# What are the requirements for no arbitrage to exist in a chaotic/dynamical system?

Consider the continuous dynamical system

$$\alpha\ddot{S}+\dot{S}=\mathcal{F}(S,t),$$

such that $$\alpha\in\mathbb{R}$$ and $$\mathcal{F}$$ is real and analytic. We assume that if a solution for $$S$$ exists, then $$S$$ is deterministic. However, for a sufficiently high maximal Lyapunov exponent $$\lambda$$, when $$\frac{1}{\lambda}\ln\left(\frac{a}{\lVert\delta_0\rVert}\right)\leq\min T,$$

$$a$$ is the tolerance of prediction, and the initial parameters are uncertain to within $$\lVert\delta_0\rVert$$ from the true initial conditions, does the system permit no-arbitrage for underlying asset price $$S$$? If not, does a risk-neutral measure / no free lunch with vanishing risk exist. I would be grateful for any help or pointers to relevant literature.

• out of curiosity: where does this question comes from? Jan 3, 2022 at 17:14
• Thanks for the reply. I am wondering whether a deterministic version of option pricing could hold (since many random physical, such as Brownian motion and pilot-wave theory, are now being modelled through integro-differential equations – which have Lyapunov exponents as high as $50,60,...$). I'm also wondering how to derive a formula for the option price of this underlying, since simply using the chain rule $\dot{f}(t,S_t)=\dot{S}\frac{\partial f}{\partial S}+\frac{\partial f}{\partial t}$ doesn't seem to be sufficient when comparing to the hedge portfolio. Jan 4, 2022 at 3:40
• I do not understand why you put the second derivative w/r to time in this ODE? financial maths do use integer-differentiel equations for long. Moreover, I do not see what would reflect your dynamics for the price dynamics? Jan 4, 2022 at 5:16