A Price Lattice Rebuilds Pricing Theory Without Hidden Variables

If option prices have always seemed to depend on invisible market states, this paper argues for a cleaner rule: use only observable price jumps. Instead of hidden diffusions or latent factors, it models prices on a discrete price lattice and lets shift operators track transitions from one price level to the next. That observable shift algebra is then used to build a completely positive, translation-covariant Lindblad semigroup, a mathematically structured way to evolve prices over time. Under the risk-neutral condition, the framework gives a nonlocal pricing equation that is diagonal in Fourier space, and in the small-mesh diffusive limit it collapses back to the classic Black-Scholes-Merton operator. The author says this is not a new parametric model, but a foundation for building models from measured transitions. The paper also claims the jump-intensity ledger should shape tail behavior and short-maturity smiles, suggesting testable links between extreme-event probabilities and implied-volatility wings.

Picture an option desk that refuses to guess hidden market moods. It watches only price moves. This model says that is enough to build pricing rules. The setup places prices on a ladder of levels, called a price lattice. A lattice is just a grid of allowed price steps. On that grid, every jump becomes a visible move, not a hidden cause. That shift matters because finance often leans on unseen state variables. Here, the surprise is that visible moves already carry the needed structure. The model does not start with a secret market story. It starts with the trail left by prices themselves. You can see the logic before any theory enters.

A ladder of prices, not a cloud of guesses

At the center sits a price operator. It tags each level with its price. The model also uses unit shifts. They move one step up or down the ladder. A frequency map, or score rule, turns prices into a changing score. That score creates a return ledger. The ledger records gaps between price levels. Jump operators, or move-rate rules, add move rates between states. Together, they make a time rule that keeps probabilities valid. It also treats every price level the same. Under the risk-neutral condition, where gains match the market rate, the model gives a nonlocal pricing equation. In plain terms, the next price depends on more than the nearest step. In wave space, which breaks a signal into waves, the equation becomes simple. When the price steps get very small, the classic Black-Scholes-Merton rule returns. That rule is the standard way to price options.

How shifts do the heavy lifting

The key move is simple. The model drops hidden diffusions, or smooth random drifts, and hidden state variables. It keeps only observable price transitions. Each move on the ladder acts like a shift operator. A shift operator is a rule that slides one price level to the next. Those shifts do not always swap cleanly. That mismatch creates noncommutativity, meaning order matters. The odd quantum-style structure comes from the visible shift rules themselves. It does not get pasted on from outside. The jump part builds a Lindblad semigroup, a time rule that keeps probabilities valid. That gives the framework a clean math spine.

Why this matters

This matters because it changes what counts as input. The model asks for observed jumps, not hidden moods. That makes the setup feel less like guesswork and more like accounting. The jump-intensity ledger, meaning the rates of visible moves, may shape two things. It may shape the far tail of price moves. It may also shape the edges of the market's volatility curve for near-dated options. That gives traders and model builders a new link to test. They can look for a direct path from rare jumps to option curves. The framework is not a ready-made model. It is a base for building one.

What comes next

The next test is direct. Do the jump rates line up with rare moves in real markets? This link should reach the far tail and the edges of the volatility curve. If it holds, model building gets a new starting point. Builders would not need hidden state stories first. They would begin with measured price moves. They could then grow to many assets, or add flow-based rules, inside the same shift rules. That is the real twist here. Observable steps, not secret states, may become the first brick in pricing theory.