Closed Quantum Black-Scholes: Quantum Drift and the Heisenberg Equation of Motion
CCLOSED QUANTUM BLACK-SCHOLES: QUANTUM DRIFTAND THE HEISENBERG EQUATION OF MOTION.
WILL HICKS
Abstract.
In this article we model a financial derivative price as an observ-able on a market state function, with a view to understanding how some ofthe non-commutative behaviour of the financial market impacts the dynam-ics. We integrate the Heisenberg Equation of Motion, by using a Riemannianmetric, and illustrate how the non-commutative nature of the model intro-duces quantum interference effects that can act as either a drag or a booston the resulting return. The ultimate objective is to investigate the nature ofquantum drift in the Accardi-Boukas quantum Black-Scholes framework ([1])which involves modelling the financial market as a quantum observable, andintroduces randomness through the Hudson-Parthasarathy quantum stochas-tic calculus ([15]). In particular we aim to differentiate between randomnessthat is introduced through external noise (quantum stochastic calculus) andrandomness that is intrinsic to a quantum system (Heisenberg Equation ofMotion).
1. Introduction1.1. The Accardi-Boukas Quantum Black-Scholes:
The quantum Black-Scholes framework of Accardi & Boukas (see [1]) derives a partial differentialequation for the value of a derivative security by applying the quantum stochas-tic calculus of Hudson & Parthasarathy (see [15]). This is a generalisation of thestandard Ito stochastic calculus based on modelling the relevant traded underlyingas a quantum observable. We note the following: • The traded underlying is modelled as an K valued stochastic process inthe Hilbert space: L ( R + ) ⊗ K . • Random noise is added through the Boson Fock space: Γ( L ( R + ) ⊗ K )(we write Γ). • One quantizes the traded asset price as an operator valued observable X on the space: K ⊗ Γ. • Derivative payouts are considered as operator valued functions of the quan-tum observable.The benefits of this approach are that one can introduce diffusions within non-commutative spaces which mirror many of the real effects of the financial markets,that are difficult to model using ‘classical’ Ito processes (for example see [12], [13],
Mathematics Subject Classification.
Primary 81S25; Secondary 91B70.
Key words and phrases.
Heisenberg Equation of Motion, Quantum Black-Scholes, QuantumStochastic Calculus, Quantum Drift. a r X i v : . [ q -f i n . M F ] J a n WILL HICKS and [14]). However, we will see that this particular choice of quantization meansthat the internal dynamics of the quantum system K governed by the Hamiltonianˆ H are lost, and have no impact on the derivative price.The time evolution of the underlying quantum state, that arises from this Hamil-tonian, drops out of the analysis in the same way as the classical drift does in theclassical Black-Scholes. If the traded underlying price evolves under geometricBrownian motion: dS = µSdt + σSdW P (1.1)Then µ is replaced by the funding cost in our chosen Martingale measure: Q . Forexample, if we choose a deposit account, bearing interest rate: r , as our risk freeasset, then the risk neutral process used for derivative pricing becomes: dS = rSdt + σSdW Q (1.2)Working in the Heisenberg interpretation of Quantum mechanics, the unitary timedevelopment operator: U t , which is usually given by: dU t = (cid:16) − i ˆ Hdt (cid:17) U t (1.3)becomes, after adding random noise through the operators dA t , dA † t , d Λ t , that fillup the Boson Fock space, combined with bounded linear operators L and S , where S is unitary: dU t = (cid:16)(cid:0) − i ˆ H − L ∗ L (cid:1) dt + L ∗ SdA t − LdA † t + (1 − S ) d Λ t (cid:17) U t (1.4)In the same way that charged particles interact with a quantized electromagneticfield via the creation of photons in a Boson Fock space, the financial marketinteracts with external environmental noise, through the creation of ‘noise packets’in a Boson Fock space. The external environment creates ‘financial photons’.The observable: X that acts on the underlying state to return the current pricefor the traded asset, evolves in time under the unitary operator described by 1.4: j t ( X ) = U ∗ t XU t dj t ( X ) = dU ∗ t XU t + U ∗ t XdU t + dU ∗ t XdU t (1.5)This in turn can be simplified using a quantum version of Ito’s lemma, describedby [15] Theorem 4.5. This result gives a multiplication table that can be used tocombine the operators in equation 1.5:- dA † t d Λ t dA t dtdA † t d Λ t dA † t d Λ t dA t dt dA t dt LOSED QUANTUM BLACK-SCHOLES 3
Using this, we get the following dynamics for X : dj t ( X ) = j t ( α † ) dA † t + j t ( α ) dA t + j t ( λ ) d Λ t + j t ( θ ) dtα = [ L ∗ , X ] Sλ = S ∗ XS − Xθ = i [ H, X ] −
12 ( L ∗ LX + XL ∗ L − L ∗ XL ) (1.7)It is important to note the following two things: • Under L = 0, we have a system where no financial photons are beingcreated. There is no environmental noise, and equation 1.7 simplifies tothe Heisenberg equation of motion. • Under L (cid:54) = 0, the Hamiltonian driving the dynamics of the underlyingHilbert space ( K ), ˆ H , impacts only the drift of the quantum observable.To derive the quantum Black-Scholes equation, following Accardi & Boukas ([1]),first we expand values for ( dj t ( X )) k using 1.6:( dj t ( X )) k = j t ( λ k − α † ) dA † t + j t ( αλ k − ) dA t + j t ( λ k ) d Λ t + j t ( αλ k − α † ) dt (1.8)Next, we quantize the derivative valuation, by considering operator valued func-tions of the quantum observable: F : [0 , T ] × B ( K ⊗ Γ) −→ B ( K ⊗
