Pseudo-Hermiticity, Martingale Processes and Non-Arbitrage Pricing
PPseudo-Hermiticity, and Removing Brownian Motion fromFinance
Will HicksSeptember 2, 2020
Abstract
In this article we apply the methods of quantum mechanics to the study of the financial markets.Specifically, we discuss the Pseudo-Hermiticity of the Hamiltonian operators associated to the typicalpartial differential equations of Mathematical Finance (such as the Black-Scholes equation) and howthis relates to the non-arbitrage condition.We propose that one can use a Schr¨odinger equation to derive the probabilistic behaviour of thefinancial market, and discuss the benefits of doing so. This presents an alternative approach toreplace the use of standard diffusion processes (for example a Brownian motion or Wiener process).We go on to study the method using the Bohmian approach to quantum mechanics. We considerhow to interpret the equations for pseudo-Hermitian systems, and highlight the crucial role playedby the quantum potential function.
The majority of mathematical models, used to capture the dynamics of the financial markets, are basedon the underlying concept of Brownian motion. This is in many ways quite natural. If one assumes thatthe log returns (for example 1 day returns) of the price of a financial asset are independent, and (at leastin the short term) identically distributed, then the central limit theorem implies that the distributionof returns will converge to a normal distribution. Furthermore, the variance will increase linearly withtime. Thus the representation of the random variables, that drive the market price, as Brownian motionsseems an obvious first step.Also, by basing the random behaviour of the financial market on Brownian motion, one ends up withpartial differential equations, such as the Black-Scholes equation, that can be mapped via changes in thecoordinate space, to the standard heat equation (for example see [13] chapter 4). Crucially, this meansthat a wide array of analytic, and numerical, methods are available to find solutions.However, the limitations of these simple approaches are well known. In this article we wish to presentthe means by which one can replace Brownian motion, and derive the equations of finance, such as theBlack-Scholes equation, using the framework of quantum probability. The ultimate goal, is to show howthis methodology can be applied, so that future research can be directed towards the modelling of more1 a r X i v : . [ q -f i n . M F ] S e p omplex market dynamics.The application of quantum probability to problems in finance has been studied in a number of sources.For example links between the Black-Scholes partial differential equation and a non-Hermitian Schr¨odingerequation have been investigated by Baaquie in [2]. Haven also discusses the implications of the modellingof derivative prices as state functions of a Schr¨odinger equation in [10] and [11]. Baaquie goes on to showhow one can apply the path integral approach of quantum mechanics, to derivative pricing in [3] and [4].In the articles discussed above, and many others on quantum finance, the links between the Black-Scholes equation of finance, and the Schr¨odinger equation of quantum mechanics, are applied to derivingnew mathematical techniques, as well as to providing new ways of understanding the behaviour of themarkets.In this article, we develop these ideas by showing how one can derive equations of finance withoutresorting to standard building blocks such as the Martingale Representation Theorem. Rather thanusing quantum methods as a means to study the equations of finance, that have been derived usingclassical approaches, we seek to develop the models entirely within the quantum framework.The intention is that this opens the door, in future research, to the application of exotic Hamilto-nian functions that can be used to apply non-Brownian drivers to finance.The approach is focused on representing the financial market as a quantum state. There is an alter-native quantum approach, suggested by Segal & Segal in [27], that is based on focusing on quantumobservables. Accardi & Boukas develop a quantum Black-Scholes by introducing randomness using theHudson-Parthasarathy quantum stochastic calculus (see [1], [19]). Further analysis and development onoperator focused approaches has been presented in [14]-[17]In some respects the Accardi-Boukas approach is the dual of the approach considered in this article.In [15], the author shows how one can generate solutions to the Accardi-Boukas quantum Black-Scholesby building on the path integral methods introduced by Baaquie ([3], [4]). The principals outlined above,apply in this case as well. One can build a model directly using the Schr¨odinger equation, rather thanby using Hudson-Parthasarathy stochastic calculus, and exploiting the purely algebraic simularities toSchr¨odinger equations.With this in mind, in section 