Fixed-Point Iteration

Many equations that arise in pricing or calibration — a bond's yield to maturity given its price, an implied volatility, a nonlinear model parameter — have no formula you can solve for algebraically. Fixed-point iteration is one of the simplest ways to home in on the answer numerically: rearrange the equation into the form $x = g(x)$, pick a starting guess, and repeatedly feed the output of $g$ back in as the next guess, hoping the sequence settles down.

Concretely, given $x = g(x)$, start with an initial guess $x_0$ and compute $x_1 = g(x_0)$, then $x_2 = g(x_1)$, and so on. If the sequence converges — that is, if $x_n$ stops changing meaningfully as $n$ grows — the value it settles on, called a fixed point, satisfies $x = g(x)$ by construction, which is exactly the equation you were trying to solve.

As an example, solving $x = \cos(x)$ (in radians) by iteration: starting from $x_0 = 1$, $x_1 = \cos(1) \approx 0.540$, $x_2 = \cos(0.540) \approx 0.858$, $x_3 = \cos(0.858) \approx 0.653$, and continuing this bouncing pattern converges after a couple dozen steps to approximately $x \approx 0.739$ — the equation's unique solution, reached with nothing more than repeated evaluation of cosine.

Whether the iteration converges at all, and how fast, depends on how "steep" $g$ is near the fixed point — a rearrangement where $g$'s slope has magnitude below 1 near the answer will converge, often slowly, while a poorly chosen rearrangement of the same underlying equation can diverge entirely, which is why picking the right form of $g(x)$ (not just any algebraic rearrangement) matters in practice, and why faster alternatives like Newton-Raphson are often preferred when derivatives are available.