Γ). Writing j t ( X ) = x , we have: F ( t, x ) = (cid:88) n,k ≥ n ! k ! ∂ n + k F∂t n ∂x k (1.9)In order to derive the final quantum Black-Scholes, Accardi & Boukas show howto derive a value process that, by holding units of the risky underlying j t ( X ), anda chosen numeraire asset, pays out the final derivative payout with probability 1. This approach is extremelypowerful, since it allows us to extend the existing classical approach to Mathe-matical Finance to a noncommutative setting. The financial markets are in factnaturally noncommutative: • It is largely impossible to determine the price one can trade at, with infiniteprecision, in advance. For example, when a trader submits a ‘bid’ orderto an exchange mechanism, they do not usually know what ‘offer’ priceswill be submitted, and therefore what price the ‘bid’ will get filled at. • One can only really identify the true price of a traded underlying by sub-mitting a ‘bid/offer’ order, and in so doing, impacting the very state ofthe market one is trying to measure. • As markets are more and more dominated by algorithms, and executiontimes fall correspondingly, this effect will become more and more pro-nounced.The properties of the resulting quantum Black-Scholes equations are furtherdiscussed in [12], [13], and [14]. It has also been noted (for example by Havenin [10], and discussed further by Melnyk and Tuluzov in [16]) that the risk in
WILL HICKS trading financial derivatives cannot be fully hedged in a ‘true’ quantum model.When executing the required strategy for the value process, it is not possible todetermine the exact price of the traded underlying j t ( X ). Furthermore, if ourderivative is to be considered a quantum observable in its own right, rather thana function of a quantum observable, then we cannot measure the risk sensitivities: ∂F/∂j t ( X ) with full precision, and therefore cannot evaluate how much of theunderlying we need to hold. Moreover, our numeraire asset drifts at a constantrate: dV = rV dt . However, constant drift at a pre-specified rate is not somethingthat can be realised in a fully quantum model of the financial market.These considerations do not invalidate in any way the resulting quantum Black-Scholes. Whilst in the quantum case, executing the value process that replicatesa derivative payout is not possible, this does not prevent the resulting valuationbeing a unique no arbitrage price for the derivative.However, as noted above, the impact of trading activity on the ‘state’ of themarket is one of the key reasons why a non-commutative approach makes sense.Therefore, in an ideal world we would like to build a model that is capable ofincorporating the impact of measurement on the market state. The impact oftrading activity on the evolution of the market state has been discussed previously,for example see [18]-[20] where the authors highlight how different market effectscan be explained using the non-commutativity of quantum mechanics.Furthermore, whilst the quantization chosen by Accardi & Boukas allows thederivation of the quantum Black-Scholes equation, a natural alternative is to con-sider derivative payouts as themselves quantum observables, rather than operatorvalued functions of a quantum observable. Whilst in some respects this distinctionis a philosophical choice around the boundary separating the quantum & classicalregimes, we find in this article that it can have a real impact on the model.Therefore, the long term goal is to build a model of the financial market basedon the following ingredients: A A quantum market state, with the dynamics controlled by a Hamiltonianfunction. B A set of observables, tradeable instruments, by which one can interactwith the market. C A means by which we can allow the introduction of environmental noise.For example, through the quantum stochastic calculus discussed above.In this article we make a start with A and B , by attempting to develop the ideaof quantum drift. In essence, when building a model based on classical stochasticcalculus, in the event that no external noise is added (no diffusion term), then thefinal payout from a derivative security can be determined with full precision. Ina quantum framework this is no longer the case. There are in fact 2 sources ofrandomness. An external source based on quantum stochastic calculus, such asthat introduced by Accardi & Boukas, and purely quantum internal randomnessthat we investigate here. LOSED QUANTUM BLACK-SCHOLES 5
We start in section 2, by deriving the dynamics for quantized financial observ-ables using the Heisenberg equation of motion. We apply a geometric techniqueto deriving eigenfunctions, before discussing how recursive techniques (for exam-ple see [2]) can be used to generate power series solutions to the closed quantumBlack-Scholes. This technique involves introducing a Riemannian metric in orderto simplify the integration, and functional form for the eigenfunctions.We go on to discuss the near classical limit in section 3. In section 4, we investi-gate some numerical examples, in order to understand the behaviour of solutions,and discuss potential applications of the techniques beyond those discussed above.Finally, we draw overall conclusions in section 5.