2, we start by explaining the classical derivation of equations, such asthe Black-Scholes equation. One crucial ingredient being the assumption that the financial markets arefree of arbitrage, and the consequent use of Martingale probability measures.We go on to show how to replace the classical derivation, using quantum probability in section 4. Weshow how, by basing the underlying random variables on a Hermitian Schr¨odinger equation, the finalequations end up with the pseudo-Hermiticity property linked to the Black-Scholes equation in [21].Finally, in section 5, we show how one can view the methods shown in section 4 from a Bohmian perspec-tive (see [18]), and discuss some of the insights this can give us. In particular we consider the meaning ofthe quantum potential that arises in the Bohmian approach, and how the form of this quantum potentialimpacts the properties of the diffusion. 2 The Martingale Representation Theorem, and the ClassicalApproach to Financial Modelling
In this section, we outline the fundamental building blocks to the classical approach to MathematicalFinance, based primarily on stochastic calculus. In the next section, we go on to show how one can re-place the use of classical stochastic calculus entirely using quantum probability, and discuss the benefitsof doing so. This is in contrast with some other approaches to quantum finance, where the form of theequations defining models (such as the Black-Scholes equation) are taken as given, having been derivedusing the classical approach, and quantum methods are then applied in the study of solutions.The most fundamental building block for models of the financial market is the Martingale. To definethis concept, we start with a probability space: (Ω , F , P ). Ω represents the space of possible outcomeswhich we are trying to model (real number line of asset prices for example), and F represents a sigmaalgebra of subsets of Ω, each subset representing a measurable event. For example, if Ω is mapped to R ,then a measurable event U ∈ F =] a, b [ would represent the event that our price X , is to be found in therange a < X < b . Finally, P is a probability measure on F . In other words a function: P : F → [0 , • P (Ω) = 1 • P ( ∅ ) = 0 • For a collection of disjoint sets: { A i } , P ( ∪ i A i ) = (cid:80) i P ( A i ).A stochastic process is given by a sequence of sigma algebras: F t , and a sequence of real valued functions: X t : F t → R n . We generally require that if s < t then F S ⊂ F t .Finally, a Martingale is a stochastic process: M t such that: • E P [ | M t | ] < ∞ for all t . • E P [ M t | M s ] = M s , for all s ≤ t . • For any function f : Ω → R n , we have E P [ f ] = (cid:82) ω ∈ Ω f ( ω ) dP ( ω ). • See [25] for more detail.The first assumption for most models of the financial market is that the model should be free of arbitrage.The first fundamental theorem of mathematical finance (see for example [5] theorem 3.9) roughly statesthat a model is arbitrage free if and only if there exists an equivalent Martingale probability measure: Q .For an alternative way to think about this, let Π t represent a trading strategy consisting of holding Π t units of the asset X t , and time t . Informally speaking, the trading strategy is considered self-financingin the event that the change in the value of the strategy is driven completely by changes in the assetprice X t . In other words, after day 1, no further cash is injected into (or taken out from) the portfolio.Our model is then arbitrageable, if there exists a self-financing strategy ∆ t such that:3 P (∆ t X t ≥ ∆ s X s ) = 1 for all s < t . • P (∆ t X t > ∆ s X s ) > Q such that the risky asset: X t , is a Martingale.The link to the Black-Scholes equation, and to the other partial differential equations of finance, isprovided by the Martingale representation theorem (see [25] theorem 4.3.4). Using this theorem, werepresent the Martingale: X t using an Ito integral: X t = X + (cid:90) t f ( s, ω ) dB ( s ) (1)Thus we see that: • Whilst many financial models have complex dynamics, the fundamental processes at the heart ofthe random behaviour are largely assumed to be driven by Brownian motion. Behaviour such as‘fat-tails’ and skew must be introduced exogenously. • The Brownian motions are introduced largely based on mathematical tractability. • Complex market dynamics are generally introduced either via the function f ( s, ω ) or by introducingnew stochastic processes, also driven by Brownian motions, with correlated Wiener processes. Webriefly touch on these, in section 3, below • The quantum approach outlined in this paper allows one to build the complex behaviours intothe fundamental random processes themselves, by selection of different Hamiltonian operatorsin the Schr¨odinger equation. Essentially, the Schr¨odinger equation does the job the MartingaleRepresentation Theorem does, in the classical approach.
