Black Scholes - Quantum Calculus

It usually takes a few decades until some mathematical object slides into Pop culture and eventually becomes kitsch. The Black-Scholes differential equation $f_t = f - x f_x - x^2 f_{xx}$ origin from the 1970ies. It is derived from stochastic process model. It became Pop culture in 2000, when NOVA made a TV show about it. It became infamous during the mortgage crisis of 2008 and indirectly appeared in the Hollywood movies “Margin Call” from 2011 (featuring Tucci, Spacey, Irons, Quinto, Moore) and the “Big short” from 2015 (featuring stars like Bale, Gosling, Pit, Carell, Robbie). We educators usually then jump onto the band wagon and ride a topic to its death. I myself used to mention Black Scholes in 21a courses since 2013. I think I used Black Scholes PDE in Homework since 2015 [PDF] at least (and always included it since then until 2025). It is a great problem to ask students for finding solutions to this equation. I usually did not reveal solutions so that students can not just look it up. Solutions which have come up over the years were $e^t + x, e^{-3t} x^2, 1/(2x) + (x/2) \exp(t) \log(x)$ . Jun Hou Fung (who had been a graduate student here and taught with me a couple of times in multi-variable calculus) found the family $exp(at) (b x^{\sqrt(1+a)} + c x^{\sqrt(1-a)} )$ . The pop culture textbook writers and youtubers are then the ones who carry the already dead corps to its grave. Derek Muller who runs Veritasium, did two years ago a great piece on the trillion dollar equation. This brings the topic into the “lava lamp” stage of kitsch. You can hardly cover it any more afterwards, even so it is beautiful. It is the lava lamp phenomenon. Lava lamps are a beautiful thing illustrating fluid dynamics but having it in your living room has become a sign of “bad taste”. Here is the three parameter Jun Hou Fung Solution

f[x_, t_] := Exp[a t] (b x^Sqrt[1 - a] + c x^-Sqrt[1 - a]);  
FullSimplify[D[f[x, t], t] == (f[x, t] - x D[f[x, t], x] -  x^2 D[f[x, t], {x, 2}]) ]

In order to make more sense of the modified wave equation $u_{tt} + D^2 u + (q-1)[ u_t/t-u/t^2]=0$ , I decided to combine the time derivative part and define a modified acceleration $D_{tt} = u_{tt} + (q-1)[ u_t/t-u/t^2]$ so that the modified wave equation $D_{tt} u = -D^2 u$ looks like the usual wave equation. I am excited about this, because this modified wave equation features sharp wave fronts in all dimensions, unlike the usual wave equation for which the strong Huygens principle fails in even dimensions. Losing the strong Huygens principle does not look terrible (it seems like a minor technical issue) but for a physicist who likes to work in 4-dimensional space time for example, this is a disaster. I saw once a speculation that this should be the reason why we have to introduce higher dimensional Kaluza-Klein models. Lets look at this modified acceleration a bit more closely. It is not defined at t=0, but we do do not want to look at arbitrary small distances anyway. Let me restate the theorem: (there had been a typo on the board as there was one t too much. Here is the correct statement:

Theorem (2026): On a q-manifold with exterior derivative $d$ and Dirac operator $D=d+d^*$ , the modified wave equation $(D_{tt} + D^2) u=0, u(0)=0, u_t(0)= df$ has the solution $u(t)= d_t f$ . Here $D_{tt} = u_{tt} + (q-1)[ u_t/t-u/t^2]$ is a modified acceleration and $d_t f = t \phi_{q+2}(tD) df$ is a modified exterior derivative and $\phi_q$ is a Bessel solution for the q-Bessel equation $u''(r)+(q-1) u'(r)/r+u(r)/r^2=0$ with $\phi_q(0)=0,\phi'_q(0)=1$ . The new exterior derivative $d_t$ is a bounded operator. The solution curve $d_t f$ is defined for all elements in the Hilbert space of $L^2$ forms. We have $\lim_{t \to 0} d_t f/t \to df$ but the limiting initial velocity $u_t(0)=df$ only makes sense for smooth differential forms. The solution $d_t f$ satisfies the strong Huygens principle: $d_tf(p)$ only invokes knowing $f$ for points in distance $t$ from $p$ . To be technically correct, $t$ must be smaller than the radius of injectivity. Otherwise, we have not to refer to distance but to the wave front $W_t(p)$ .

The prototype is the one dimensional case $q=1$ , where $D_{tt} f = f_{tt}$ and $D=i d/dx$ and $\phi_3(r) = \sin(r)/r=[\exp(i r)-\exp(-ir)]/(2ir)$ and $u(t,x) = d_t f(x) = [f(x+t)-f(x-t)]/2$ and $d_t f/t \to f'(x)$ in the limit $t \to 0$ . The strong Huygens principle holds because $u(t,x)$ only invokes values of the initial condition at distance $t$ away from $x$ . In the 2- dimensional case $q=2$ already, the strong Huygens principle fails for the traditional wave equation $u_{tt} = - D^2 u$ . There is still the nice d’Alembert solution $u(t,x) = \phi_1(tD) u(0) + \phi_3(tD) u'(0)$ (like in all dimensions q) but the Kirchhoff solution formula shows that in even dimensions it now invokes initial values also inside the disk of radius $t$ . There are wakes. It would be a total disaster to have such wakes in three dimensions. It would mean that if we shine out light with a flashlight in vacuum and turn it off, that we will still see some light (even if there is nothing that can bounce back) It would be as if photons moving away would radiate backwards also. (We would see “tail lights” of the photons). When using radio broadcasting, we would hear echos even without radio waves bouncing back. It would be nice to deduce some consequences which would follow from a lack of the Huygens principle. It is most likely that we could not exist as humans, maybe because every star would get flooded and so cooked in its own radiation. Invoking the cheap antropic principle would give a reason “why” we need a strong Huygens principle after all or give an other reason (there are topological reasons why we can not live in 2D and dynamical reasons why can not live in 4D).