2. Closed Quantum Black-Scholes
The approach starts with a Hilbert space: H , that describes the state of themarket. For example, we could select L ( R ). In this case, we could define theoperator: X to measure the expected price for a market transaction that occursright now ( t = 0). So: X ψ ( x ) = xψ ( x ) (2.1) E [ X ] = (cid:104) ψ | Xψ (cid:105) = (cid:90) R x | ψ ( x ) | dx (2.2)Following the usual principals of quantum mechanics, in the Heisenberg interpreta-tion, time development of the price observable is controlled by a unitary operator: U ( t ) = e i ˆ Ht , where ˆ H is the Hamiltonian operator for our system. Now, at time t (cid:54) = 0, we have: X ( t ) = e i ˆ Ht X e − i ˆ Ht (2.3) X ( t ) ψ ( x ) = e i ˆ Ht xe − i ˆ Ht ψ ( x ) (2.4)The expected price for a transaction in the future is now: E t [ X ( t )] = (cid:104) ψ | X ( t ) ψ (cid:105) = (cid:104) e − i ˆ Ht ψ | X e − i ˆ Ht ψ (cid:105) = (cid:90) R x | ψ ( x, t ) | dx (2.5)Where: ψ ( x, t ) = e − i ˆ Ht ψ ( x ). This framework can be applied to any observable onthe state of the financial market. For observable A , we have: A ( t ) = e i ˆ Ht Ae − i ˆ Ht (2.6) E t [ A ( t )] = (cid:104) ψ | A ( t ) ψ (cid:105) = (cid:104) e − i ˆ Ht ψ | Ae − i ˆ Ht ψ (cid:105) = (cid:90) R x | ψ ( x, t ) | dx (2.7)Differentiating equation 2.6, we get the Heisenberg equation of motion for a generalobservable, (where [ A, B ] = AB − BA ): dA ( t ) dt = i [ ˆ H, A ( t )] (2.8) WILL HICKS
Let X ( t, T ) represent the market price for a tradedasset that is fixed at time t , and exchanged at maturity T . We consider theobservable relating to the price for a derivative contract that pays out an amount F ( X ( T, T )), depending on the final observation of the price: X ( T, T ).We let U ( t, x ) represent the value for this contract at time t , contingent on thecurrent price: X ( t, T ) = x . If X ( t, T ) can be modelled by some probability space:(Ω , P , F ), we have from the first and second fundamental theorem of mathematicalfinance (see for example [4], Theorems 10.14, 10.17, 10.18): • The market is arbitrage free if and only if there exists a Martingale mea-sure:
Q ∼ P . • The market is complete, so all derivative contracts U ( t, x ) can be replicatedby trading in X ( t, T ), if and only if the Martingale measure is unique. Thisis the case, assuming the number of underlying random variables that drivemarket prices, is the same as the number of traded assets. In this simplecase: 1. • The value of the derivative contract is given by the discounted expectationin the Martingale measure.Therefore, if we have an interest rate: r , and discount factors: e − r ( T − t ) , we have: U ( t, x ) = e − r ( T − t ) E Q [ F ( X ( T, T ))] (2.9)We now turn back to equation 2.8, and consider the requirements for this equationto represent a pricing equation for a quantum observable for the derivative payout: F . We write the observable X ( t, T ) = X , and insert into equation 2.8: dXdt = i [ ˆ H, X ] (2.10)We assume our Hamiltonian operator incorporates a potential energy component, V , which acts by pointwise multiplication by x . In other words:ˆ Hψ ( x ) = − σ ∂ ψ∂x + V ( x ) ψ ( x ) (2.11)So: i [ ˆ H, X ] ψ ( x ) = − iσ ∂ψ∂x (2.12)If we define a ‘momentum’ operator: (cid:0) ˆ P ψ (cid:1) ( x ) = − i ∂ψ∂x , we get: dXdt = σ ˆ P (2.13)Therefore, classically speaking , the relevant Martingale measure is that which driftswith the market price. I.e, the measure under which the momentum is zero. Froma quantum perspective we have: d ˆ Pdt = i [ ˆ H, ˆ P ] = − dVdX (2.14)So, in the absence of any potential function, the expected rate of change in themomentum operator is zero. In our quantum framework, we cannot insist the LOSED QUANTUM BLACK-SCHOLES 7 momentum is zero. However, if we take as an initial condition that the expectedmomentum is zero, then we will have E [ X ] = E T [ X ], and our measure will bea Martingale measure. Thus, even though we cannot apply the delta hedgingargument used in the original derivation of the Black-Scholes partial differentialequation (see [5],[17]), we can still construct a Martingale measure, and conse-quently an arbitrage free price.In other