The Black-Scholes model is based on the stochastic process below: X t = X + σ (cid:90) t X s dB ( s ) (2)where for simplicity X t represents the forward price (which is a Martingale) for the traded asset. In thissection, we briefly discuss some of the most common ways in which to enrich this simple model usingclassical approaches.Whilst each method has benefits and drawbacks, the compromise is generally modelling complex dy-namics at the expense of computational efficiency in generating solutions. Furthermore, it is to beexpected that no single approach can offer a perfect representation of the random behaviour of financialmarkets.Therefore, by applying completely different theoretical building blocks, the quantum approach can offernew insights and new mathematical techniques. The path integral technique shown by Baaquie in [2]-[4],being a key example. 4 .1 Local Volatility: The first natural extension of this model is to allow the volatility σ to itself depend on the current valuefor X t and the time t . This was first introduced by Dupire (see [7]), and crucially allows one to adaptthe function σ ( X t , t ) so that the stochastic process simultaneously generates observed prices for vanillaoptions at different strikes & maturities. This is in contrast to the Black-Scholes model, whereby differentvalues for the constant σ are obtained by calibrating it to different strikes. The local volatility modelling approach described above remains a common work-horse model for manypractitioners in the financial markets. However, the model is designed as a simple means by which tomatch market prices, rather than a model to address the well known deficiencies of Black-Scholes.One key issue is that the single factor Ito process which reproduces all observed vanilla option prices, isunique. There are no degrees of freedom that can be used to ensure the underlying process has reasonabledynamics. For example, the evolution of the prices for short maturity options, or the implied correlationbetween the equity spot price, and the Black-Scholes implied volatility cannot be controlled, and areboth generally unrealistic in local volatility models.The very fact that from one business day to the next, the local volatilities implied by the traded pricesfor vanilla options change, leads practitioners to consider stochastic volatility models.A general stochastic volatility model may be defined as follows (for example see [13] chapter 6): X t = X + (cid:90) t ν s σ ( X s , s ) X s dB ( s ) ν t = ν + (cid:90) t a ( ν s , s ) ds + (cid:90) t b ( ν s , s ) dB ( s ) dB ( t ) dB ( t ) = ρdt (3)By defining suitable functions: a ( ν t , t ) and b ( ν t , t ), one is able to calibrate a stochastic process to observedoption prices, whilst also retaining dynamics that make sense. However, this generally comes at thefollowing cost: • These solutions can replicate behaviours seen in the past. However, there is often a trade offbetween developing models based on financial principals, versus developing models specifically tofit historical observations. • Generating solutions using available numerical methods (such as Monte-Carlo techniques) is oftenprohibitively computationally intensive.It should be noted that the quantum approach will also experience similar types of issues. However, bystarting from a completely different set of ideas: that of the wave-function rather than that of the Itodiffusion process, one is likely to develop complex models with different strengths and weaknesses.5 .3 Other Classical Approaches:
In the sections above, we have introduced important classical approaches, applied by practitioners, inorder to resolve some difficulties associated with the Black-Scholes model. There are a number of otherextensions,some of which which we mention below: • When one measures the Hurst exponent ( H ) for the daily log returns of most traded equities, onedoes not generally find H = 1 /
2. This has lead to the application of fractional Brownian motion toproblems in finance (see for example [26]). In these models incremental daily returns are no longerindependent and the models are not arbitrage free. Incorporating fractional Brownian motion usinga Schr¨odinger equation approach is an important potential future direction for research. • The approaches discussed above, incorporate all the random behaviour to the traded equity price.In some cases it is important to incorporate some randomness to interest rates, and the discountfactors applied to future cashflows. We discuss these ideas further in section 4.5. • Some authors have investigated the introduction of random jumps in the stock price, using L`evyprocesses (see for example [22]). Investigating quantum representations for Levy processes is an-other possible future avenue for research.
To illustrate how the quantum approach can replace the Martingale representation theorem discussedabove, and the use of Brownian motions & Wiener processes in Mathematical Finance, we first illustratehow this can be achieved using the example of the simple Black-Scholes equation.Looking at the problem from another perspective, the key property that allows the use of the quantum ap-proach to study the Black-Scholes equation is the Pseudo-Hermiticity of the Black-Scholes Hamiltonian.For further discussion on this, refer to [21].
