 Journal of Mathematical Finance, 2011, 1, 41-49 doi:10.4236/jmf.2011.13006 Published Online November 2011 (http://www.SciRP.org/journal/jmf) Copyright © 2011 SciRes. JMF Adaptive Wave Models for Sophisticated Option Pricing Vladimir G. Ivancevic Defence Science & Technology Organisation, Canberra, Australia E-mail: Vladimir.Ivancevic@dsto.defence.gov.au Received August 19, 2011; revised October 13, 2011; accepted October 22, 2011 Abstract Adaptive wave model for financial option pricing is proposed, as a high-complexity alternative to the stan- dard Black-Scholes model. The new option-pricing model, representing a controlled Brownian motion, in- cludes two wave-type approaches: nonlinear and quantum, both based on (adaptive form of) the Schrödinger equation. The nonlinear approach comes in two flavors: for the case of constant volatility, it is defined by a single adaptive nonlinear Schrödinger (NLS) equation, while for the case of stochastic volatility, it is de- fined by an adaptive Manakov system of two coupled NLS equations. The linear quantum approach is de- fined in terms of de Broglie’s plane waves and free-particle Schrödinger equation. In this approach, financial variables have quantum-mechanical interpretation and satisfy the Heisenberg-type uncertainty relations. Both models are capable of successful fitting of the Black-Scholes data, as well as defining Greeks. Keywords: Black-Scholes Option Pricing, Adaptive Nonlinear Schrödinger Equation, Adaptive Manakov System, Quantum-Mechanical Option Pricing, Market-Heat Potential 1. Introduction Recall that the celebrated Black-Scholes partial differ- ential equation (PDE) describes the time-evolution of the market value of a stock option [1,2]. Formally, for a function defined on the domain =,uuts, 000 is the standard deviation, or volatility of s, r is the short-term prevailing continuously—compounded risk—free interest rate, and is the time to matur-ity of the stock option. In this formulation it is assumed that the underlying (typically the stock) follows a geomet-ric Brownian motion with “drift” >0T and volatility , given by the stochastic differential equation (SDE)   d= dd,ststtstWt (2) where W is the standard Wiener process. The Black-Scholes PDE (1) is usually derived from SDEs describing the geometric Brownian motion (2), with the stock-price solution given by:  122=0etWtst s  . In mathematical finance, derivation is usually perfor- med using Itô lemma  (assuming that the underlying asset obeys the Itô SDE), while in physics it is performed using Stratonovich interpretation [5,6] (assuming that the underlying asset obeys the Stratonovich SDE ). The Black-Sholes PDE (1) can be applied to a number of one-dimensional models of interpretations of prices given to u, e.g., puts or calls, and to s, e.g., stocks or fu- tures, dividends, etc. The most important examples are European call and put options, defined by: Call 12,=dede ,Tusts krT (3) Put2 1,=dede ,rT Tustks (4) 1()=1 erf,22 21ln 2d= ,sTrkT  22ln 2d= ,sTrkT  42 V. G. IVANCEVIC where erf is the (real-valued) error function, k de- notes the strike price and  represents the dividend yield. In addition, for each of the call and put options, there are five Greeks (see, e.g. [9,10]), or sensitivities, which are partial derivatives of the option-price with respect to stock price (Delta), interest rate (Rho), volatility (Vega), elapsed time since entering into the option (Theta), and the sec- ond partial derivative of the option-price with respect to the stock price (Gamma). Using the standard Kolmogorov probability approach, instead of the market value of an option given by the Black-Scholes Equation (1), we could consider the cor- responding probability density function (PDF) given by the backward Fokker-Planck equation (see [6,7]). Al- ternatively, we can obtain the same PDF (for the market value of a stock option), using the quantum-probability formalism [11,12], as a solution to a time-dependent linear or nonlinear Schrödinger equation for the evolution of the complex-valued wave -function for which the absolute square, 2, is the PDF. The adaptive nonlin-ear Schrödinger (NLS) equation was recently used in  as an approach to option price modelling, as briefly re- viewed in this section. The new model, philosophically founded on adaptive markets hypothesis [13,14] and Elliott wave market theory [15,16], as well as my own recent work on quantum congition [17,18], describes adaptively controlled Brownian market behavior. This nonlinear ap- proach to option price modelling is reviewed in the next section. Its important limiting case with low interest-rate reduces to the linear Schrödinger equation. This linear approach to option price modelling is elaborated in the subsequent section. 