words, whilst we cannot know, with full precision, the nature of theself financing almost-simple strategy with which we can replicate a payout, thefundamental theorem of mathematical finance (for example see [4] Theorem 10.5)still guarantees that we cannot execute a self financing strategy h ( t ), such thatthe final payout U ( h ( T )) satisfies the following: • P (cid:0) U ( h ( T )) ≥ (cid:1) = 1 • P (cid:0) U ( h ( T )) > (cid:1) > We can now apply the Heisenberg ap-proach to derive a closed quantum Black-Scholes equation for the derivative price.Let U represent the price for our derivative contract. In order to derive the re-quired equation, we must first construct a suitable Hilbert space operator fromthis classical function.Starting from the standard position operator X , we need to invoke the spectraltheorem (see for example [8] Theorem 7.12) in order to represent U as a functionof this position operator: U ( X, t ). In fact, complications arise from the fact thatboth X , and U ( X, t ) are unbounded. For the purposes of this article, we ignorethese complications, and proceed on a formal basis.Thus we assume we can form an operator U ( X, t ), that acts on the market stateby pointwise multiplication by an unknown function of x and t . We then have: E ψ [ U ] = (cid:104) ψ ( x ) , e it ˆ H U e − it ˆ H ψ ( x ) (cid:105) = (cid:104) e − it ˆ H ψ ( x ) , U e − it ˆ H ψ ( x ) (cid:105) (2.15)So, applying the Heisenberg equation of motion: E ψ (cid:20) ∂U∂t (cid:21) = (cid:104) e − it ˆ H ψ ( x ) , i [ ˆ H, U ] e − it ˆ H ψ ( x ) (cid:105) (2.16)If we assume the potential energy component from 2.11, commutes with the de-rivative price operator, [ V, U ] = 0, then we have: i [ ˆ H, U ] = − iσ ∂U∂x ∂∂x − iσ ∂ U∂x = σ ∂U∂x ˆ P + σ P ∂U∂x (2.17)So our closed Quantum Black-Scholes becomes:
Proposition 2.1.
Let X ( t, T ) represent the forward price, observed at t for thepurchase of an asset at T , and let U ( x, t ) represent the price for the derivativepayout, at time t , contingent on the current forward price: x = X (0 , T ) . Then we WILL HICKS have: E ψ (cid:104) ∂U∂t (cid:105) = E ψ (cid:104) σ (cid:16) ˆ P ∂U∂x + ∂U∂x ˆ P (cid:17)(cid:105) (2.18) Where, for operator A : E ψ [ A ] = (cid:104) ψ, Aψ (cid:105) (2.19) Equation 2.18, will yield an arbitrage free price if and only if we have: E ψ [ ˆ P ] = 0 ,and [ ˆ H, ˆ P ] = 0 . Note the following: • The derivative price operator: U is now dependent on the state function.Now, rather than making assumptions about the stochastic process thatany traded price follows, the price can be derived by making assumptionsabout the market state, and associated Hamiltonian dynamics. • If we choose H = L ( R ), it is possible to show for reasonable cases, thatthe operator ˆ P ∂U∂x + ∂U∂x ˆ P is self-adjoint on a dense domain (see [9], [6]).This enables us to calculate associated eigenfunctions that can be used tointegrate numerically. • The potential energy component from the Hamiltonian (2.11) controls the classical drift of the system. By assuming that this commutes with U weensure that the potential does not impact the no-arbitrage price of thederivative. Expanding out theright hand side of equation 2.18, we get: (cid:68) ψ, ∂U∂t ψ (cid:69) = (cid:68) ψ, − i σ ∂ U∂x ψ (cid:69) − (cid:68) ψ, iσ ∂U∂x ∂ψ∂x (cid:69) (2.20) Naive Approach:
To find eigenfunctions, we must solve the equation (for ψ ): − iσ ∂U∂x ∂ψ∂x − iσ ∂ U∂x ψ − λψ = 0 (2.21)Rearranging, we have: 1 ψ ∂ψ∂x = 2 i λσ − ∂ U∂x ∂U∂x (2.22)Applying the integrating factor method, we can calculate the eigenfunctions: φ λ ( x ) = 1 (cid:112) ∂U/∂x exp (cid:18) i λσ (cid:90) x ( ∂U/∂x ( s )) − ds (cid:19) (2.23)The functional form for this solution is suggestive of using a new distance metric: g ( x ) = ∂U/∂x to solve the problem. We investigate this approach in the nextsection. LOSED QUANTUM BLACK-SCHOLES 9
Geometric Approach:
