Integration with quad and ODE solvers

Adding up a curve: areas and moving systems

An integral is the area under a curve. integrate.quad(f, a, b) computes $\int_a^b f(x)\,dx$ for a function you write, and returns the value together with an error estimate. It is adaptive: it samples more where the function changes fast. If you only have values at points, not a function, you add up thin slices instead. The trapezoid rule, np.trapezoid, joins neighbouring points by straight lines and adds the areas of the trapezoids. Simpson's rule, integrate.simpson, uses small parabolas and is much more accurate for smooth curves. A function with a kink, such as a call payoff $\max(S-K,0)$, breaks the smoothness that quad expects, so you tell it where the kink is with points=[K].

The second tool is for systems that move. An ordinary differential equation (ODE) says how fast something changes, for example a short interest rate pulled toward a long-run level $\theta$ at speed $\kappa$: $\dfrac{dr}{dt}=\kappa(\theta-r)$. integrate.solve_ivp(f, [0, T], [r0]) follows it forward in time. The simplest hand method is the Euler step, $r_{\text{new}}=r+h\,f(r)$, which assumes the speed stays constant over a step of length $h$. In finance, integrals turn a forward-rate curve into a discount factor: $D(T)=\exp\!\big(-\int_0^T f(t)\,dt\big)$, and they turn a payoff and a probability density into an expected value.

• Function or points: use quad when you can write the function, and trapezoid or Simpson when you only have values.
• Tell quad about kinks: a payoff such as $\max(S-K,0)$ needs points=[K] for a reliable answer.
• Integrals price things: area under a forward curve gives a discount factor, and payoff times density gives an expected value.

Lab · what's graded

• A trapezoid area10 pts
• An Euler step10 pts
• The exact solution15 pts
• A discount factor15 pts
• The right integrator15 pts
• Pricing a call by integration10 pts
• Why tell quad about the kink25 pts