Stochastic Calculus Basics for Future FSAs

From Deterministic to Stochastic

Stochastic calculus extends ordinary calculus to handle functions driven by random processes. The central object is the Ito integral, which defines integration with respect to Brownian motion. Unlike ordinary calculus, the chain rule changes: Ito's lemma includes an extra second-order term that accounts for the quadratic variation of Brownian motion. This seemingly small difference has profound implications for pricing financial instruments, modeling interest rates, and valuing insurance products with embedded options.

Applications in Actuarial Science

FSA candidates encounter stochastic calculus in the Investment and Financial Markets (IFM) content and in quantitative finance modules. Stochastic differential equations (SDEs) model stock prices (geometric Brownian motion), interest rates (Vasicek, CIR models), and mortality intensity (stochastic mortality models). The Girsanov theorem enables switching between real-world and risk-neutral probability measures, a technique essential for pricing variable annuity guarantees and other insurance products with market-linked payoffs. Building intuition for these tools requires working through many examples and understanding the financial interpretation of each mathematical step.