The analysis above, suggests using a distance metric: g ( x ) to solve 2.1. The general form for a Laplacian operator on a 1 dimensionalRiemannian manifold is given by (see for example [11] chapter 4):∆ g = g − / (cid:16) ∂∂x + A x (cid:17) g − / (cid:16) ∂∂x + A x (cid:17) + Q ( x ) (2.24)where A x represents the components of an Abelian connection, and Q ( x ) a sectionof a real vector bundle over our manifold. If we select A x = Q ( x ) = 0, we end upwith the Laplace-Beltrami operator: σ (cid:112) g ( x ) ∂∂x (cid:18) (cid:112) g ( x ) ∂∂x (cid:16) ... (cid:17)(cid:19) (2.25)However, our objective is to use geometry to simplify the problem, to enable us toderive eigenfunctions: • By introducing a distance metric: g ( x ) we are translating simple Euclideanspace into a Riemannian manifold. The Euclidean notion of distance isstretched by different amounts at different points. • By choosing A x , Q ( x ), we are choosing a convenient coordinate systemon our manifold. Choosing nonzero A x , Q ( x ) is the equivalent for theRiemannian manifold, to choosing curvilinear coordinates in Euclideanspace.Expanding out the general Laplacian, 2.24, we get:1 g ( x ) ∂ ∂x + (cid:18) g ( x ) − / ∂ ( g ( x ) − / ) ∂x + A x g ( x ) (cid:19) ∂∂x + (cid:18) A x g ( x ) + 1 g ( x ) ∂ ( A x ) ∂x + Q ( x ) (cid:19) (2.26)Therefore by choosing: A x = − g / ∂ ( g ( x ) − / ) ∂xQ ( x ) = − A x g ( x ) − g ( x ) ∂ ( A x ) ∂x (2.27)The operator is simplified to: ∆ g = 1 g ( x ) ∂ ∂x (2.28)Under this coordinate system our Hamiltonian becomes (we use a slight abuse ofnotation, by still writing the independent variable as x ):ˆ Hψ ( x ) = − σ g ( x ) ∂ ψ∂x (2.29)Equation 2.18 now becomes: (cid:68) ψ, ∂U∂t ψ (cid:69) = (cid:68) ψ, − i σ g ( x ) ∂ U∂x ψ (cid:69) − (cid:68) ψ, iσ g ( x ) ∂U∂x ∂ψ∂x (cid:69) (2.30) Setting g ( x ) = ∂U/∂x , we get: (cid:18) − iσ g ∂g∂x − λ (cid:19) = iσ ψ ∂ψ∂x (2.31)Multiplying by − i/σ , and integrating we get: (cid:112) g ( x ) ψ ( x ) = e iλx/σ (2.32) We write the Heisen-berg equation of motion as: ∂U∂t = i [ ˆ H, U ] (2.33)A formal solution to this equation can be written (see [2]): U ( t ) = exp ( iLt ) U (0) LU = [ ˆ H, U ] (2.34)Power series solutions to this equation are discussed below. However, we can usethe eigenfunctions to solve for the short time behaviour for U ( t ). The first step isto expand the market state over the eigenfunctions (2.32): (cid:112) g ( x ) ψ ( x ) = 1 σ (cid:90) R ˜ ψ ( λ/σ ) e iλx/σ dλ (2.35)Applying the spectral theorem at a purely formal level (for example, see [8] The-orem 10.9, 10.10) we get: E ψ (cid:2) U ( t ) (cid:3) ≈ (cid:90) R U ( x ) | ψ ( x ) | dx + 1 σ (cid:16) (cid:90) R λ | ˜ ψ ( λ/σ ) | dλ (cid:17) t + O ( t ) (2.36)Strictly speaking, g ( x ) will vary with time, as the function U = U ( x, t ) varies.However, 2.34 represents an expansion about the initial value: g ( x, Phase Factor Choice, and Quantum Interference.
One of the requirementsfor the model to be arbitrage free, is that the expected momentum is zero. There-fore we require: (cid:90) R k | ˜ ψ ( k ) | dk = 0˜ ψ ( k ) = (cid:90) R ψ ( x ) e − ikx dx (2.37)Having a real valued market state function: ψ ( x ) will ensure this is the case. Inthis event, equation 2.36 will result in a distribution: | ˜ ψ ( λ/σ ) | that is an evenfunction of λ/σ . Therefore we end up with:1 σ (cid:16) (cid:90) R λ | ˜ ψ ( λ/σ ) | dλ (cid:17) = 0 E ψ (cid:2) U ( t ) (cid:3) ≈ (cid:90) R U ( x ) | ψ ( x ) | dx + O ( t ) (2.38)So, modulo the higher order terms, whilst the quantum framework introducesunhedgeable uncertainty, the expected return on the derivative is not impacted.However, we can meet the condition 2.37 by using a real function multiplied by a LOSED QUANTUM BLACK-SCHOLES 11 phase factor: exp ( iφ ( x )) where φ ( x ) is an even function of x . For example (withnormalising constant C , and real α ): ψ ( x ) = Cexp (( iα − x / σ ). In section 4,we will see that this introduces a non-zero expected return on the derivative thathas no classical counterpart.1 σ (cid:16) (cid:90) R λ | ˜ ψ ( λ/σ ) | dλ (cid:17) (cid:54) = 0 (2.39)In these instances, there is a non-zero expected return on the derivative, even wherethe underlying is Martingale and there is no source of external noise. This returncan be positive where the quantum interference boosts the derivative valuation, orit can be negative, whereby the non-commutativity yields a holding cost associatedwith the derivative. Brief Comment on the Topology of the Transformation.
