MachineLearning – Order & Chaos

Zeta-Mobius Arithmetic Volatility: An Option-Pricing Extension

Classical option pricing usually begins with a diffusion model for the underlying asset price:

$\displaystyle dS_t = S_t\mu_t\,dt + S_t\sigma_t\,dW_t$

The central modeling question is not only the drift or the payoff. It is the volatility input. In the classical Black-Scholes model, volatility is constant or estimated from historical data. In stochastic-volatility and rough-volatility models, volatility becomes its own random process. The extension developed here introduces a different state variable: arithmetic residual energy generated by a zeta-Mobius transform.

The purpose is not to replace arbitrage-free pricing. The purpose is to replace the volatility estimate inserted into a standard pricing map.

1. The Source Construction

For ${\rm Re}(s) > 1$ , the Riemann zeta function is:

$\displaystyle \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$

Euler’s product formula expresses the same object through the primes:

$\displaystyle \zeta(s)=\prod_{p}\frac{1}{1-p^{-s}}$

Taking the reciprocal gives:

$\displaystyle \frac{1}{\zeta(s)}=\prod_p(1-p^{-s})$

Expanding this product produces the Mobius function:

$\displaystyle \frac{1}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}$

The Mobius function takes three possible values:

$\mu(1)=1$ .
$\mu(n)=(-1)^k$ when $n$ is the product of $k$ distinct primes.
$\mu(n)=0$ when $n$ is divisible by the square of a prime.

This is the mathematical source of the model. The sequence $\mu(n)$ is deterministic, but it behaves like a structured cancellation sequence. It alternates across square-free prime products and deletes terms containing repeated prime factors. In financial language, it becomes a finite arithmetic sieve over lagged market observations.

2. The Causal Zeta-Mobius Transform

Let $X_t$ be a market signal observed at time $t$ . This signal could be returns, volume, realized variance, spread, order-flow imbalance, funding rate, or implied-volatility skew.

For truncation depth $N$ and exponent $s > 0$ , define the finite transform:

$\displaystyle Z_s[X](t)=\sum_{n=1}^{N}\frac{\mu(n)}{n^s}X(t-n)$

A normalized version can also be used:

$\displaystyle \widetilde{w}_n^{(N,s)}=\frac{\mu(n)n^{-s}}{\sum_{k=1}^{N}|\mu(k)|k^{-s}}$

$\displaystyle Z_t^{(N,s)}[X]=\sum_{n=1}^{N}\widetilde{w}_n^{(N,s)}X_{t-n}$

This version is causal because it uses only observations available before time $t$ . The normalization keeps the scale comparable when $N$ or $s$ changes.

3. The Arithmetic-Sieve Residual Theorem

Theorem. Let $X_t$ be a locally square-integrable discrete financial signal. The causal zeta-Mobius transform $Z_t^{(N,s)}[X]$ is a finite arithmetic-sieve residual of $X_t$ . It extracts the part of the recent path that survives Mobius cancellation.

The interpretation is simple. Ordinary local noise should partly cancel under the arithmetic sieve. Structured crowding, liquidity stress, volatility clustering, hidden periodicity, or adversarial bursts may survive as residual energy.

This leads to the residual-energy statistic:

$\displaystyle E_Z(t)=\sum_j\left|Z_{s_j}[X](t)\right|^2$

In a multi-channel setting, this becomes:

$\displaystyle E_Z(t)=\sum_{\ell=1}^{d}\sum_{N}\sum_s\omega_{\ell,N,s}\left(Z_t^{(N,s)}[X^{(\ell)}]\right)^2$

This quantity is not merely realized volatility. It is arithmetic residual energy: the energy left over after a prime-factorization-driven cancellation filter has acted on the market signal.

4. From Residual Energy to Volatility

The simplest arithmetic-volatility specification is:

$\displaystyle \sigma_t^2=\sigma_0^2+\lambda E_Z(t)$

where $\lambda$ controls how strongly zeta-Mobius residual energy affects variance.

For empirical option pricing, it is often more stable to start with a base volatility estimate and apply the arithmetic adjustment multiplicatively:

$\displaystyle \sigma_{arith}(t)=\sigma_{base}(t)\sqrt{1+\lambda E_Z^+(t)}$

Here $E_Z^+(t)$ is a positive transformation of residual energy. It may be a positive z-score, a softplus transformation, or another positive link function.

