Integral Calculus and Differential Equations › Summary

This chapter developed integral calculus and differential equations as tools for quantitative finance. The key concepts and their financial applications include:

Integral Calculus:

• The definite integral accumulates cash-flow rates and instantaneous continuously compounded rates; the latter produce log gross returns that must be exponentiated to obtain gross return factors
• Under the Fundamental Theorem's hypotheses, a known antiderivative gives the definite integral by endpoint subtraction
• Integration techniques (substitution, parts) appear throughout derivative pricing and risk calculations

Continuous Compounding:

• The limit of discrete compounding leads to the continuous formula $V(t) = V_0 e^{rt}$
• This formula arises as the solution to the differential equation $dV/dt = rV$
• Time-varying discounting integrates the instantaneous forward curve: $D(0,T) = e^{-\int_0^T f(0,s) ds}$. A continuously compounded zero rate instead gives $D(0,T)=e^{-R(0,T)T}$, while par yields must first be transformed into zero rates or forward rates

Differential Equations:

• First-order linear ODEs model portfolio growth, the drift or conditional mean in a mean-reverting model, and asset depreciation
• Separation of variables solves many fundamental financial models
• Logistic equations capture bounded growth in market adoption and saturation
• Tolerance-controlled numerical methods can approximate equations without closed-form solutions, subject to solver and floating-point limits

Connection to Advanced Topics:

• The Black-Scholes PDE extends ODE concepts to option pricing
• Stochastic differential equations add randomness to the deterministic models studied here
• Under stated no-arbitrage, measure, payoff, and discounting conditions, risk-neutral prices are discounted expectations

These foundations support continuous-time finance models used in quantitative practice. The next chapter develops numerical methods and algorithms for financial problems; later stochastic-process and derivatives chapters add randomness and pricing applications.