2. Nonlinear Adaptive Wave Model for General Option Pricing 2.1. Adaptive NLS Model The adaptive, wave-form, nonlinear and stochastic option- pricing model with stock price s, volatility  and interest rate r is formally defined as a complex-valued, focusing (1+1)-NLS equation, defining the time-dependent op-tion-price wave function =,st, whose absolute square 2,st represents the probability density func- tion (PDF) for the option price in terms of the stock price and time. In natural quantum units, this NLS equation reads: 21=,=2tssii1,m (5) where denotes the adaptive market-heat po- tential (see ), so the ter=,rw 2=Vthe represents -dependent potential field. In the simplest nonadap- tive scenario  is equal to the interest rate r, while in the adaptive case it depends on the set of adjustable syn- aptic weights ijw as: 21=1 3,= erfiniiiws .r wwrw (6) Physically, the NLS Equation (5) describes a nonlinear wave (e.g. in Bose-Einstein condensates) defined by the complex-valued wave function ,st of real space and time parameters. In the present context, the space-like variable s denotes the stock (asset) price. The NLS Equation (5) has been exactly solved using the power series expansion method [20,21] of Jacobi elliptic functions . Consider the -function descri- bing a single plane wave, with the wave number k and circular frequency :  i,=e ,ks tst (7) with =andskt . Its substitution into the NLS Equation (5) gives the nonlinear oscillator ODE:   231=0.2'' k   (8) We can seek a solution  for (8) as a linear func-tion  01=snaa, where sn,= snssm are Jacobi elliptic sine func- tions with elliptic modulus 0,1m, such that sn= sin,0ssand sn,1= tanhss. The solution of (8) was calculated in  to be  =sn,for 0,1, mm =tanh,for =1.m This gives the exact periodic solution of (5) as    0,1m221i12,=sn e,kstm ksts ktwm1for ; (9)   21i222for ,=1,tanh e,=kst ksts ktwm (10) where (9) defines the general solution, while (10) defines Copyright © 2011 SciRes. JMF 43V. G. IVANCEVICthe envelope shock-wave1 (or, “dark soliton”) solution of the NLS Equation (5). Alternatively, if we seek a solution  as a linear function of Jacobi elliptic cosine functions, such that cn,0= cosss and cn,1= sechss2,  01=cnaa , then we get    221i1223,=cn e,for 0,1;kstmkstms ktwm(11)   21i124, =seche,for =1,kst ksts ktwm (12) where (11) defines the general solution, while (12) de- fines the envelope solitary-wave (or, “bright soliton”) so- lution of the NLS Equation (5). In all four solution Expressions (9), (10), (11) and (12), the adaptive potential is yet to be calculated us- ing either unsupervised Hebbian learning, or supervised Levenberg-Marquardt algorithm (see, e.g. [23,24]). In this way, the NLS Equation (5) becomes the quantum neural network (see ). Any kind of numerical analy- sis can be easily performed using above closed-form solutions as initial conditions. w=1, ,4 , st iiThe adaptive NLS-PDFs of the shock-wave type (10) has been used in  to fit the Black-Scholes call and put options (see Figures 1 and 2). Specifically, the adap-tive heat potential (6) was combined with the spatial part of (10)  2=tanhssk,t (13) while parameter estimates where obtained using 100 it- erations of the Levenberg-Marquardt algorithm. Figure 1. Fitting the Black-Scholes call option with (w)- adap- tive PDF of the shock-wave NLS-solution (10). Figure 2. Fitting the Black-Scholes put option with (w)- adap- tive PDF of the shock-wave NLS 2(s, t) solution (10). Notice the kink near s = 100. As can be seen from Figure 2 there is a kink near . This kink, which is a natural characteristic of the spatial shock-wave (13), can be smoothed out (Figure 3) by taking the sum of the spatial parts of the shock-wave solution (10) and the soliton solution (12) as: = 100s 212=d tanhd sech.sskt skt (14) 1A shock wave is a type of fast-propagating nonlinear disturbance that carries energy and can propagate through a medium (or, field). It is characterized by an abrupt, nearly discontinuous change in the charac-teristics of the medium. The energy of a shock wave dissipates rela-tively quickly with distance and its entropy increases. On the other hand, a soliton is a self-reinforcing nonlinear solitary wave packet that maintains its shape while it travels at constant speed. It is caused by a cancelation of nonlinear and dispersive effects in the medium (or, field). 