The Unified Law of Quantitative Finance

After some rather extensive but necessary reflections in the previous two sections, we can now formulate the unified law of quantitative finance.

I haven't come across it, but the following formulation seems both concise and comprehensive:

Any return in financial markets results from the interaction of risk, time, and uncertainty.

Asset prices follow stochastic processes aimed at eliminating arbitrage, where the expected risk premium is proportional to volatility and temporal fluctuations.

Despite its hypothetical nature, this law is the most universal.

Quantitative finance, asserting itself as a scientific discipline with important practical relevance, necessarily relies on mathematical and statistical methods to analyze the behavior of financial assets, model market processes, and develop practical tools using financial engineering approaches.

The most frequently used tools today for solving such problems include stochastic processes, computational risk assessment algorithms (VaR, CVaR), and—considering current trends—machine learning techniques capable of rapidly analyzing large volumes of data to uncover non-obvious patterns.

However, despite the large number of models available, none of them is entirely universal. Each market and each category of assets has its own specific characteristics. This naturally leads to the desire to find a unifying structure or law that describes the common features of financial markets, regardless of their nature. It is from this idealistic pursuit that the concept of a unified law emerged.

Core Postulates of the Unified Law of Quantitative Finance

The foundation of the unified law of quantitative finance can be based on the idea that all market processes follow fundamental principles, derived from the following postulates/assumptions:

• Stochastic Principle: All financial assets are subject to random influences, which can be described by stochastic processes. This means the price of any asset at any moment is a random variable influenced by both internal and external factors.

• Arbitrage Principle: In the long run, arbitrage opportunities in the market tend to zero. Short-term arbitrage, which corrects imbalances, is one of the key mechanisms maintaining market stability.

• Risk-Reward Principle: Any opportunity for profit comes with a certain level of risk. The higher the potential return, the greater the risk to the investor. This balance can be formalized using concepts like risk premium or the beta coefficient (as in the CAPM model).

• Temporal Variability Principle: All processes on financial markets change over time. Changes in volatility, interest rates, spreads, and other market parameters can be modeled using time series and dynamic models.

Mathematical Formulation

To give the unified law of quantitative finance a strict mathematical form, the following components are essential:

• Random Walk Model: The market price of an asset can be described by a stochastic differential equation (SDE):

dS (t) = μS (t)dt + σS (t) dW (t)

Where: μ is the mean return, σ is the volatility, W (t) is a Wiener process (standard Brownian motion or random walk).

• No-Arbitrage Principle: Formally, this principle can be expressed as:

E [S (T)] = S (0) ⋅ e^rT

Where: E [S (T)]  is the expected price of the asset at future time T, r is the risk-free interest rate.

• Risk-Return Relationship: Based on CAPM, expected return on an asset can be expressed as:

E [R_i] = R_f + β_i (E [R_m] − R_f )

where:
E[R_i] is the expected return of the asset;
R_f is the risk-free rate;
E[Rm] is the expected market return;
βi is the beta coefficient, which characterizes the sensitivity of the asset to market risk

This formula is used to evaluate the return of an asset, taking into account its risks compared to the market.

Interaction with Real Markets

Applying the unified law of quantitative finance in real-world conditions requires accounting for various characteristics of market systems:

1. Information Asymmetry: In real financial systems, information is distributed unevenly. Market participants with better information have advantages. The unified law must consider this through adjustments in pricing models.
2. Market Microstructure: Real markets consist of many participants with different strategies, liquidity needs, and preferences. This affects pricing dynamics, volatility, and spreads. The unified law must consider such interactions.
3. Regulatory Constraints: Laws and regulations impact arbitrage, liquidity, and volatility. Regulations often aim to maintain systemic stability, and this must be reflected in the unified law.

Possible Applications

If the unified law of quantitative finance can be strictly defined, it opens many practical applications:

1. Algorithmic Trading: Automated trading systems could use the law to build strategies based on forecasts of asset prices, volatility, and key parameters.
2. Risk Assessment: Companies could build more accurate risk models, enhancing credit, market, and operational risk management.
3. Financial Derivatives: The unified law could serve as the foundation for fair pricing and scenario analysis of derivatives like options and futures.
4. Hedging Models: Market participants could develop more effective hedging strategies that minimize loss risk while maintaining performance.

Conclusion

The unified law of quantitative finance is a concept aimed at bringing together all aspects of financial markets into a single coherent mathematical model. Based on the principles of statistically, absence of arbitrage, and the balance of risk and return, it could become a powerful tool for solving a wide range of practical problems.

However, its realization requires further research and careful consideration of the unique features of different markets and the potential deviations from theoretical assumptions.

Thus, the future of financial research may lie in the development of this universal law, which could underpin new approaches to asset management, risk assessment, and the construction of sophisticated financial strategies.