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The magic of arbitrage: How to calculate derivative prices using the no-arbitrage principle?
In financial economics, asset pricing is a crucial concept that encompasses the formal treatment and development of various market pricing principles. These pricing principles typically exist in two interrelated forms: general equilibrium asset pricing and rational pricing. The common challenge faced by these theories and models is how to accurately assess the value of derivatives in a market environment where demand and supply are constantly changing.
Asset pricing models are not only used to determine the required rate of return on a specific asset, but are also a powerful tool for hedging risks.
General Equilibrium Asset Pricing
According to general equilibrium theory, asset prices are determined by demand and supply in the market. These models are derived from Modern Portfolio Theory, of which the Capital Asset Pricing Model (CAPM) is considered as the prototype. Asset prices meet the requirement in this process that the supply and demand for each asset must be equal at that price, achieving what is known as market clearing.
In practice, calculating the value of an investment or share requires the following steps: first, making a financial forecast for the relevant business or project; second, discounting the expected cash flows according to the rate of return reflected in the selected model; and finally , these present values are aggregated to obtain the final value.
General equilibrium pricing is widely used to evaluate diverse investment portfolios and can create a unified price for many assets.
Rational pricing
Compared with general equilibrium theory, rational pricing methods focus more on the calculation of derivatives so that they remain arbitrage-free with the underlying equilibrium asset price. In this approach, the risk price is unique for each asset, making these models generally “low dimensional”.
When calculating the price of a derivative, it is necessary to consider a model based on the asset price behavior, i.e. the chosen asset pricing model, and calibrate its parameters to the observed prices. This approach relates the expected value of cash flows to possible payments over different price ranges. Although classic pricing models such as the Black-Scholes model assume that the return process is logarithmic, there are still many other models that incorporate factors such as mean reversion and volatility.
Rational pricing also applies to fixed-income instruments, such as bonds, where it is necessary to ensure that the yield curve is arbitrage-free for the prices of various individual instruments.
Interactions between principles
These asset pricing principles are related to each other through the fundamental asset pricing theorem. In the absence of arbitrage, the market imposes a probability distribution over possible market scenarios, called a risk-neutral or equilibrium measure, and this probability measure determines the market price by discounting its expected value.
These approaches can be considered as a means to support financial decision making, especially in the dynamic environment of capital markets. We can see that whether it is general equilibrium or rational pricing, the logic behind it points to a reasonable price formation mechanism. However, the basis for constructing these models often covers complex economic variables and market behaviors.
All of this points to a core question: In the complex and diverse financial markets, can we find a universally applicable pricing model to predict future investment directions?