It is clear fromabove, that where the option delta: ∂U/∂x = 0, the Laplacian used is not defined.The transformation squashes all points on the real number line representation ofthe asset price where the option delta is zero, to a single point. Furthermore,where ∂U/∂x = 0 is negative, the direction of the distance metric also reverses.So for example: • For a call option payout at final maturity ( t = T say), the full real numberline is squashed to the positive real number line, with all points below thestrike to zero. • For a straddle payout, where the delta is negative below the strike andpositive above the strike, the real number line is bent back in on itself.For these reasons, we must split any payout, or hedging strategy, into the fol-lowing: • Prices along the real number line, where we are a buyer ( ∂U/∂x > • Prices along the real number line, where we have no interest in buying orselling ( ∂U/∂x = 0). • Prices along the real number line, where we are a seller ( ∂U/∂x <
Building a Power Series Solution to the Closed Quantum Black-Scholes.
Following the analysis in [2], we set L as the Liouvillian superoperator, so that forgeneral operator Q ( t ) we have: LQ ( t ) = [ ˆ H, Q ( t )]. Then, conditional on convergence, the solution to 2.33 can be written: U ( t ) = (cid:88) k ≥ ( it ) k k ! U k U k = L k U (0) (2.40)In this way, we can broadly proceed as follows: • Set g ( x ) in 2.35 based on the final option payout, and carry out a changein variables from t to τ = T − t , for final maturity T . • Decide on the market probability density function, and associated statefunction: ψ ( x ). • Calculate ˜ ψ ( λ/σ ) from the inverse Fourier transform of 2.35.So for example, set U ( x,
0) = ( x − K ) + for a vanilla call option, and further set τ = T − t for final maturity T . From equation 2.35, we get:˜ ψ ( λ/σ ) = (cid:90) R x>K ψ ( x ) e − iλx/σ dx = (cid:90) ∞ K ψ ( x ) e − iλx/σ dx (2.41)This value for ˜ ψ ( λ/σ ) can then be inserted into 2.36, to generate a first order callprice expansion, valid for small time to maturity. To calculate the higher orderterms in the expansion, one would have to work out eigenfunctions of the operator: L k U . For each consecutive term in the expansion 2.40, we must expand the marketstate function over a new eigenfunction, and use the resulting transform (denoted˜ ψ k ( λ ) say) to calculate U k . We can apply the same geometric technique for eachof the terms in the recursive solution.In general, to obtain the eigenfunctions which we will use to get the k th orderterm, we expect to end up with a differential equation to solve as follows: k (cid:88) j =0 p j ( x ) ∂ j ψ∂x j − λψ = 0 (2.42)Where the functions p j ( x ) derive from the payout in question: U , and its deriva-tives. Writing PDE: 2.42 in standard form (for example see [3] chapter 3) weget: ∂ k ψ∂x k + k − (cid:88) j =0 p j ( x ) p k ( x ) ∂ j ψ∂x j − λp k ( x ) ψ = 0 (2.43)Equation 2.43 is likely singular, in the event that there are zeros in the function p k ( x ). Assuming U and its derivatives are not polynomials of finite order, theequation is also irregular, since it will remain singular after multiplying by x N forall N . LOSED QUANTUM BLACK-SCHOLES 13