The first step is to decide what we wish to model. For example, we may decide that we wish to modellog returns for a listed equity. This seems reasonable, since relative movements in the listed share priceseem more fundamental than the absolute amount. In this case, we write the stock price as: S = e x (4)where x represents the random variable we wish to model. In fact, for the majority of liquid listedequities, one can observed forward prices directly in the market, rather than constructing a forwardcurve using a discount curve, forecasting dividends etc. Therefore, we ignore dividends & interest rates,and assume that S represents the market price for the forward. In this step, we will encode the dynamics for the random variable x into a Schr¨odinger equation. Theform of the Schr¨odinger equation will control what kind of variable we get. For example, fat-tailed,6kewed, Markovian etc.There are different ways of interpreting the use of the Schr¨odinger equation in this way. Equation 5describes the evolution of a wave-function, and whilst real world measurements associated to this wave-function will have random outcomes, the wave-function evolution is deterministic. We discuss differentfinancial interpretations in more detail in section 4.4.For the time being, and for the sake of simplicity, we use a straight forward Hamiltonian function: i ∂ψ∂t = ˆ Hψ (5)ˆ H = − σ ∂ ∂x + Cψ ( x, t ) represents the wave function for our random variable: x . We define the position operator: X inthe usual way: E ψ [ X ] = (cid:90) R x | ψ ( x, t ) | dx E ψ [ X ] = (cid:90) R x | ψ ( x, t ) | dxC represents an operator that can be used to enforce the Martingale condition. For example, in this casewe show below how we can use a constant potential to ensure e x is a Martingale. This will be addressedin section 4.3. So far, we have defined a random variable using the Schr¨odinger equation, and obtained a random variablewith the correct dynamics. However, we still do not have a partial differential equation with respect tothe original market observable: S = e x . We can construct a valid Schr¨odinger equation using the changeof variables: S = e x , although the resulting Hamiltonian is generally Pseudo-Hermitian, rather thanHermitian (see [2]). The wave-function ψ is defined as belonging to a Hilbert space. For example, we could define: ψ ∈ L ( R ).For this choice, the Hilbert space inner product is given by: (cid:104) φ | ψ (cid:105) = (cid:90) R φ ( y ) ψ ( y ) dy (6)In this case, the model defined by equation 5, will conserve probability in the event that:ˆ H † = ˆ H
7n fact, as detailed in [24], we can define a different inner product (and consequently a different Hilbertspace), using a linear Hermitian automorphism: η . We define: (cid:104) φ | ψ (cid:105) η = (cid:90) R φ ( y )( ηψ )( y ) dy (7)Now, our Hamiltonian: ˆ H will conserve probability, and have real valued spectrum, in the event that:ˆ H † = η ˆ Hη − (8)ˆ H † η = η ˆ H Equation 8, defines a Pseudo-Hermitian Hamiltonian.
In our case, when we transfer from one coordinate system to another, we must conserve the innerproducts. We have: dSdx = e x So, changing variables: f ( S ) = φ ( x ), and g ( S ) = ψ ( x ), we get: (cid:104) φ | ψ (cid:105) = (cid:90) R φ ( x ) ψ ( x ) dx = (cid:90) R f ( S ) g ( S ) dS = (cid:90) R f ( e x ) g ( e x ) e − x dx So, by writing: ( ηψ )( x ) = e − x ψ ( x ), we can translate from a Schr¨odinger equation defined with respect to x , to one with respect to S . First assume we can find a positive operator square root for η . Ie, a positiveoperator ρ , such that ρ = η . Then, we write ˆ K = ρ − ˆ Hρ . We have (since ˆ H , and ˆ ρ are assumed to beHermitian): ˆ K † ρ = ρ ˆ Hρ − ρ = ρ ˆ Hρ = ρ ˆ K Therefore, if we can find such an operator: ρ , then the Hamiltonian we require is given by: ρ − ˆ Hρ . Inour case, we have: ρ ( x ) = e − x/ , and so:ˆ Kψ = e x/ (cid:18) − σ ∂ ( e − x/ ψ ) ∂x + Ce − x/ ψ (cid:19) = − σ ∂ ψ∂x + σ ∂ψ∂x + (cid:18) C − σ (cid:19) ψ (9)We must now adjust the operator: C to ensure the traded underlying: S = e x is a Martingale under theHamiltonian: ˆ K . 8 .3.3 The Martingale Requirement: Note that if ψ is a solution to the Schr¨odinger equation defined by ˆ H (equation 5), and if ρ representspointwise multiplication by an x function (in this case ρ ( x ) = e − x/ ), then: ρ − ψ is a solution to theSchr¨odinger equation defined by Hamiltonian: ˆ K (equation 5).We require that S = e x is a Martingale under the measure defined by: η , and the Hamiltonian ˆ K given by 4.3.2. Therefore, the Martingale requirement in this case translates to: E η [ e x | x ] = (cid:104) δ x | e x | e x/ ψ (cid:105) η = e x If we write the solution to 5 as: ψ ( x, t ) = 1 √ πσ t exp (cid:18) − x σ t − Ct (cid:19) (10)Writing (without loss of generality) x = 0,and utilising 7 with ηψ = η ( x ) ψ ( x, t ), we get: E ψ [ e x | x = 0] = (cid:90) R e y e y/ ψ ( y, t ) e − y dy = e x e σ t/ e − Ct Therefore in this case we must set C as a constant, with value σ /
