Root finding
Used constantly to back out implied volatility from an option price.
Bisection - always converges if the root is bracketed, but slowly (linear). Robust.
Newton-Raphson - quadratic convergence, so it roughly doubles the correct digits each step. But it needs the derivative and a decent starting point, and it can diverge.
For implied vol the derivative is vega, which is available analytically, so Newton is the standard choice - usually with a bisection fallback for robustness. Being able to say why you would combine them is a strong answer.
Monte Carlo
Simulate many paths, average the payoffs.
The key fact: standard error falls like 1/sqrt(N). Four times the samples for half the error - the same square-root law as everywhere else.
Crucially, this rate is independent of dimension, which is why Monte Carlo beats grid methods for high-dimensional problems even though it is slow in one dimension.
Variance reduction
More valuable than more samples, and a good thing to raise unprompted:
- Antithetic variates - use each random draw and its negative, exploiting symmetry.
- Control variates - subtract a correlated quantity whose exact value you know.
- Importance sampling - sample more where the payoff is non-zero, essential for deep out-of-the-money options where almost all paths contribute nothing.
Floating point
Interviewers do test this.
- Never use exact equality. Compare with a tolerance.
- 0.1 + 0.2 does not equal 0.3 in binary floating point.
- Catastrophic cancellation: subtracting two nearly equal numbers destroys precision. The classic case is the quadratic formula, where one root should be computed via the alternative form.
- Summing many small numbers into a large accumulator loses precision; Kahan summation fixes it.
Numerical differentiation
Finite differences for Greeks when no analytic form exists. Central differences are second-order accurate and preferable to forward differences. The step size trades truncation error against floating-point noise, with an optimum around the square root of machine epsilon.
Practise in programming and DSA.