Symmetry of the Market

Markets continually change their character. Interest rates move, currencies respond to one another, and the price of protection changes even when the underlying market seems quiet. A useful model must allow this freedom while preserving the relationships that make prices financially consistent.

Symmetry offers a way to build that structure into the model from the beginning. The idea is to identify what should remain consistent as the description changes, then use those relationships to organize the uncertainty that remains. Algebraic market models pursue this approach through geometry and number systems extending beyond the real line.

The market is larger than its current prices

Imagine two markets with the same yield curve: bonds of every maturity have the same prices in both. In one, option prices reflect relatively quiet conditions ahead. In the other, they reflect substantial uncertainty about future rates and their relationships.

The yield curve alone does not describe all the uncertainty. Volatility and correlation can carry information of their own. When their state cannot be recovered from the current underlying prices, they are called unspanned. When their risks also cannot be replicated by trading the underlying instruments, hedging those instruments leaves a residual exposure.

A model should have room for this distinction. It should be possible to describe today's underlying prices and the future evolution of uncertainty without forcing one to determine the other completely.

Symmetry as a construction principle

An exchange rate is a relative value. A bond can serve as a unit in which to express other assets. Changing the reference asset changes the numbers and the corresponding pricing probabilities, while the financial relationships must remain consistent. Projective geometry gives this familiar idea a natural mathematical language: it organizes quantities through their ratios.

The symmetry is in the model's relationships across equivalent descriptions. It leaves room for unequal currencies, asymmetric risks, and a changing market.

The construction begins with families of asset-value functions compatible with a range of random motions. Harmonic functions provide the original example. Familiar from potential theory, they have a local averaging property that connects naturally with martingale pricing: suitably expressed values have no predictable local gain under the pricing measure. Full pricing consistency also requires appropriate boundary and integrability conditions.

The functions representing underlying values can retain their form while uncertainty changes. This analytical unspanning can hold even when volatility depends entirely on the existing state. Additional stochastic factors allow volatility or correlation to carry information beyond current underlying prices.

A richer language for uncertainty

Complex numbers and quaternions combine real components using rules that encode geometric relationships. A quaternion has four real components; its multiplication captures a structure that an arbitrary list of four numbers does not. These number systems supply useful ways to organize transformations and compatible patterns of random movement. John Baez's The Octonions describes the wider mathematical setting.

In the original conformal construction, a common volatility scale can vary while the asset-value functions remain fixed. Relative volatilities and correlations are more tightly constrained. The algebraic extension seeks a larger, structured family of possible covariance dynamics.

The quaternionic construction allows additional volatility and correlation components to vary without rebuilding the underlying valuation functions. Quiet short-term rates need not force long-term rates or exchange rates to become quiet as well. All observable prices remain ordinary real numbers.

The exceptional octonionic case suggests a further mathematical frontier. It requires a separate construction and remains exploratory.

Where tractability comes from

Unspanned risk supplies flexibility; analytical structure makes that flexibility manageable. Their combination is the objective.

Some harmonic constructions have explicit distributions for where a process first reaches a boundary. Such results can replace a calculation over many interior paths with a more direct calculation on that boundary. They give a concrete example of geometry doing computational work.

Richer constructions use an evolving boundary to organize the state and its permitted motion. Contact geometry describes the allowed directions along that boundary. These models lead toward structured simulation methods, with exact results in simpler cases providing useful foundations and checks. General derivative prices still depend on the chosen dynamics and may require numerical calculation.

The broader aim is to preserve useful analytical relationships as the model gains flexibility. Separating the representation of underlying values from parts of the covariance dynamics creates a route to more organized calibration, including fitting information across maturities. The practical advantage must be established for each specification.

A common structure for market realities

Interest rates, currencies, volatilities and correlations interact. A useful framework should accommodate that interaction while allowing their behaviour to change across maturities and over time. Time-dependent geometric parameters provide one way to represent term structures; additional algebraic coordinates offer a systematic way to add factors within the same structural family.

This is a programme for constructing and testing models. Its appeal lies in bringing market flexibility and mathematical discipline into the same design. Particular models must still be specified fully, checked for pricing consistency, calibrated to observations, and tested for numerical accuracy and behaviour beyond the calibration sample.

The ambition is to find structures in which realistic uncertainty has room to move, and in which the relationships needed for valuation remain accessible. Symmetry supplies the organizing principle. Algebra broadens the possibilities. Market evidence determines which constructions earn their place.

From market dynamics to contract values

Our research on financial clocks and rough volatility approaches realistic uncertainty through its timing, persistence and dependence on market conditions.

SHTAR specifies what a contract observes, retains and delivers. A valuation provider connects those terms to market assumptions and pricing methods, making the relationship between the contract and its model explicit.