Theoretical Research
DOMAIN: QUANTITATIVE FINANCE FRAMEWORK: CONTINUOUS-TIME STATUS: EMPIRICALLY VERIFIED
01. Continuous-Time Asset Pricing Paradigm
We systematically reject the Markovian assumption inherent in the standard Black-Scholes manifold. The intuition here is absolute: independent increments assume the market possesses no memory of its own structural formation. The incumbent geometric Brownian motion model relies on a standard Wiener process:
\[ dS_t = \mu S_t dt + \sigma S_t dW_t \]
This is theoretically inferior because limit order book microstructure is fundamentally driven by auto-correlated order flow. To capture this infinite-dimensional memory, we must discard the standard physical probability measure and map the asset trajectory onto a fractional geometry, where the past rigorously bounds the future.
02. The Rough Volatility Framework
To model the persistent memory and the explosive collapse of liquidity at the microsecond level, standard stochastic volatility models are mathematically insufficient. We deploy a fractional Brownian motion framework, parameterized strictly by the Hurst exponent. The intuition is that when the Hurst exponent is less than one-half, the variance process becomes rough, perfectly modeling the anti-persistent, rapidly mean-reverting nature of high-frequency liquidity vacuums.
We define the rough volatility process analytically via the Riemann-Liouville fractional integral, abandoning heuristic smoothing entirely:
\[ v_t = v_0 + \frac{1}{\Gamma(H + \frac{1}{2})} \int_0^t (t-s)^{H - \frac{1}{2}} \lambda(v_s) dW_s \]
This formalization provides an exact analytical proof for the power-law explosion of the implied volatility skew as maturity approaches zero, resolving the theoretical failure of incumbent diffusion models.
03. Fractional Martingale Pricing Boundary
Under the fundamental theorem of asset pricing, the discounted asset price must exist as a local martingale under the risk-neutral measure to prevent theoretical arbitrage. The intuition here presents a profound mathematical barrier: fractional Brownian motion is not a semi-martingale, meaning standard Itô calculus collapses.
To enforce the martingale boundary, we do not approximate. We isolate the deterministic drift component by deploying a Wick-Itô-Skorohod integral, mapping the fractional noise onto an orthogonal space. This yields the exact fractional Black-Scholes partial differential equation:
\[ \partial_t V(t,S) + r S \partial_S V(t,S) + H t^{2H-1} \sigma^2 S^2 \partial_{SS} V(t,S) - rV(t,S) = 0 \]
04. The Atanasovski Fractional Skew Measure
Standard probability density functions fail fundamentally because they are symmetric or rely on finite variance. In sub-microsecond latency environments, the limit order book exhausts asymmetrically, characterized by infinite-variance jumps and heavy-tailed kurtosis.
To solve this, my mentor and I formulated the Atanasovski Fractional Skew Measure. The intuition behind this measure is that it physically models the exact continuous-time collapse of the order book as a space-fractional diffusion process. It is a dynamically shifting topological distribution, parameterized by fractional decay and directional execution skew.
The measure-theoretic density is analytically proven and defined by its characteristic integral over a Sobolev space:
\[ \mathcal{A}(x; H, \kappa, \beta) = \frac{1}{\pi} \int_0^\infty e^{-\kappa \xi^{2H}} \cos\left( x\xi - \beta \kappa \xi^{2H} \tan\left(\frac{\pi H}{2}\right) \right) d\xi \]
In this derivation, the Hurst parameter rigorously controls the memory of the liquidity flow, while the skew operator controls the directional severity of aggressive market orders.
05. Restricted Theoretical Repositories
The underlying proofs for the measure-theoretic convergence of the Atanasovski Measure are currently isolated within our internal mathematical architecture.