Mathematical foundations of financial market modeling
Financial Markets
A financial market is an institutional framework in which financial instruments are issued and exchanged. In the primary market, newly created securities such as shares or bonds are sold for the first time by firms or governments to investors. The objective of this article is to introduce the economic role of financial markets and to present the fundamental structural properties required for coherent financial modeling.
For theoretical clarity, we adopt a discrete-time framework. Time is indexed by \[ t \in \{0,1,\dots,T\}, \] where \(T\) denotes a fixed time horizon.
Financial markets and traded assets
Financial markets encompass several types of instruments. The equity market consists of ownership claims on firms, such as common or preferred shares. Firms raise capital by issuing equity securities, and these securities are subsequently traded on organized exchanges such as Euronext Paris or the New York Stock Exchange. Prices are determined by supply and demand and reflect expectations regarding firm performance and economic conditions.
The bond market consists of debt instruments issued by governments or corporations. A bond is a contract specifying that the issuer will repay a principal amount at maturity and pay periodic interest, referred to as coupons. Bonds provide predictable income streams and are fundamental instruments for public and private financing.
The derivatives market includes financial instruments whose value depends on an underlying asset. Typical examples are options, forwards, futures, and swaps. Derivatives are used for risk management, speculative positioning, and arbitrage strategies across markets.
The primary market enables capital allocation within the economy. Firms obtain financing for investment and expansion, while investors allocate resources in expectation of future returns. Financial markets also facilitate risk transfer. For example, firms exposed to commodity price risk may use derivative contracts to hedge adverse fluctuations.
Structural properties of a financial market
For a financial market to function efficiently, several properties are desirable. Liquidity ensures that assets can be traded without significant price impact. Transparency guarantees that relevant information is accessible to participants. Fairness implies equal access to market information. Efficiency requires that prices reflect available information and that arbitrage opportunities do not persist.
The absence of arbitrage is a central assumption in modern financial mathematics and underlies pricing theory.
Mathematical Framework
We consider a financial market composed of one risk-free asset and \(q\) risky assets.
Let \(S_t^0\) denote the price of the risk-free asset. In discrete time, we assume it evolves deterministically according to \[ S_{t+1}^0 = (1 + r_t) S_t^0, \] where \(r_t\) denotes the short-term interest rate.
Let \(S_t^i\) for \(i=1,\dots,q\) denote the prices of risky assets. Their evolution is stochastic and will be specified through an appropriate probabilistic model.
Trading strategies and wealth process
A trading strategy is a predictable process \[ \delta_t = (\delta_t^0, \delta_t^1, \dots, \delta_t^q), \] where \(\delta_t^i\) represents the number of units of asset \(i\) held at time \(t\).
The associated wealth process is defined by \[ V_t = \sum_{i=0}^q \delta_t^i S_t^i. \]
A strategy is said to be self-financing if changes in wealth are solely due to variations in asset prices. In discrete time, this condition reads \[ V_{t+1} - V_t = \delta_t^0 \, dS_t^0 + \sum_{i=1}^q \delta_t^i (S_{t+1}^i - S_t^i). \] Equivalently, \[ V_t = V_0 + \delta_t^0 \, dS_t^0 + \sum_{s=1}^{t-1} \sum_{i=0}^q \delta_s^i (S_{s+1}^i - S_s^i). \]
In continuous time, the self-financing condition is expressed as \[ \begin{align} dV_t &= \delta_t^0 \, dS_t^0 + \sum_{i=1}^q \delta_t^i \, dS_t^i. \\ \Leftrightarrow V_t &= V_0 + \delta_t^0 \, dS_t^0 + \int_1^t \sum_{i=0}^q \delta_s^i \, dS_s^i. \end{align} \]
In the case where the risky assets give dividends, the self-financing condition must be modified to account for dividend payments. Let \(D_t^i\) denote the cumulative dividends paid by asset \(i\) up to time \(t\). The wealth process then satisfies : \[ dV_t = \delta_t^0 \, dS_t^0 + \sum_{i=1}^q \delta_t^i \, dS_t^i + \sum_{i=1}^q \delta_t^i \, dD_t^i. \]
One can talk in term of discounted wealth process \(\tilde V_t = \frac{V_t}{S_t^0}\), which satisfies \[ d\tilde V_t = \sum_{i=1}^q \delta_t^i \, d\tilde S_t^i, \] where \(\tilde S_t^i = \frac{S_t^i}{S_t^0} = \beta_t S_t^i\) is the discounted price of the risky asset \(i\).
Arbitrage
An arbitrage opportunity is defined as a self-financing strategy satisfying \[ V_0 = 0, \qquad V_T \ge 0 \text{ almost surely}, \qquad \mathbb{P}(V_T > 0) > 0. \] Such a strategy generates profit without risk or initial investment. Markets admitting arbitrage are economically inconsistent, and therefore we assume the market is arbitrage-free.
Risk-neutral measure
Under the absence of arbitrage, there exists an equivalent probability measure \(\mathcal{Q}\) under which discounted asset prices are martingales. Let \[ \tilde S_t^i = \frac{S_t^i}{S_t^0} \] denote the discounted price process. Then \[ \mathbb{E}^{\mathcal{Q}}[\tilde S_{t+1}^i \mid \mathcal{F}_t] = \tilde S_t^i. \] This result allows derivative pricing via expectation: \[ \Pi_0 = S_0^0 \, \mathbb{E}^{\mathcal{Q}}\left[\frac{\varepsilon}{S_T^0}\right], \] where \(\varepsilon\) denotes the contingent claim payoff at maturity.
Market completeness
The market is said to be complete if every contingent claim \(\varepsilon\) can be replicated by a self-financing strategy. In a complete market, the risk-neutral measure is unique and every derivative admits a unique arbitrage-free price. In an incomplete market, multiple risk-neutral measures may exist and perfect replication is not always possible, leading to pricing and hedging challenges.
As an illustration, the payoff of a European call option with strike \(K\) and maturity \(T\) is given by \[ \varepsilon = \max(S_T - K, 0). \]
Financial markets play a fundamental role in capital allocation, risk transfer, and price discovery within the economy. From a mathematical perspective, their analysis relies on a structured framework built upon trading strategies, wealth dynamics, and probabilistic modeling.
The absence of arbitrage constitutes the cornerstone of modern financial theory. It guarantees the existence of a risk-neutral measure under which discounted asset prices behave as martingales, thereby providing a coherent foundation for derivative pricing.
Finally, the notion of market completeness determines whether contingent claims can be perfectly replicated and uniquely priced. These structural concepts form the basis of quantitative finance and underpin the development of more advanced pricing and hedging models.