5. Proposition: Positivity and Modularity

Proposition. If $\sigma_{base}(t)>0$ , $E_Z^+(t)\geq 0$ , and $\lambda\geq0$ , then

$\displaystyle \sigma_{arith}(t)=\sigma_{base}(t)\sqrt{1+\lambda E_Z^+(t)}$

is a positive volatility estimator. It can be inserted directly into a standard option-pricing equation without changing the payoff function.

Positivity: the square-root form preserves nonnegative volatility.
Modularity: the zeta-Mobius term changes the volatility input, not the option payoff.
Interpretability: the volatility premium is driven by structured arithmetic residual energy.

6. Black-Scholes with Arithmetic Volatility

The Black-Scholes call price is:

$\displaystyle C_{BS}(S_t,K,\tau,r,\sigma)=S_tN(d_1)-Ke^{-r\tau}N(d_2)$

with

$\displaystyle d_1=\frac{\log(S_t/K)+(r+\frac{1}{2}\sigma^2)\tau}{\sigma\sqrt{\tau}}$

$\displaystyle d_2=d_1-\sigma\sqrt{\tau}$

The zeta-Mobius extension keeps this pricing map but replaces $\sigma$ with $\sigma_{arith}(t)$ :

$\displaystyle C_{ZM}(S_t,K,\tau,r)=C_{BS}(S_t,K,\tau,r,\sigma_{arith}(t))$

The interpretation is that zeta-Mobius residual energy supplies a volatility state variable. It does not replace no-arbitrage pricing. It modifies the volatility estimate that enters the no-arbitrage pricing equation.

7. Calibrated Arithmetic Volatility

A more practical version estimates the volatility map from data:

$\displaystyle \log(\widehat{\sigma}_{t,\tau})=\alpha_{\tau}+\beta_{\tau}\log(\sigma_{base}(t))+\gamma_{\tau}z_{ZM}(t)$

where $z_{ZM}(t)$ is a standardized zeta-Mobius residual feature. This form allows the data to determine whether arithmetic disorder increases or decreases expected future volatility at a given horizon.

The corresponding option price is:

$\displaystyle C_{ZM}=C_{BS}(S_t,K,\tau,r,\widehat{\sigma}_{t,\tau})$

8. Interpretation

The economic interpretation is that market volatility is not only Brownian, stochastic, or rough. It may also contain arithmetic structure produced by discrete market activity. Markets are made of clustered trades, orders, cancellations, liquidity shocks, and heterogeneous horizons. The zeta-Mobius transform is designed to test whether some part of that structure survives arithmetic cancellation.

Brownian volatility captures continuous random fluctuation.
Rough volatility captures persistent irregularity and memory.
Zeta-Mobius arithmetic volatility captures structured residual disorder after Mobius cancellation.

The model is therefore best viewed as an additional volatility-state variable. It can be used alongside realized volatility, implied volatility, EWMA volatility, GARCH, HAR-RV, rough volatility, or other volatility estimators.

9. Practical Pricing Hypothesis

The practical hypothesis is:

Option prices improve when the volatility input includes arithmetic residual energy that anticipates future realized variance better than rolling volatility alone.

This gives a concrete testing procedure:

Compute a market signal $X_t$ , such as returns or realized variance.
Apply the zeta-Mobius transform across several values of $N$ and $s$ .
Construct $E_Z(t)$ or a standardized feature $z_{ZM}(t)$ .
Estimate $\sigma_{arith}(t)$ or $\widehat{\sigma}_{t,\tau}$ .
Insert the resulting volatility into Black-Scholes or another pricing map.
Compare out-of-sample volatility and option-price error against rolling volatility, EWMA, GARCH, HAR-RV, and implied-volatility baselines.

10. Summary

The zeta-Mobius option-pricing extension can be summarized in three equations:

$\displaystyle Z_s[X](t)=\sum_{n=1}^{N}\frac{\mu(n)}{n^s}X(t-n)$

$\displaystyle E_Z(t)=\sum_j\left|Z_{s_j}[X](t)\right|^2$

$\displaystyle \sigma_{arith}(t)=\sigma_{base}(t)\sqrt{1+\lambda E_Z^+(t)}$

In calibrated form, the volatility model becomes:

$\displaystyle \log(\widehat{\sigma}_{t,\tau})=\alpha_{\tau}+\beta_{\tau}\log(\sigma_{base}(t))+\gamma_{\tau}z_{ZM}(t)$

The broader point is that financial markets are discrete, clustered, adversarial, and multi-scale. The Mobius function provides a deterministic arithmetic sieve for extracting residual structure from such systems. In option pricing, that residual structure becomes a candidate volatility state variable.

This is the central idea of zeta-Mobius arithmetic volatility.