Numerical Methods and Algorithms in Finance › Key Parameters

The key parameters for the numerical methods covered in this chapter are:

• tol (tolerance): A threshold whose units depend on the solver: yield half-width for bisection, yield or spread step for the Newton examples, and monetary residual for implied volatility. The Newton examples also require a separate price_tol. A smaller parameter step alone does not certify price accuracy; check output stability and account for inner numerical error and floating-point limits.
• max_iter: Maximum number of iterations before algorithms terminate. Choose max_iter large enough for tested cases and handle non-convergence explicitly.
• a, b (interval bounds): Ordered finite endpoints for bisection, in the financial yield domain $-1<a<b$ here. Either endpoint can be an exact computed root; otherwise opposite residual signs supply a bracket for continuous $f$. Compare signs without multiplying floating-point residuals.
• x0 (initial guess): Starting point for Newton-Raphson. It must be in the model's admissible domain; local convergence also depends on a suitable starting basin. An algebraic root outside the financial domain is not an acceptable result.
• n (resolution or quadrature order): For the composite trapezoidal and Simpson's rules in this chapter, $n$ is the number of equal subintervals, so the grid contains $n+1$ points. For Gauss-Legendre quadrature, $n$ is the number of nodes. Increasing $n$ often reduces truncation error for sufficiently smooth integrands until roundoff or other numerical limits dominate. Classical composite-rule orders are $O(1/n^2)$ for the trapezoidal rule and $O(1/n^4)$ for Simpson's rule when their smoothness assumptions hold.
• vol (volatility): Positive finite rate-shift standard deviation in the toy callable-bond illustration. It is not Black-Scholes stock volatility or a calibrated rate-process parameter; zero dispersion is not supported by this continuous-density implementation.
• oas (option-adjusted spread): Model-dependent constant spread over a specified benchmark curve that makes an option-adjusted model price match an observed price.