Stochastic calculus can look like a seemingly complex topic in finance, particularly for those delving into option pricing and other financial models that involve randomness. Mastering this field requires a strong foundation and the right resources.
Here are three tips to help you deepen your understanding of stochastic calculus, complete with recommendations for prerequisites and resources.
1. Build a Strong Foundation in Calculus
Before diving into stochastic calculus, it's essential to understand basic calculus concepts in a deterministic, non-stochastic context. Focus on grasping the following key ideas:
Limits: Understanding how functions behave as they approach a specific point.
Derivatives: Comprehending the rate at which a function changes at any given point.
Integrals: Learning to calculate the area under a curve, which is crucial for understanding accumulation and area.
A great starting point for these topics is Khan Academy, which offers comprehensive and accessible lessons on calculus.
2. Understand the Differences Between Deterministic and Stochastic Spaces
A crucial part of mastering stochastic calculus is understanding how computation changes when moving from deterministic to stochastic spaces. This involves:
Recognizing how randomness influences derivative and integral calculations.
Exploring when a function is continuous and differentiable, and how these properties change in a stochastic environment.
Deterministic Calculus
In deterministic calculus, Taylor's expansion is typically used to approximate functions as:
Thus, the differential df(x) of some deterministic, continuous and differentiable function f(x) becomes:
And, in the limit as x approaches a, the (x-a) term tends to zero, hence we neglect all terms after the first order term, so that:
Stochastic Calculus
In stochastic calculus, when we write the expansion of the differential of some function of a stochastic process, we have:
However, in stochastic calculus, the second-order term (dXₜ)² is crucial because (dXₜ)² ≈ dt when Xₜ is a stochastic process. This is due to the property of Brownian motion increments, where (dWₜ )² = dt.
And why is that? When Xₜ is a stochastic process driven by Brownian motion, (dXₜ)² ≈ dt because the square of the infinitesimal change in Brownian motion (dWₜ )² captures the variance of the process and is proportional to the small time increment dt.
All terms after the second-order term are considered negligible, so that (dWₜ )³ ≈ 0, as well as all following terms. Thus, the differential df(Xₜ) of some function f(Xₜ) of a stochastic process Xₜ reduces to:
3. Explore Stochastic Concepts with Video Resources
Once you have a handle on basic calculus, delve into the stochastic aspects. John H. Cochrane’s videos on YouTube are particularly helpful. They offer clear explanations of complex concepts in stochastic calculus, making them more approachable.
There is also an incredible “Topics in Mathematics with Applications in Finance” class available online on MIT OpenCourseWare whose quality blew my mind.