Lesson 1.1: The Limitations of Classical Calculus for Stochastic Processes

The concept of a 'smooth' function is mathematically synonymous with differentiability. Consider, for example, the orbital path of a planet or the deterministic growth of a fixed-interest-bearing account. These paths are continuous and, crucially, differentiable.

Differentiability implies that upon sufficient magnification (i.e., 'zooming in') at any given point, the function's curve locally converges to a straight line—its tangent.

This property of local linearity is the fundamental basis of classical calculus.

It is this 'straightening out' that permits the definition of a single, unambiguous 'slope' at that point. We formally call this 'slope' the derivative ($df/dt$).

The entire analytical toolkit from Module 0, including derivatives, integrals, and Taylor expansions, is predicated on the existence of this well-defined limit.

Herein lies the central challenge. An empirical examination of a financial asset price, such as a stock chart, reveals a path that is not smooth. It is, by contrast, jagged, chaotic, and highly irregular.

This irregularity is not merely superficial 'noise' on an otherwise well-behaved function; it represents a fundamentally different class of mathematical path. The source of this stochastic (or 'wiggly') behavior is persistent, high-frequency randomness, driven by factors such as stochastic news arrivals, algorithmic trading activity, and the aggregation of millions of independent market decisions.

This leads to the critical observation that invalidates classical calculus:

In a classical differentiable function (like $f(x) = x^2$), progressive magnification causes the curve to appear flatter and flatter, ultimately approximating a straight line.

On a stock chart, a contrary property is observed.

• A 1-year chart displays a high degree of irregularity.
• If one 'zooms in' to a single day, the 1-day chart displays a comparable level of jaggedness and unpredictability.
• A further magnification to a 1-minute chart still reveals a similar chaotic structure.

This property, where the statistical characteristics of the path appear invariant to the scale of observation, is known as self-similarity. It is the visual signature of a mathematical object called a fractal.

A random path of this nature never converges to a local straight line, regardless of the level of magnification.

This self-similarity poses an insurmountable problem for classical calculus. Consider the formal attempt to find the derivative (the 'slope') of a stock price $S$ at a precise instant, $t$. The derivative is defined as the limit of the difference quotient as the time interval $\Delta t$ approaches zero:

In a 'smooth' world, this ratio converges to a single, finite number. However, for a self-similar random path, this convergence fails. As the time interval $\Delta t$ is reduced (from 1 minute, to 1 second, to 1 millisecond), the observed volatility of the price $S$ does not decrease proportionally. The ratio $\frac{\Delta S}{\Delta t}$ does not settle on a single value; rather, it continues to fluctuate wildly and, in fact, diverges.

The mathematical conclusion is stark: the limit does not exist.

• A derivative exists if and only if the path exhibits local linearity (i.e., 'straightens out').
• A random financial path does not exhibit local linearity.
• Therefore, a random path possesses no derivative at any single point.

If no derivative $df/dt$ exists, the entire classical calculus toolbox—including the standard chain rule and Taylor expansions—is invalid for analyzing such processes. We are thus unable to model the 'Delta' or 'Gamma' of an option, as these are defined as derivatives. We are at an impasse.