The link to Black-Scholes comes up because the homogeneous structure of the derivative reminded me of the homogeneous structure of the derivatives appearing in Black Scholes. Indeed, we can write Black-Scholes as $f_t = S^2 D_{SS} f$ . Written like this, it looks a bit like a “driven heat equation”. It is illustrated pretty well in the “Margin call” movie, how delicate it can be to balance risk in the case of systems that can show run-away solutions. Like any model, also Black-Scholes only is valid in certain ranges. If the margins are reached, one gets a “margin call”. The models breaks down for many reasons now, not at least once the confidence into the system is gone. The movie “margin call” showed how quickly one needed to react and organize a fire sale over night in order to save the “firm” from collapse. Both the “Margin Call” movie and the “Big Short” movie were fantastic and also a bit depressive (even so they probably distort quite a bit for dramatic reasons). An other nice movie in the finance sector is “The wolf of wall street” from 2013 featuring diCaprio (a movie that was was not based on the 2008 crash but more on traditional fraud and money laundering.)

Lets look at the basic Newton type questions: what happens if the acceleration is zero? We have the ordinary differential equation $u'' + (q-1) [ u'/t - u t^2]=0$ . Newton found that an object moves at a constant velocity unless acted upon by a nonzero net force.

Invertia law of Newton: For any $q$ , the modified acceleration $D_{tt} u=0$ and $u(0)=0, u'(0)=v$ has the unique solution $u(t)=tv$ . There is no solution if u(0) is not zero.

Now start turning on acceleration. Lets look at constant acceleration a, this means that the system is subject of a constant force. We still have a quadratic growth but the is some damping of the order of q.

Second law of Newton: For any $q$ the modified acceleration $D_{tt} u=a$ and $u(0)=0, u'(0)=v$ has the unique solution $u(t)=tv+ a t^2/(q+1)$ . There is no solution if u(0) is not zero.

In the case q=1, we have the usual acceleration and the get $tv+a t^2/2$ . Again, also here in general, we need u(0)=0, so that we have a unique solution. The case when u(0) is not zero does not make sense. Also in the case of the wave equation, where in the classical case $u(t)= \phi_1 (tD) u(0) + \phi_3(tD) u'(0)$ is the d’Alembert solution in all dimensions (written usually in the form of pseudo differential operators (defined via Fourier theory) $D=-\sqrt{-\Delta}$ in PDE texts, I prefer to take $D=d+d^*$ and simultaneously look at solutions in all form sectors. This by the way is mathematically equivalent to define “Dirac matrices”. Mathematically, the trivial Clifford algebra defined by zero quadratic forms is equivalent to the algebra of differential forms. As vector spaces they are the same.

Lets look at an other example, where we look at a driven Differential equation $D_{tt} u = sin(t)$ as we do in intro linear algebra courses with the usual acceleration $u_{tt} = sin(t)$ , a case of resonance. Here is a link to a 1 minute video about driven differential equations. But now:

Driven oscillator example: For any q, the differential equation $D_{tt} = \sin(t)$ has bounded solutions with u(0)=0,u'(0)=-1/q. For a general initial velocity v, the solution u(t)+(-1/n-v) is bounded. The initial velocity -1/n just beats the resonance. This generalizes the n=1 case, where $u''=sin(t) with u(0)=0,u'(0)=-1$ has the bounded solution -sin(t).

This is already interesting for q=2, where $-\sin(t) +1/t-\cos(t)/t$ is the bounded solution or for q=3, where $-\sin(t) - 2 \cos(t)/t +2 \sin(t)/t^2$ is the unique bounded solution. In general, for any q, we have a unique bounded solution of the driven equation which is of the form $-sin(t) + R(t)$ where $R(t) t^{q-1}$ is a $A(t) \cos(t) + B(t) \sin(t)$ with polynomials A,B. The modified acceleration not only forced the initial position to be zero, it also forces the initial velocity if one asks for a unique bounded solution.

Uniqueness of driven modified oscillator: the equation $D_{tt} u = \sin(t)$ has a unique bounded solution. The requirement of existence forces the initial position, the requirement for boundedness forces the initial velocity!

This can be generalized with a rather general driving term, but it is a phenomenon for differential equations, I have not seen so far. We deal with second order differential equation for which existence and boundedness forces a unique initial condition. By the Piccard uniqueness theorem for differential equations such a feature is only possible if the differential equations are singular at t=0 (if a second order differential equation is Lipschitz at t=0, one can give both initial position and initial velocity and has a unique solution). Coming back to the modified wave equation $(D_{tt} + D^2) f=0$ , we have unique solutions $u(t) = d_t f$ if $u_t(0)= df$ is the initial velocity. By the way, I just see that on the black board of the presentation yesterday, there is on t too much. With the definition $d_t f = t \phi_{q+2}(Dt) df$ to the left, there should be no $d_t v(p)$ and not $t d_t v(p)$.

Author: oliverknill