2A closely related solution of an anharmonic oscillator ODE:  3=0'' ss s is given by 22=cn1 ,12 12mmThe adaptive NLS-based Greeks (Delta, Rho, Vega, Theta and Gamma) have been defined in , as partial derivatives of the shock-wave solution (10). 2.2. Adaptive Manakov System Next, for the purpose of including a controlled stochastic volatility3 into the adaptive-NLS Model (5), the full bidi-rectional quantum neural computation model  for option-price forecasting has been formulated in  as a self-organized system of two coupled self-focusing NLS .ssmmm 3Controlled stochastic volatility here represents volatility evolving in a stochastic manner but within the controlled boundaries. Copyright © 2011 SciRes. JMF 44 V. G. IVANCEVIC Figure 3. Smoothing out the kink in the put option fi t, by com- bining the shock-wave solution with the soliton solution, as defined by (14). equations: one defining the option-price wave function =,st and the other defining the volatility wave function =,st: Volatility NLS: 221i= ,2tssrw,  (15) Option price NLS: 22i= , .2tssrw1  (16) In this coupled model, the -NLS (15) governs the ,st-evolution of stochastic volatility, which plays the role of a nonlinear coefficient in (16); the -NLS (16) defines the (,)st-evolution of option price, which plays the role of a nonlinear coefficient in (15). The purpose of this coupling is to generate a leverage effect, i.e. stock volatility is (negatively) correlated to stock returns4 (see, e.g. ). This bidirectional associative memory effec-tively performs quantum neural computation , by giving a spatio-temporal and quantum generalization of Kosko’s BAM family of neural networks [28,29]. In ad- dition, the shock-wave and solitary-wave nature of the coupled NLS equations may describe brain-like effects frequently occurring in financial markets: volatility/price propagation, reflection and collision of shock and soli- tary waves (see ). The coupled NLS-system (15)-(16), without an em- bedded learning (i.e., for constant w=r-the inter-est rate), actually defines the well-known Manakov sys-tem,5 proven by S. Manakov in 1973  to be com- pletely integrable, by the existence of infinite number of involutive integrals of motion. It admits “bright” and “dark” soliton solutions. The simplest solution of (15)-(16), the so-called Manakov bright 2-soliton, has the form re- sembling that of the sech-solution (12) (see [34-40]), and is formally defined by:  222i 22solψ,= 2sech24e,atasbtstbb satc(17) where sol,ψ,= ,ststst, is a unit vector such that 12=,Tccc2212=1cc. Real-valued parameters a and b are some simple func-tions of ,,k, which can be determined by the Le- venberg-Marquardt algorithm. I have argued in  that in some short-time financial situations, the adaptation effect on  can be neglected, so our option-pricing Mo- del (15)-(16) can be reduced to the Manakov 2-soliton Model (17), as depicted and explained in Figure 4. 3. Quantum Wave Model for Low Interest-Rate Option Pricing In the case of a low interest-rate , we have 1r1r, so 0,V and therefore Equation (5) can be approximated by a quantum-like option wave packet. It is defined by a continuous superposition of de Broglie’s plane waves, ‘physically’ associated with a free quantum particle of unit mass. This linear wave packet, given by the time-dependent complex-valued wave func- tion ,=stˆ, is a solution of the linear Schrödinger equation with zero potential energy, Hamiltonian opera- tor H and volatility  playing the role similar to the Planck constant. This equation can be written as: Figure 4. Hypothetical market scenario including sample PDFs for volatility 2 and 2 of the Manakov 2-soli- ton (17). On the left, we observe the evolution of stochastic volati lity: we have a collision of two volatility com- ponent-solitons, ,st ,1Sst and ,2Sst , which join together into the resulting soliton ,2Sst , annihilating the ,1Sst component in the process. On the right, we observe the ,st -evolution of option price: we have a collision of two option component-solitons, ,1Sst and ,2Sst , which pass through each other without much change, except at the collision point. Due to symmetry of the Manakov system, volatility and option price can exchange their roles. 