8. Finally, writing ˆ K in terms of thenew variable S = e x , we find: i ∂ψ∂t = ˆ H BS ψ (11)ˆ H BS = − σ S ∂ ∂S Equation 11 transforms to the standard Black-Scholes equation, using the mapping it → τ . This trans-formation, from real time to complex time, is often referred to as a “Wick rotation”. A similar Wickrotation can be used to tranform the standard (Gaussian) Schr¨odinger equation to the heat equation. Inthe same way that the Wick rotation transforms a Gaussian wave-function to a Gaussian heat diffusion,the Black-Scholes wave-function is transformed to a Black-Scholes diffusion process.Under equation 11, randomness only enters the model at the point of making a measurement of themarket price. After carrying the Wick rotation, randomness is introduced through diffusion. The objective in section 4 is to find a new method to build Martingales without relying on the repre-sentation using Ito integrals. This way, one is able to build complex market dynamics into the randomprocesses themselves, and provides the opportunity for studying new models of the financial market.Care though should be taken when applying equation 11 in a financial context. For example, thefollowing are possible interpretations: A ψ could represent the value process for a self-financing trading strategy. Then one could useequation 11 for derivative pricing, assuming an initial condition defined by the final payout. Onecan ensure ψ is real valued by carrying out the Wick rotation, although the financial interpretationof doing this is not clear. 9 One could interpret derivative prices as observables on the market state function: ψ . This is theinterpretation discussed in [17]. In this instance, derivative prices will be real valued, althoughstrictly positive payouts, such as a call option payout, will not always have strictly positive value.In order to ensure a strictly positive payout has positive value, one must incorporate interactionbetween the closed market quantum system & noise from the environment. C One can simply interpret the solution ψ as giving the statistical properties of the market. Thenintegrate over a (Martingale) probability function to derive derivative prices.For the time being, in this article we are applying the quantum framework to derive the statisticalbehaviour of the market, rather than assuming a fully quantum approach. This approach can fit withininterpretations A or C , which we discuss in more detail below. In interpretation A , ψ represents the value function for a financial derivative. For this reason, | ψ ( S, t ) | dS no longer represents a probability measure. However, if one considers the solutions relating to Arrow-Debreu securities, one can see that the interpretation of ψ as a wave-function in the conventional senseis still necessary.The Arrow-Debreu security is defined as the derivative that pays out $1, in the event that the tradedunderlying is exactly S T at maturity T . Risk neutral pricing theory requires that the derivative value ψ is given (for t < T , and Martingale probability measure Q ) by: ψ = E Q [ δ S T ] = (cid:90) R δ ( y − S T ) dQ ( y ) = p ( S T , T | S t , t )Alternatively, if our quantum state starts in the Dirac state: δ S t , and ψ represents the solution to theSchr¨odinger equation at time T , then we have: p ( S T , T | S t , t ) = (cid:104) δ S t | ψ (cid:105) = ψ ( S t , t )The other alternative, interpretation C , is that we represent the financial market using a quantum state:the ‘quantum market’. Equation 11 then represents the Schr¨odinger equation for the quantum market,in which case the interpretation of ψ as a wave-function in the conventional sense is required, and theconventional initial conditions applied when solving a Black-Scholes