Wild Randomness, and the application of Hyperbolic Diffusion in Financial Modelling
aa r X i v : . [ q -f i n . M F ] J a n Wild Randomness, and the application of Hyperbolic Diffusion inFinancial Modelling
Will HicksJanuary 13, 2021
Abstract
The application of the Cauchy distribution has often been discussed as a potential model of thefinancial markets. In particular the way in which single extreme, or “Black Swan”, events can impactlong term historical moments, is often cited. In this article we show how one can construct Martin-gale processes, which have marginal distributions that tend to the Cauchy distribution in the largevolatility limit. This provides financial justification to the approach investigated in [19], and high-lights an example of how quantum probability can be used to construct non-Gaussian Martingales.We go on to illustrate links with hyperbolic diffusion, and discuss the insight this provides.
In [15], Mandelbrot distinguishes between the concepts of ‘mild’ randomness, for example that describedby a normal distribution, which consists of fluctuations about a well defined mean, and ‘wild’ randomness,for example that described by a Cauchy distribution. The Cauchy distribution has undefined moments,including an undefined mean, and describes a random variable whereby single extreme observations canhave a significant effect on the the overall statistical properties.Mandelbrot also presents evidence, in [15], that financial market returns are often better describedby wild randomness, in the sense that ‘black swan’ events occur from time to time that significantlydistort the mean return observed over a historical period. This is despite the fact that, for reasons ofpragmatism, models applied by practitioners in the financial services industry are invariably based onmodels of ‘mild’ randomness.The Black-Scholes model is derived using the principals of non-arbitrage, and through applying theMartingale Representation Theorem (see for example [18]) to represent the financial market price as anIto process.In [19] the authors attempt to generalise the Black-Scholes equation using relativistic quantum me-chanics. The authors map the Klein-Gordon equation back to a new equation in the same way thatthe Schr¨odinger equation is mapped to the conventional Black-Scholes (for example see [2]). The keymotivation for doing so, is to investigate the behaviour of the market in the large volatility limit.Under these extreme conditions, the authors find that the processes driving the log returns for a stock1rice tend to a Cauchy distribution, and they note that this is in line with analysis carried out on thechaotic behaviour of the financial market by Mandelbrot in [15]. Therefore, the motivation for studyingsuch models is clear.However, despite the appeal of the approach, there are issues to overcome. Generally, when derivingthe partial differential equations of mathematical finance, one starts by expanding the function repre-senting the derivative price ( u ( x, t )) to first order in dt (see for example [13]): dx = σ ( x, t ) dWdu = (cid:18) ∂u∂t + σ ( x, t ) ∂ u∂x (cid:19) dt + ∂u∂x σ ( x, t ) dW (1)Then, taking expectations in the Martingale measure, we end up with a differential equation that is 2ndorder with respect to the risky underlyings (eg the forward price: x ) and first order with respect to time.Therefore, this approach cannot be used to to justify the use of an equation such as the Klein-Gordonequation, which is second order with respect to time, for modelling in finance.In this article we investigate these issues using quantum probability. First in section 2, we discusssome of the principals in relation to this quantum approach. Then in section 3, we show how the pseudo-Hermiticity of the Klein-Gordon Hamiltonian (see [16], [17]) can be used to enforce the Martingalecondition and the extent to which a non-arbitrage model can be constructed. Finally, in section 4, weshow how the resulting equation can be equated to hyperbolic diffusion equations, and discuss insightsthat can be obtained from this comparison. The application of quantum methods to mathematical finance has been investigated by a number ofsources. One example approach is suggested by Accardi & Boukas