We can resolve this issue, by using a judicious choice of Riemannian metric,turning p k ( x ) into a constant. After we have done this, if we choose a payout U with continuous derivatives that do not blow up, then we will be able to solve theresulting equation, and find appropriate eigenfunctions for the operator: L k U .It should be noted that for many payouts that are traded (including vanilla calloptions, European digital options etc), there are discontinuous derivatives, and sostrictly further work is still required. Noting that caution is required, we move onand show how to calculate the second order term: L U , for illustration purposes.We have: L U ψ = [ ˆ H, [ ˆ H, U ]] ψ . We write the Hamiltonian as: ˆ H = − σ D , foran elliptic operator D yet to be determined. For now, we write ˆ H = a ( x ) ∂ /∂x + a ( x ) ∂/∂x + a ( x ) and end up with:[ ˆ H, U ] = a ( x ) ∂ U∂x ψ + 2 a ( x ) ∂U∂x ∂ψ∂x + a ( x ) ∂U∂x (2.44)And, we have:[ ˆ H, [ ˆ H, U ]] = a ( x ) ∂ ∂x (cid:16) a ( x ) ∂ U∂x ψ + 2 a ( x ) ∂U∂x ∂ψ∂x + a ( x ) ∂U∂x ψ (cid:17) + a ∂∂x (cid:16) a ( x ) ∂ U∂x ψ + 2 a ( x ) ∂U∂x ∂ψ∂x + a ( x ) ∂U∂x ψ (cid:17) + a ( x ) (cid:16) a ( x ) ∂ U∂x ψ + 2 a ( x ) ∂U∂x ∂ψ∂x + a ( x ) ∂U∂x ψ (cid:17) − a ( x ) ∂ U∂x (cid:16) a ( x ) ∂ ψ∂x + a ( x ) ∂ψ∂x + a ( x ) ψ (cid:17) − a ( x ) ∂U∂x ∂∂x (cid:16) a ( x ) ∂ ψ∂x + a ( x ) ∂ψ∂x + a ( x ) ψ (cid:17) − a ( x ) ∂U∂x (cid:16) a ( x ) ∂ ψ∂x + a ( x ) ∂ψ∂x + a ( x ) ψ (cid:17) (2.45)From equation 2.45 we can collect together terms in: ∂ k ψ/∂x k . For ∂ ψ/∂x , weget: (cid:16) a ( x ) ∂U∂x − a ( x ) ∂U∂x (cid:17) = 0 (2.46)So the coefficient of ∂ ψ/∂x is zero.The coefficient for ∂ ψ/∂x is given by:4 a ( x ) ∂ U∂x + 2 a ( x ) ∂a ( x ) ∂x ∂U∂x (2.47)This can be rearranged to get:4 a ( x ) ∂∂x (cid:16) a ( x ) ∂U∂x (cid:17) − a ( x ) (cid:16) ∂a ∂x ∂U∂x (cid:17) (2.48)If we assume, that our Laplacian operator takes the form: a ( x ) ∂ ∂x , as is the casein section 2.3.2, then we have a ( x ) = a ( x ) = 0. If this is the case, the coefficient for ∂ψ/∂x is given by:2 a ( x ) ∂∂x (cid:16) a ∂ U∂x (cid:17) + 2 a ( x ) ∂ ∂x (cid:16) a ∂U∂x (cid:17) (2.49)Now, if we insert the Riemannian metric from above, so that: a ( x ) = − σ (cid:18) ∂U∂x (cid:19) − (2.50)We find 2.49 can be written: 2 a ( x ) ∂∂x (cid:16) a ∂ U∂x (cid:17) = σ (cid:16) ∂U∂x (cid:17) − ∂∂x (cid:18)(cid:16) ∂U∂x (cid:17) − ∂ U∂x (cid:19) (2.51)Similarly, 2.48 becomes: − a ( x ) (cid:16) ∂a ∂x ∂U∂x (cid:17) = σ (cid:16) ∂U∂x (cid:17) − (cid:18)(cid:16) ∂U∂x (cid:17) − ∂ U∂x (cid:19) (2.52)Finally, the zero order term in 2.45, with a = a = 0, is given by: σ (cid:16) ∂U∂x (cid:17) − ∂ ∂x (cid:18)(cid:16) ∂U∂x (cid:17) − ∂ U∂x (cid:19) (2.53)Pulling this all together, we find the 2nd order eigenvalue equation for L U is aSturm-Liouville problem (for example see [7] chapter 7.7).( L U ) ψ − λψ = 0 becomes: ∂∂x (cid:18) p ( x ) ∂ψ∂x (cid:19) + (cid:16) ∂ p∂x (cid:17) ψ − (cid:16) ∂U∂x (cid:17) λσ ψ = 0 p ( x ) = ∂ U/∂x ∂U/∂x (2.54)Finally, following [7], we can write this equation in a more amenable shape byusing the transformation: φ ( x ) = h ( x ) ψ ( x ), h ( x ) = (cid:0) p ( x ) ∂U/∂x (cid:1) / , and thecoordinate function: s = (cid:82) xa (cid:0) ∂U/∂x (cid:1)(cid:0) ∂ U/∂x (cid:1) − / dx , for arbitrary a . Underthis transformation 2.54 can be written: ∂ φ∂s + (cid:16) λσ − Q ( s ) (cid:17) φ = 0 Q ( s ) = 1 h ∂ h∂s + ∂ p∂x ∂U/∂x (2.55)Equation 2.55 can then be tackled using the WKB approximation methods out-lined in [8] chapter 15. We defer this detailed analysis to a future work. LOSED QUANTUM BLACK-SCHOLES 15