partial differential equation (ie thefinal pay-out) have no meaning. In many ways, this is the cleaner interpretation in the sense that thereis a single quantum state (the financial market) which one can measure in many different ways (differentderivative instruments). Rather than considering each different derivatives as different quantum states.Either interpretation is acceptable mathematically: • Interpretation A : Arrow-Debreu securities are quantum states, and we integrate over these to derivethe value for conventional derivatives. • Interpretation C : There is a single quantum state. Derivatives operate at the classical level, in thesense that by trading a derivative, one is making a measurement of the quantum market.In either case, an important next step is to consider the impact of frequent measurement to the timedevelopment of the quantum state. In [23], the authors investigate the application of the continuous10uantum measurement method to derivative pricing. We defer further investigation of this to a futurework. The main objective of the current work is to apply the methods of quantum mechanics in a purelystatistical sense. C , and Ideas for Further Development of theMethod: In the example studied in this article (the simple Black-Scholes case), the operator C is a constantpotential and has the effect of adjusting the drift in order to ensure that the traded underlying isa Martingale. However, one can use non-constant C to enforce the Martingale condition in modelsincorporating more exotic effects. In this section we suggest some ideas for potential future developmentin this direction. We are representing our traded underlying as S = e x . In this case, S represents the forward contract fora particular maturity. If S represents the spot price for an equity, S T represents the forward price formaturity: T , and r represents a funding rate, then non-arbitrage principals (ignoring dividends) require: S T = S e rT If holding the stock pays a dividend: D at time t d < T , then again this cash flow should be substractedfrom the price we can achieve at time T , in order for the forward price to be arbitrage free, so that: S T = S e rT − De r ( T − t d ) In general, especially for the major equity indices (FTSE100, Euro Stoxx50, S&P500) as well as the mostliquid individual stocks, there are market makers for forward contracts and dividend forecasts are backedout from these. Furthermore, the volatility of the differentials between forward prices at different matu-rities is of a lower order of magnitude compared to the volatility of the spot price. For this reason, mostmodels of the equity markets, used by practitioners, only incorporate the randomness of S . Interestrates, and funding costs, are treated as deterministic variables. Dividend cash flows are either treated asfixed, or treated as a fixed percentage of the spot price: S .There are various classical approaches to modelling the randomness of a term structure of discountfactors. For example one can model stochastic interest rates using the Heath Jarrow Morton framework(see [12]). However, classical models have a number of drawbacks. In general one must compromisebetween the fact that a continuous discount curve is an infinite dimensional object, and the tractabilityof methods for calibration and generating solutions.In fact, one can use the operator C to incorporate some of the randomness of these factors into themodel. Whilst the difficulties mentioned with classical models will also impact quantum models, thesedo offer different angles from which to approach the issue. For example, we assume: • The dividends & interest rates can be incorporated into a single discount factor. We write this: S = DF e x . 