Integral Calculus and Differential Equations › Integral Calculus and Differential Equations
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A deposit schedule tells you how quickly money enters an account, but how much arrives over five years? A changing discount curve creates the reverse problem: what is a future cash flow worth today? Integrals turn specified rates into accumulated amounts, while differential equations describe how balances evolve under those rates.
These tools support many continuous-time finance models. For a time-varying instantaneous continuously compounded rate, integrating the rate and exponentiating gives the gross accumulation factor. Differential equations describe models in which asset prices drift and diffuse over time. The Black-Scholes framework, developed later in the book, includes a partial differential equation for an option's model value.
This chapter develops integral calculus and differential equations through financial applications. You will compute accumulated cash flows and continuously compounded growth factors, then model continuous growth, decay, and mean reversion.