4The hypothesis that financial leverage can explain the leverage effect was first discussed by F. Black . 5Manakov system has been used to describe the interaction between wave packets in dispersive conservative media, and also the interaction between orthogonally polarized components in nonlinear optical fibres (see, e.g. [32, 33] and references therein). Copyright © 2011 SciRes. JMF 45V. G. IVANCEVIC2ˆˆi=,where=2 tsHH.s (18) Thus, we consider the -function describing a single de Broglie’s plane wave, with the wave number k, linear momentum =pk, wavelength =2π,kk angular frequency 2=2,kk and oscillation period 2=4=2Tkkk. It is defined by (compare with [41,42,12]) 2i2i22,=e =e=cosisin ,2kkks tks tkst AAkk2AkstAkst (19) where A is the amplitude of the wave, the angle 2=2kkks tkst represents the phase of the wave k with the phase velocity: ==kkvkk2.The space-time wave function ,st that satisfies the lin- ear Schrödinger Equation (18) can be decomposed (using Fourier’s separation of variables) into the spatial part s and the temporal part ietas:  ii,=e=e.Ettst ss  The spatial part, representing stationary (or, amplitude) wave function, satisfies the linear har- monic oscillator, which can be formulated in several equivalent forms: i=e,kssA2222=0, =0,2=0, =0.'' '''' ''kkkpkEv  (20) Planck’s energy quantum of the option wave k is given by: 21==2kkEk . From the plane-wave expressions (19) we have: i,=epsEtkkst A —for the wave going to the “right” and i,=epsEtkkst A —for the wave going to the “left”. The general solution to (18) is formulated as a linear combination of de Broglie’s option waves (19), compris- ing the option wave-packet:  =0,=, , (with).nikiistc stn (21) Its absolute square, 2,,st represents ability density function at a time t. n optionbythe prob- The group velocity of a wave-packet is given : =dd .gkvk It is related to the phase velocity v of a plane wave as: k=dd.gkkkkvv v Closely related is the center of the option wave-pack t (the point of maximuem amplitude), given: by=d d.kstk The following quantum-motivated assertions can be stated: y 1) Volatilit has dimension of financial action, or energy  time. 2) The total rgy E of an option wave-packet is (in the case of simil r plane waves) given by Planck’s su- perposition of theenea energies kE of n individual waves: 2== ,2knEn k  =Lnwhere denotes the angular momentum of the option wave-packet, representing the shift between its and decaygrowth, and vice versa. 3) The average energy E of an option wave-packet is given by Boltzmann’s partition function: =0e==,kTknknE EE =0ee1bnE EkkbT bTnwhere b is the Boltzmann-like kinetic constant and T is the market temperature. nE4) The energy form of the Schrödinger Equation (18) reads: =i tE. 5) The eigenvalue equation for the Hamiltonian op- erator ˆH is the stationary Schr ödi nger equation:  2ˆ=,or =,2ssHsEs Ess  which is just another form of the harmonic oscillator (20). It has oscillatory solutions of the form: ii2212=ee,EsEskkEsc c called energy eigen-states with energies and deno- ted by: kEEkEˆ=.HsE s The Black-Scholes put and call options have been fit-ted with the quantum PDFs (see Figures 5 and 6) given byt opti the absolute square of (21) with =7n and =3n, respectively. Using supervised Levenberg-Marquardt al- gorithm and Mathematica 7, the following coefficients were obtained for the Black-Scholes puon: Copyright © 2011 SciRes. JMF 46 V. G. IVANCEVIC Figure 5. Fitting the Black-Scholes put option with the quan- tum PDF given by the absolute square of (21) with n = 7. Figure 6. Fitting the Black-Scholes call option with the quan- tum PDF given by the absolute square of (21) with n = 3. Note that fit is good in the realistic stock region:Using the same algorithm, the following coefficients were obtained for the Black-Scholes call option: Now, given some initial option wave function, []75,140s. **=0.0031891, =0.0031891, =2.62771, =2.62777, =2.65402,tkkk12345671234567=2.61118, =2.64104, =2.54737, =2.62778, =1.26632, =1.26517,=2.74379, =1.35495, =1.59586, =0.263832, =1.26779,withkkccccccc**=94.0705, =31.3568.BS BStt kk**123123**=11.9245, =11.9245, =0.851858, =0.832409,=0.872061, =2.9004, =2.72592, =2.93291,with0.0251583, =0.00838609.BStkkkccctt ,0 =,0ss a solution to the initial-value problem for the linear Schrö- dinger Equation (18) is, in terms of the pair of Fourier transforms 1,, given by (see )  2i1i1200,st=e =e.ktt   (22) p- tion wave-function at time t = 0 givby the complex- valued