11 This discount factor is driven by the same combination of factors (both specific to the equity inquestion and general) that drive the spot price. The discount factor is therefore represented usinga function: DF ( x ). So that we now write our forward price: DF ( x ) e x . • The majority of the time the dividends and interest are stable. In other words we have DF ( x ) ≈ x . • Without loss of generality we can incorporate the initial spot price ( S say) into DF ( x ) so thatthe starting value of x is 0. • By modelling interest rates & dividends using the same underlying variable ( x ) as the main marketprice, the intention is to keep the model simple from the perspective of calibration & generatingsolutions, whilst simultaneously incorporating factors such as cuts to dividends after a marketcrash.Now we require DF ( x ) e x to be a Martingale. Consider, for example, that we wish to take account ofthe possibility of a cut in interest rates or a cut in the dividend yield under extreme market conditions.This would lead to respectively an increase or a decrease in the relevant discount factor. Then we couldapply: DF ( x ) = 11 + εx In this instance, positive ε would lead to a reduction in the discount factor under extreme marketconditions, corresponding to a cut in the dividend yield expectations. Alternatively, a negative valuefor ε would lead to an increase in the discount factor under extreme market conditions. This wouldcorrespond to a cut in interest rates. Figure 1 shows a chart of the discount factor against the driver: x .Now in order to ensure: DF ( x ) e x is a Martingale we require a non-constant potential C ( x ). Althoughwe can no longer necessarily write out a closed form solution (such as equation 10) we can still applythe Feynman-Kac formula (see for example [9] Theorem 20.3) and use Path integral methods to findsolutions. One could relax the above assumption that the random behaviour is driven by a single random variable,by simply extending our Hilbert space from L ( R ) to L ( R n ), where n represented the number of randomvariables. For example, we could model a forward price: S T = DF ( y ) e x , whereby our Hilbert space couldbe: L ( R ). Now, the financial market is represented by wave function: ψ ( x , t ), where x = ( x, y ), and x was the random variable driving the equity markets, and y the random variable driving the interest rates.Now, we must use the operator C to ensure DF ( y ) e x is a Martingale, and therefore C must againbe a non-constant potential: C ( x, y ). Whilst modelling in a higher number of dimensions may increasethe difficulty in finding solutions, the same principals apply. One can still use Path integral methods,based on the Feynman-Kac result. 12igure 1: Discount factor versus driver for positive & negative values for ε . In this section, we briefly review the strategy presented in section 4, from the perspective of Hamilton-Jacobi theory and the Bohmian approach to quantum mechanics. For an overview of this approach, see[18]. For an example of an application to quantum finance, see [20].Start by inserting the wave function: ψ = Re iS (12)Into the Schr¨odinger equation with Hermitian Hamiltonian:ˆ H = − σ P one obtains 2 partial differential equations (see for example [18] chapter 3): ∂S∂t + σ ( ∇ S ) − σ ∇ RR = 0 (13) ∂ R ∂t + ∇ (cid:0) σ R ∇ S (cid:1) = 0 (14)Now, equation 13 can be interpreted as a Hamilton-Jacobi equation with “quantum potential”: Q = − σ ∇ RR . A classical solution S to this equation represents a particle with momentum: P = ∇ S | ψ | = R represents the probability density function for the particle.Therefore, again from a classical perspective, if we interpret R as the density ( ρ ) of a cloud of particles,then the particle flux will be defined by: (cid:126)q = σ ∇ S Finally therefore, we can interpret equation 14 as the continuity equation for the particle: ∂ρ∂t + ∇ (cid:126)q = 0 (15) We now investigate how this works for the Black-Scholes Hamiltonian. For now, we work using thepseudo-Hermitian Hamiltonian given by equation 9. We proceed by inserting the wave function 12 intothe Schr¨odinger equation defined by the Black-Scholes Hamiltonian. ∂ψ∂t = ∂R∂t e iS + iR ∂S∂t e iS ∂ψ∂x = ∂R∂x e iS + iR ∂S∂x e iS ∂ ψ∂x = ∂ R∂x e iS + 2 i ∂R∂x ∂S∂x e iS − R (cid:18) ∂S∂x (cid:19) e iS So, inserting ψ = Re iS into the Schr¨odinger equation: i ∂ψ∂t = − σ ∂ ψ∂x + σ ∂ψ∂x We get: i ∂R∂t e iS − R ∂S∂t e iS = σ e iS (cid:18) ∂R∂x + R (cid:18) ∂S∂x (cid:19) − ∂ R∂x (cid:19) + iσ e iS (cid:18) R ∂S∂x − ∂R∂x ∂S∂x (cid:19) We now divide by e iS , and collect together real & imaginary terms: R ∂S∂t + σ (cid:18) ∂S∂x (cid:19) + σ ∂R∂x − σ ∂ R∂x = 0 i ∂R∂t + iσ ∂R∂x ∂S∂x − i σ R ∂S∂x = 0Finally, we divide the real equation by R , multiply the imaginary terms by R , and replace ∂/∂x withthe more general ∇ to get: ∂S∂t + σ ( ∇ S ) σ ∇ RR − σ ∇ RR = 0 (16) ∂R ∂t + ∇ (cid:18) σ R ∇ S (cid:19) − σ R ∇ S = 0 (17)14ow, equation 16 can still be interpreted as a Hamilton-Jacobi equation, with new