∂x ) ∂ ( g ( x,(cid:15) ) p ( x,(cid:15) )) ∂x for n = 1, and:( − n a n − x + 2 n ∂a n − x ∂x + n ( n − ∂ a nx ∂x n ! ∂ n ( g ( x, (cid:15) ) p ( x, (cid:15) )) ∂x n (23)for n ≥
2. Similarly, for (cid:15) we have:( − n a n − (cid:15) + 2 n ∂a n − (cid:15) ∂(cid:15) + n ( n − ∂ a n(cid:15) ∂(cid:15) n ! ∂ n ( g ( x, (cid:15) ) p ( x, (cid:15) )) ∂(cid:15) n (24)We now make the assumption that H is a normalised probability distributionwith expectation zero for x and (cid:15) . Ie, ∂a ∂x = 0, a x = 0, and a (cid:15) = 0. Theseassumptions ensure the coefficients with n = 0 , In this section, we give a brief overview of McKean stochastic differential equa-tions, before introducing how the particle method, discussed in the book byGuyon & Henry-Labord`ere: [8], can be used in their simulation. We then go onto present numerical results from the bid-offer model discussed above, placingparticular emphasis on understanding how quantum effects become apparentthrough small transformations applied to a classical Black-Scholes system.14 .1 McKean Stochastic Differential Equations
McKean nonlinear stochastic differential equations were introduced in [16], andrefer to SDEs, where the drift & volatility coefficients depend on the underlyingprobability law for the stochastic process. Following notation from [8] we have: dX t = b ( t, X t , P t ) dt + σ ( t, X t , P t ) dW t These are then related to the nonlinear Fokker Planck equation: ∂p∂t = 12 (cid:88) i,j ∂ ( σ i ( t, x, P t ) σ j ( t, x, P t ) p ( t, x )) ∂x i ∂x j − (cid:88) i ∂ ( b i ( t, x, P t )) ∂x i (25)In this case, we can write equation (11) in this form. We have for r = 0, b ( t, x, (cid:15), P t ) = b ( t, x, (cid:15), P t ) = 0 and σ ( t, x, (cid:15), P t ) = (cid:115) g ( x, (cid:15) ) E p (cid:20) H ( x − y x ,(cid:15) − y (cid:15) | x,(cid:15) ) p ( x,(cid:15),t ) (cid:21) , σ ( t, x, (cid:15), P t ) = (cid:115) g ( x, (cid:15) ) E p (cid:20) H ( x − y x ,(cid:15) − y (cid:15) | x,(cid:15) ) p ( x,(cid:15),t ) (cid:21) .Therefore, we can simulate the solution to equation (11) by first calculating thefunction H ( x − y x , (cid:15) − y (cid:15) ) using a moment matching algorithm, and then simulat-ing the following McKean SDE, with uncorrelated Wiener processes dW , dW : dx = (cid:115) g ( x, (cid:15) ) p ( x, (cid:15), t ) E p ( y ) (cid:2) H ( x − y x , (cid:15) − y (cid:15) | x, (cid:15) ) (cid:3) dW d(cid:15) = (cid:115) g ( x, (cid:15) ) p ( x, (cid:15), t ) E p ( y ) (cid:2) H ( x − y (cid:15) , (cid:15) − y (cid:15) | x, (cid:15) ) (cid:3) dW (26)The simulation of the above SDE relies on the particle method outlined in Guyon& Henry-Labord`ere’s book Nonlinear Option Pricing chapters 10, 11 (cf: [8]).Each path ( x i , (cid:15) i ) now interacts with the other paths: ( x j , (cid:15) j ) , j (cid:54) = i during thesimulation process, and the convergence of the method relies on the so called propagation of the chaos property. This states: Definition 4.1.