Gaussian function: he width p is the av- erage momentum of the wave. Its Fourier transform, For example (see ), suppose we have an initial oen 2/2 i,0=ee,as kss where a is tof the Gaussian, while 0ˆ=,0ks, is given by 220eˆ=.kpakaThe solution at time t of the initial value problem is given by  22i221,=ee d,2π kk pks taastk awhich, after some algebra becomes   222i iexp 21 i,=,with =.1iasspp tatstpat kiven by the real-valued Gaussian function, As a simpler example,6 if we have an initial option wave- function g224e,0 =,ss the solution of complex-valued (18) is given by the - function,  24exp 21 i,= .1istst t 6An example of a more general Gaussian wave-packet solution of (18) is given by:  2200001ii22,= exp,1i1ias sptpssast at at where 00,spt are initial stock-price and average momentum, while ais the width of the Gaussian. At time the `particle' is at rest around , its average momentum 0. The wave function spreads with time while its maximum decreases and stays put at the origin. At time the wave packet is the complex-conjugate of the wave-packet at time t. =0t=0p=0sCopyright © 2011 SciRes. JMF 47V. G. IVANCEVICFrom (22) it follows that a stationary option wave-packet is given by:  i1ˆˆ=ed,where=[2ks ].skkk s As 2s is the stationary stock PDF, we can cal-culate the expectation values of the stock and the wave number of the whole option wave-packet, consisting of n measured plane waves, as:  2ˆ=dand=2d.sss skk kk  (23) red around the mean values (23). The width of the distribution of the recorded The recordings of n individual option plane waves (19) will be scattes- and -values are uncertainties ks and respectively. The satisfy th Heisenberg-type uncertainty relation: ,k ye,2nsk  which imply the similar relation for the total option en-ergy and time: .2 4. A New Stock-Market Research Program Based on the above wave stock-market anEt nalysis, I pro-pose a new financial research program as follows. Firstly, define the general adaptive wave model for option pricing evolution as a (linear) combination of the (5). The three wave-components of this general model are: 1) the linear wave packet previously defined particular solutions to the adaptive NLS-Equation pack ,etst, given by (21); 2) the shock-wave ,shockst, given by (10); and 3) the soliton soliton ,st, given by (12). Formally, the general adaptive wave model is defined by: 2123sec ,ks t kAs2igeneral 1=01i222,= etanhekks tiikst kstA cAskt 1ih ektw dentudes of the t inSecondly, we need to find the most representative fi- nancial index or contemporary markets data that clearly show in their evolution both the efficient markets hy- pothesis  and adaptive markets hypothesis . Once we find such a representative data, we need to fit it using our general wave Model (24) and the powerful Len-berg-Marquardt fitting algorithm. I remark here that, based on my empirical experience, the general wave Model (24) is capable of fitting any financial data, provided we use appropriate number of fitting coefficients (see  for al details). l have a model that can be used fowaves. For the purpose of fitting a, the Levenberg-Marquardt algo- nhere , =1,iAi ote adaptive ampli, 5hree waves, while the other parameters are defined the previous section. vetechnicOnce we have successfully fitted the most representa- tive market data we wilr prediction of many possible outcomes of the current global financial storm. 5. Conclusions I have proposed an adaptive-wave alternative to the stan- dard Black-Scholes option pricing model. The new model, philosophically founded on adaptive markets hypothesis [13,14] and Elliott wave market theory [15,16], describes adaptively controlled Brownian market behavior. Two ap- proaches have been proposed: 1) a nonlinear one based on the adaptive NLS (solved by means of Jacobi elliptic functions) and the adaptive Manakov system (of two coupled NLS equations); 2) a linear quantum-mecha- nical one based on the free-particle Schrödinger equation nd de Broglie’s plane athe Black-Scholes datrithm was used. The presented adaptive and quantum wave models are spatio-temporal dynamical systems of much higher com- plexity  then the Black-Scholes model. This makes the new wave models harder to analyze, but at the same time, their immense variety is potentially much closer to the real financial market complexity, especially at the time of financial crisis. 6. References  F. Black and M. 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