3. Impact of Quantum Interference & Discussion of the Classical Limit3.1. Near Classical Limit.
Equation 2.36 represents the spread of returns λ that result from purely internal quantum effects. In a Martingale model, theclassical limit, given by: lim σ →∞ , results in the following: E ψ (cid:2) U ( t ) (cid:3) ≈ (cid:90) R U ( x ) | ψ ( x ) | dx + lim σ →∞ (cid:18) σ (cid:16) (cid:90) R λ | ˜ ψ ( λ/σ ) | dλ (cid:17) t (cid:19) = (cid:90) R U ( x ) δ ( x − x ) dx = U ( x ) (3.1)So we have that the classical limit represents a model with zero randomness (in-ternal or external) and so the expected payout is given by the payout at the initialvalue. In other words, the classical system isolated from external noise results inzero uncertainty in the final payout. The quantum system isolated from externalnoise, has a degree of (positive or negative) time value given by a purely quantuminterference.In the near classical limit, where σ is large in comparison to the varianceembedded in the market state: σ >> (cid:90) R x | ψ ( x ) | dx (3.2)We note the following: • The presence of σ in the Fourier transform term ˜ ψ ( λ/σ ) will ensure theexpected return is more and more localised. • We intuitively expect the average quantum rate of return: µ λ = σ (cid:16) (cid:82) R λ | ˜ ψ ( λ/σ ) | dλ (cid:17) to be O (1 /σ ). This would imply that in the limit of large σ (relativeto the variance of the state function), the quantum term will becomeincreasingly localised around zero. • The k th order term in the expansion 2.40, will result in a term in eigen-values of order: λσ k , originating from k applications of the Hamiltonianfunction. This implies that in the near clasical limit of large σ , we canignore higher order terms. • If we write: σ ψ = (cid:82) R x | ψ ( x ) | dx , then the ratio σ ˆ H /σ ψ , where σ ˆ H is thevolatility parameter in the Hamiltonian, represents a scale parameter thatdetermines the importance of the quantum interference terms. Where σ ˆ H /σ ψ is small, it is more likely the quantum interference will be lessimportant, relative to the external sources of noise. • In reality, the length of time between observations of the system (executedtrades) is likely to be small in most cases. This is further reason to ignorethe higher order terms in the expansion 2.40.We find overall that a localised market state function, or a low volatility Hamil-tonian function, results in a model that is more classical in the sense that the internal quantum interference has less impact. A less localised market state, andhigh volatility Hamiltonian function results in a model that is more quantum, inthe sense that the internal quantum interference has more impact.
4. Numerical Examples
In this section we use three numerical examples to illustrate the points madeabove.
Where the wave function is real valued,the expected return distribution; ˜ ψ ( λ/σ H ) will be an even function. Therefore,although there will be quantum interference in the time development of the deriv-ative price observable: U , the expected impact is zero. In the chart below, we set ψ ( x ) = (1 / (cid:112) πσ S ) exp ( − x / σ S ). In this case, σ S = σ ˆ H = 0 .
2. We find that set-ting g ( x ) = ∂U/∂x = 1 x> increases the uncertainty in the quantum interference.However, the expected interference is still zero. Figure 1.
The chart shows ˜ ψ ( λ/σ H ) for g ( x ) = 1, and g ( x ) =1 x> . In both cases we have set the wave function to ψ ( x ) =(1 / (cid:112) πσ S ) exp ( − x / σ S ). We now apply a phase factor to the wavefunction: ψ ( x ) = (1 / (cid:112) πσ S ) exp (( i − x / √ σ S ). We show the resulting distri-butions in the return: ˜ ψ ( λ/σ H ), for different values of σ ˆ H . In each of the examplesin this section g ( x ) = 1 x> . LOSED QUANTUM BLACK-SCHOLES 17
Figure 2.
The chart shows ˜ ψ ( λ/σ ) for σ ˆ H = 0 .
2, and σ ˆ H = 0 . σ ˆ H moves the system closer to the near classical limit. The distribution becomestighter, and the expected value closer to zero.The final chart shows the results for σ ˆ H = 0 .
2, and the wave function set to: ψ ( x ) = (1 / (cid:112) πσ S ) exp (( αi − x / √ σ S ), with α = + / −
1. This highlights thefact that the expected value for the return can be positive or negative, dependingon the wave function.
Figure 3.
The chart shows ˜ ψ ( λ/σ ) for α = +1, and α = −
5. Conclusion
In this article we have investigated the randomness that can be introducedthrough the non-commutativity of the quantum framework, rather than throughan external source of noise. We have also shown that, with a real valued marketwave function, this randomness will not lead to positive time value in a vanilla op-tion. However, once a non-trivial phase factor: exp ( iφ ( x )) is applied, the internalrandomness can lead to positive or negative time value.Furthermore, although a real valued wavefunction will mean the expected timevalue for the option is zero, the spread of potential returns will introduce uncertaintime value, once quantum measurement is thrown into the mix.Going forward, further work is required in a number of areas. For example, thisincludes, but is not limited to:a) Deciding on how to model trading activity that disturbs the state of themarket. For example through frequent measurement of the quantum ob-servables.b) Incorporating external environmental noise, for example through quantumstochastic calculus.c) The simplest approach may be to combine noise that arises from purelyexternal sources of noise (for example geopolitical events, release of eco-nomic data etc) and trading activity, through the methods of quantumstochastic calculus. However, an alternative approach may be to treatthese 2 separately.d) Adding more mathematical rigour to the informal geometric approach ofsection 2.3.2. References [1] Accardi, L.; Boukas, A: The Quantum Black-Scholes Equation,
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Will Hicks, Investec Bank PLC, 30 Gresham Street, London EC2V 7QP, UnitedKingdom
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