quantum potential: Q = σ ∇ RR − σ ∇ RR . However, the interpretation of equation 17 as a continuity equation no longerworks in the same way. This is a consequence of the fact that the pseudo-Hermitian Hamiltonian givenby equation 9, does not conserve probability in the standard Hilbert space inner product.However, we know from [21] and [24], that the Hamiltonian conserves probabilities under the innerproduct 7, with η = e − x . Therefore, the probability density is now given by: ρ = e − x R . Inserting thisinto the classical continuity equation 15 we get: e − x ∂R ∂t + ∇ (cid:0) e − x σ R ∇ S (cid:1) = 0 (18)After multiplying through by e x we get back to equation 17. To start with, consider the Hamilton-Jacobi equation for a particle, obeying classical mechanics withzero potential (see [18]). The partial differential equation is given by: ∂S∂t + σ ( ∇ S ) S , which is itself determined by the relevant initial conditions, determines the motion ofthe free particle through the relationship: P = −∇ S In the absence of any potential, the particle simply moves at constant velocity. If we introduce a largenumber of particles, each with different initial conditions, then each particle will move with a differentvelocity, depending on the initial condition. In the absence of an external potential, these velocities willbe constant.Therefore, consider the situation of N such particles moving freely along the real number line, hav-ing started at x = 0. The position of the particle: i , with velocity: v i , after time: t , will be given by: X i = v i t . Therefore the variance of the particles after time: t is given by: E [ X ] = 1 N N (cid:88) i =1 ( v i t ) E [ X ] = 1 N N (cid:88) i =1 v i t E [ X ] − E [ X ] = t N (cid:18) N (cid:88) i =1 ( v i ) − (cid:16) N (cid:88) i =1 v i (cid:17) (cid:19) (20)We note the following: • The variance increases with t . Therefore, this cannot represent a diffusion with independentincrements. 15 In fact, each time step for one of the particles is 100% correlated, with the previous time-step. Thisis natural, since each particle is moving freely under constant velocity. • If we wish the N particles to represent particles moving under a random diffusion, such as Brownianmotion, then we must introduce a potential function.Thus, if we interpret the wave-function as defining the probability distribution for a classical diffusion,then the quantum potential Q defines the potential function that will ensure the classical particles havethe correct statistical properties.Put another way, consider N particles moving under quantum potential: Q . These particles, whilstobeying the laws of classical mechanics, and depending on the initial conditions, would have a probabil-ity distribution consistent with the Schr¨odinger wave-function. In some senses one could interpret therandomness as arising simply from the variation in the initial conditions. Once the particles were set inmotion, the system is deterministic.In fact, the setting of the initial conditions, can be defined by the moments one wishes the probabilitydensity function to have. This is the equivalent to specifying the mean and variance of the diffusionprocess: lim N →∞ N (cid:88) i =1 v i → σ lim N →∞ N (cid:88) i =1 v i → µ The fundamental solution to equation 5 with C = 0, is given by: ψ ( x, t ) = K t ( x ) ∗ (cid:102) ψ ( p ) K t ( x ) = 1 √ πσ it exp (cid:16) ix σ t (cid:17) (21)Where f ∗ g represents the convolution: (cid:82) f ( x − y ) g ( y ) dy , and (cid:101) h ( p ) the Fourier transform of h ( x ). Wenote, that under the Wick rotation, τ = it we get back to a heat kernel: K τ ( x ) = 1 √ πσ τ exp (cid:16) − x σ τ (cid:17) Following the analysis in [18] chapter 3, we set: R = ( ψψ ) / = 12 πσ τ exp (cid:16) x σ τ (cid:17) Q = − σ ∇ RR = σ (cid:18)(cid:16) xσ t (cid:17) − σ t (cid:19) (22)So finally, the Quantum potential in this case acts as a quadratic potential, that becomes shallower overtime. Therefore, it is clear from the analysis above, that one potential future avenue of research is to designdiffusion processes, to be applied to finance, by starting with the base Hamilton-Jacobi equation for afree particle. The simplest case being equation 19, before choosing the quantum potential: Q . References [1] Accardi L, Boukas A: The Quantum Black-Scholes Equation,
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