in [1]. The authors develop a quan-tum Black-Scholes equation using the methods of quantum stochastic calculus (see [11]), in the placeof the standard Ito calculus. In this approach, the derivative price is modelled as an observable actingon the Market, which is represented by a state that exists within a Fock space. The authors developoperators that define the evolution of the market state, by adding noise to the Fock space. This noise isthe equivalent of the usual Ito diffusion within a classical space.In [9], the author discusses the nature of drift in such a quantum model. The related notion of the closed quantum Black-Scholes is suggested. In this model, it is found that, unlike randomness introducedfrom an external source, the intrinsic randomness associated to the quantum state does not necessarilyintroduce the usual property of strictly positive time value for the holder of an option contract (such asa vanilla call option contract).In this article, we take a different approach. We model the price of a financial instrument using awave-function that is described by a Schr¨odinger equation. For example, if we wished to model a finan-2ial underlying that followed a Gaussian process, we could use the following equation: i ∂ψ∂t = ˆ Hψ (2)ˆ H = ∂ ∂x (3)Under this model, the probability density for the market price x , at time t , given the solution ψ ( x, t ) to2, is given by: p ( x, t | ψ ) = h δ x | ψ i = ψ ( x, t )Under this interpretation, one can apply the mapping it → τ (often referred to as a Wick rotation),under which 2 is mapped to the standard heat equation. This link has been investigated by a number ofsources. For example, Haven discusses the implications of modelling a derivative price as the state func-tion in a Schr¨odinger equation in [6], [7]. In addition, the path integral approach to financial modellinghas been investigated in [2]-[4], [14].In some senses this approach is justified on mathematical rather than financial grounds. We are modellingusing a closed system, and there is no external source of randomness. As discussed in [9], it is generallysuch external randomness that drives classical diffusion like properties and associated positive time-valuefor option contracts. However, the approach allows us to define a probability measure with the desiredstatistical properties, and which can interpreted as a classical diffusion, via the Wick rotation. In relativistic classical mechanics, the energy of a free particle is specified by the energy-momentum4-vector. With the speed of light, c , set to 1, this is given by: ( E, p x , p y , p z ). The theory of specialrelativity implies that the scalar product (in the Minkowski metric) of this vector with itself gives thesquare of the rest mass: E − | p | = m . Finally, by quantizing in the usual way: E → i∂/∂t , and p x → − i∂/∂x , we get the Klein-Gordon equation (in one spatial dimension): ∂ ψ∂t = ∂ ψ∂x − m ψ (4)In theory, one could attempt to use the methods outlined in section 4 of [10], to provide a non-arbitragejustification for a generalised Black-Scholes equation which was second order in time. This is one advan-tage of the approach given, over the classical approach to mathematical finance. However, this cannotbe achieved by working directly with the Klein-Gordon equation.One can write 4 as follows (see [5], section 4.1): ∂∂t (cid:18) ψ ψ (cid:19) = (cid:18) ∇ x − m (cid:19) (cid:18) ψ ψ (cid:19) Since the matrix operator is not skew-adjoint, unlike the Schr¨odinger equation, the Klein-Gordon equationis not Hermitian. Therefore, further work is required to provide the necessary risk neutral justification.We must first investigate the pseudo-Hermiticity of the Klein-Gordon equation.3 .1 Pseudo Hermiticity of the Klein-Gordon Equation:
A detailed study on the pseudo-Hermiticity of the Klein Gordon operator can be found at [16] and [17].In this section we simply quote the results we wish to apply in a financial modelling context.To start with, write equation 4 as a Schr¨odinger equation on the Hilbert space, H = L ( R n ) ⊕ L ( R n )(with arbitrary λ ∈ R − i ∂ Ψ( t ) ∂t = ˆ H Ψ( t ) (5)Ψ( t ) = (cid:18) ψ ( t ) + iλ ˙ ψ ( t ) ψ ( t ) − iλ ˙ ψ ( t ) (cid:19) ˆ H = 12 (cid:18) λD + λ − λD − λ − − λD + λ − − λD − λ − (cid:19) In [16] Mostafazadeh shows that 5 is Pseudo-Hermitian, under the automorphism: η + = 18 (cid:18) λ + D − λ − D − λ − D − λ + D − (cid:19) (6) D = − ∂ ∂x + µ Since η + is positive definite, this implies that the Hamiltonian defined by 5 has a real valued spectrum,and conserves probability under the positive definite inner product:Φ = (cid:18) φ + iλ ˙ φφ − iλ ˙ φ (cid:19) Ψ = (cid:18) ψ + iλ ˙ ψψ − iλ ˙ ψ (cid:19) h φ | ψ i η + = 12 (cid:16) h φ | ψ i + Z R dx Z R dy ˙ φ ( x ) G ( x − y ) ˙ ψ ( y ) (cid:17) G ( u ) = exp ( − µ ( | u | ) / π | u | (7) If, instead of the relation given in [16], we use:Ψ( t ) = (cid:18) e − it/λ ψ ( t ) + iλe − it/λ ˙ ψ ( t ) e − it/λ ψ ( t ) − iλe − it/λ ˙ ψ ( t ) (cid:19) the Schr¨odinger equation defined by the new Hamiltonian (5) is given by: Dψ + iλ ˙ ψ = − iψλ + 2 i ˙ ψ − λ ¨ ψ − Dψ + iλ ˙ ψ = − iψλ + λ ¨ ψ λ ∂ ψ∂t − i ∂ψ∂t + Dψ = 0If we set: D = − σ ∂ ∂x + µ , then we get: i ∂ψ∂t − λ ∂ ψ∂t = − σ ∂ ψ∂x + µ ψ (8)In the next sections, we review the following: • The limiting case λ →
0, section 3.3. • The Cauchy distributed solution in the case µ →
0, section 3.4. • Enforcing the Martingale condition in section 3.5. • Links to the hyperbolic heat conduction equation, section 4. λ → . First define: L λ ψ = λ ∂ ψ∂t − σ ∂ ψ∂x + µ ψ So that we have: i ∂ψ∂t = L λ ψ Furthermore, we denote: ˆ Hψ = − σ ∂ ψ∂x + µ ψ (9)If we use the norm defined by: || ψ || = p h ψ | ψ i , then we have: || L λ ψ − ˆ Hψ || = λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ψ∂t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Now strictly speaking, since L λ and ˆ H are unbounded operators they can only be defined on a densesubset of the Hilbert space L ( R ). However, assuming ψ is in the intersection of the domains of ˆ H , L λ and ∂ /∂t , then we have, for λ → || L λ ψ − ˆ Hψ || → H BS ψ = − σ ∂ ψ∂x + σ ψ Therefore, we find that in the limiting case of λ →
0, if µ → σ /
8, then the random variable will matchthe quantum version of the arbitrage free Black-Scholes equation: i ∂ψ∂t = ˆ H BS ψ .4 The Cauchy Distributed Solution. Now let ψ be a solution to equation 8. We have seen that for small λ , the solution will be similar to theBlack-Scholes model. In this section we consider the model, in the limit µ →
0, and for large λ .Let ψ be a solution to equation 8. Further, let f ( t ) represent a differentiable complex function oftime. Then we have: ∂ ∂t ( f ( t ) ψ ) = ∂ f∂t ψ + 2 ∂f∂t ∂ψ∂t + f ( t ) ∂ ψ∂t ∂ ∂x ( f ( t ) ψ ) = f ( t ) ∂ ψ∂x Therefore, if we set f ( t ) = exp ( − it/ λ ), then we can write equation 8 as: λf ( t ) ∂ ( f ( t ) ψ ) ∂t = − f ( t ) D ( f ( t ) ψ ) D = − σ ∂ ∂x + (cid:18) µ + 14 λ (cid:19) (10)After carrying out the Wick rotation τ = − it , taking the limit µ →
0, and multiplying both sides by f ( t ), we get: λ ∂ ( f ( τ ) ψ ) ∂τ + σ ∂ ( f ( τ ) ψ ) ∂x = (cid:18) λ (cid:19) f ( τ ) ψ (11)Next we make the substition z = x p λ/σ to get: ∂ ( f ( τ ) ψ ) ∂τ + ∂ ( f ( τ ) ψ ) ∂z = (cid:18) λ (cid:19) f ( τ ) ψ (12)As shown in [19], the equation: ∂ ( f ( τ ) ψ ) ∂τ + ∂ ( f ( τ ) ψ ) ∂z = 0Has a Cauchy distributed solution: f ( τ ) ψ C ( z, τ ) = 1 π τz + τ Feeding this back into equation 12, we find: • f ( τ ) → λ → ∞ . • The right hand side of 11 reduces with O ( λ − ) as λ → ∞ . • The coefficients on the left hand side are independent of λ Therefore, in the limit of large λ , we find our solution tends to a Cauchy distribution.6 .5 The Martingale Condition: We assume that the Martingale condition can be written: E ψ [ X | x ] = x (13)The solution to 5 yields two independent functions: ψ ( x, t ), and ˙ ψ ( x, t ). As noted in [16] inner product7 is both positive definite, and time invariant. It can therefore be used to construct a probabilityinterpretation. To do this, using 7, we assume the probability density function, conditional on the valueat t = 0, is given by: p ( x, t | x ,