For all functions φ ( x, (cid:15), t ) ∈ C ( R ) : N N (cid:88) j =1 φ ( x j , (cid:15) j ) N →∞ −−−−→ (cid:90) R φ ( x, (cid:15), t ) p ( x, (cid:15), t ) dxd(cid:15) (27)In our case, the SDE (26), is a McKean-Vlasov process, and we have fromGuyon, Henry-Labord`ere (cf: [8] Theorem 10.3), and originally Sznitman (cf:[22]), that the propagation of the chaos property holds.15 .2 Particle Method The first step is to discretize the SDE: (26), as follows: dx i = (cid:18) N (cid:88) j =1 H ( x j − x i , (cid:15) j − (cid:15) i ) P ( x j , (cid:15) j ) P ( x i , (cid:15) i ) g ( x i , (cid:15) i ) (cid:19) . dW ,i d(cid:15) i = (cid:18) N (cid:88) j =1 H ( x j − x i , (cid:15) j − (cid:15) i ) P ( x j , (cid:15) j ) P ( x i , (cid:15) i ) g ( x i , (cid:15) i ) (cid:19) . dW ,i (28)Where P ( x j , (cid:15) j ) represents a suitably discretized probability function. The al-gorithm then proceeds as follows:1. Solve for the moments of the “blurring” function H ( x − y x , (cid:15) − y (cid:15) | x, (cid:15) )using propositions 3.2, and 3.3.2. Choose a parameterised distribution to approximate H ( x − y x , (cid:15) − y (cid:15) | x, (cid:15) ),and fit the parameters using the calculated moments. For example, ap-proximate H ( x − y x , (cid:15) − y (cid:15) | x, (cid:15) ) as a univariate/bivariate normal distribu-tion.3. Simulate the 1st time step, t , using the value of H (0 , | x , (cid:15) ), for startingpositions x , (cid:15) .4. After each simulation, allocate the simulated paths into discrete probabil-ity buckets: P ( x j , (cid:15) j ), for paths j = 1 to N .5. Proceed from the t k − to t k timestep, using (28), the value of H ( x − y x , (cid:15) − y (cid:15) | x, (cid:15) ), and the discrete buckets at t k − .6. Iterate steps 4 & 5 until the final maturity: t F . We can see from (28), that small translations, will lead to a variance scalingfactor: (cid:80) Nj =1 H ( x j − x i , (cid:15) j − (cid:15) i ) P ( x j ,(cid:15) j ) P ( x i ,(cid:15) i ) This will have the impact of reducing the volatility of those paths which liein the middle of the “bell curve”, owing to the negative curvature of the proba-bility law at these points - probability mass is spread by the “blurring” functionto lower probability points.Similarly, at the extremes of the probability density curve where the curva-ture is positive, probability mass is spread to areas with net higher probability.In essence the market memory of a recent extreme event, will lead to a highermarket volatility at the next time step.16his effect differs from the negative skew observed in local volatility models(for example the work by Dupire: cf [6]), and from stochastic volatility models(for example Heston: [11]), in the sense that the increase in volatility is linkedto recent random moves in the tail of the probability distribution, rather thanto the level of the stochastic volatility or a static function of the price, and time.To highlight the difference, in the process given by equation (28), one couldallow for periodic rebalancing of the process. For example, one could replacethe unconditional probability, with the probability conditional on the previousstep. In this way, the level of the volatility would depend purely on a “memory”of recent price history, rather than on the absolute level of the market price,or an additional random variable. The market responds to large moves with aheightened fear factor. The study of modelling such processes with rebalancing,will involve advanced techniques for calculating the conditional probabilities,and we defer detailed study to a future work.
In this section, we simulate the one-factor process described in section 2.1, and3.1. In this case, we approximate H ( y ) using a normal distribution using themoments from proposition 3.2: N ( ε , ε ).The non-zero 1st moment, will lead to an upside/downside bias to the “marketfear factor” effect. Essentially, by introducing a translation in the negative x direction, one introduces downside ‘fear’ into the model.Figures 1 & 2 below, illustrate the results from a 2 step Monte-Carlo pro-cess, with g ( x ) = 0 . x , starting value: x = 1, 100K Monte-Carlo paths, and500 discrete probability buckets. The scatter plot shows the magnitude of theproportional return on the 1st time-step on the horizontal axis, and the secondtime-step on the vertical axis: 17igure 1: ε = 0, horizontal axis represents the proportional return for the firsttime-step, vertical axis represents the second second time-step.Figure 2: The results for ε = 0 .
02, horizontal axis represents the proportionalreturn for the first time-step, vertical axis represents the second time-step.18igure 1 shows the results for ε = 0. This is a classical Black-Scholes sys-tem, and there is no correlation between the magnitude & direction of the 1stand 2nd time-steps.Figure 2 shows the proportional returns for ε = 0 .
02 (in blue), overlaid ontop of the ε = 0 results (in orange). The volatility of the second step is reducedon those paths where the first time-step has been small. There is a slight in-creased second step volatility for those paths with large positive first steps, andsignificant second step volatility for those paths with a large negative first step.In effect, the drop in market prices has introduced “fear” into these paths.The final chart shows the probability distributions for the natural logarithmof the simulated value after 50 one day time-steps. The non-zero translationresults in a natural skewness in the distribution.Figure 3: Distribution for the natural log of the final price after 50 one day time-steps. 100K Monte-Carlo paths, and 500 discrete probability buckets. In this paper, we demonstrate how unitary transformations can be used tomodel novel quantum effects in the Quantum Black-Scholes system of Accardi& Boukas (cf [1]).We show how these quantum stochastic processes can also be modelled using19onlocal diffusions, and simulated using the particle method outlined by Guyon& Henry-Labord`ere in [8].By introducing a bid-offer spread parameter, and extending the Accardi-Boukasframework to 2 variables, we show how rotations, in addition to translations,can be applied. Thus, a richer representation of the information contained inthe current market leads to a wider variety of unitary transformations that canbe used.In section 4, using a Monte-Carlo simulation, we illustrate how introducing atranslation to the one dimensional model leads to a skewed distribution, wherebyrecent market down moves leads to increased volatility going forward. In effect,the market retains memory of recent significant moves.In [6], Dupire shows how to calibrate a local volatility to the current vanillaoption smile. This enables a Monte-Carlo simulation that is fully consistent withcurrent market option prices. Carrying out the same analysis, using the newQuantum Fokker-Planck equations, is another important next step to consideras a future development of the current work.
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