0) = 12 | ψ ( x, t ) | + 12 (cid:12)(cid:12)(cid:12)(cid:12) ˙ ψ ( x, t ) Z R dyG ( x − y ) ˙ ψ ( y, t ) (cid:12)(cid:12)(cid:12)(cid:12) Ψ( x, t ) = (cid:18) ψ ( x, t ) + iλ ˙ ψ ( x, t ) ψ ( x, t ) − iλ ˙ ψ ( x, t ) (cid:19) So finally, the Martingale condition can be expressed in this case as: x = 12 Z R x | ψ ( x, t ) | dx + 12 Z R x (cid:12)(cid:12)(cid:12)(cid:12) ˙ ψ ( x, t ) Z R dyG ( x − y ) ˙ ψ ( y, t ) (cid:12)(cid:12)(cid:12)(cid:12) dx (14) In general, classical diffusion models can be obtained by combining the equations: ∂u∂t = ∇ · q (15) q = − K ∇ u (16)Let u ( x, t ) represent the number of particles at time t in an infinitesimally small region around x , and q represent the flux of the particle flow. Then equation 15 simply represents the continuity equation: thenumber of particles in the system is constant. Equation 16, represents the fact that the movement ofparticles is purely random, and that an increased concentration of particles in one region will result inmore particles randomly flowing out of the region, than randomly flowing in.In [8], the authors show that by amending equation 16 to: q + λ ∂ q ∂t = − K ∇ u (17)one generates the hyperbolic diffusion equation: λ ∂ u∂t + ∂u∂t = K ∂ u∂x (18)Similarly, if one applies a Wick rotation; τ = it to equation 8 with µ = 0, one obtains: ∂u∂τ + λ ∂ u∂τ = σ ∂ u∂x (19)7o we can see that, in the same way that the Gaussian solutions of the conventional heat equation arelinked to the Schr¨odinger equation with Hamiltonian:ˆ H = − σ ∂ ∂x The solutions in this case, with µ = 0, are linked to the Hyperbolic Heat equation.In this model, due to equation 17, a gradient in the concentration of particles can coexist with zeroflux, when it drives instead an acceleration in the flux. In other words the flow of particles doesn’t reactinstantaneously to an asymmetric distribution of particles, but reacts over a time period characterisedby λ . A small value for λ , indicates that the flux will accelerate quickly.An explicit finite difference scheme shows that the vector at time: n + 1, is given by (where L isthe discretized second derivative operator): u n +1 = u n + (cid:18) δt − δt λ (cid:19) ∂ u n ∂t + (cid:18) Kδt λ (cid:19) L u n (20)From this it is clear that: • From 17, it is clear that by modelling using a diffusion equation that is second order in time, we aremodelling with a system whereby information flows at finite speed. Indeed, this is the motivationgiven by most authors when studying hyperbolic diffusion. The finite speed of the transfer ofinformation has clear parallels in the understanding of the dynamics of the financial market. • From the discretization 20, it is clear, that the next diffusive time-step is driven both by the currentprobability density/concentration of particles ( u n ) and its current rate of change ( ∂ u n /∂t ). Thiscan also be applied in the financial context. The dynamics of a market in a rapidly changing stateare likely to be very different to the market that has been stable over recent history. • This conclusion is consistent with the quantum approach provided above, where the general solutionto equation 8 involved 2 independent functions ψ ( x, t ), and ˙ ψ ( x, t ). • At first glance the dependence of the diffusion on the initally observed rate of change destroys thepossibility of using this approach in a non-arbitrage pricing framework. However, there are manyreal properties of the financial markets, such as a variance that doesn’t scale linearly with time,that do not fit the stringent requirements for a non-arbitrage approach. Therefore, we leave openthe question of how the relativistic quantum Black-Scholes can be applied in practice. • Lastly, if one wishes to model in the large λ limit, one must scale the diffusivity K (or σ ) accord-ingly. In this article we have sought to provide justification to the approach suggested in [19], using methodsdeveloped in [10]. The result is a model that in the small λ limit behaves like a standard Black-Scholesmodel, and in the large λ limit tends to models of wild randomness, such as the Cauchy distribution.8e have also shown how links to the hyperbolic heat equation can provide some understanding re-garding the nature of the dynamics of the models, and also numerical methods that may be applied.Further work is required to incorporate the model into a real model of the financial market, and todecide the extent it can be embedded within a no arbitrage pricing framework. References [1] Accardi L, Boukas A: The Quantum Black-Scholes Equation,
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