1. Analytic pricing – what it really buys you

Analytic = closed-form (or semi-closed) formula for price and often Greeks.

Examples:

  • Black–Scholes / Black-76 for European options.
  • Simple forwards, swaps, some caps/floors under LMM-style assumptions.
  • Certain spread options, swaptions, etc via Fourier / characteristic functions.

Advantages:

  • Speed:
    • Single formula evaluation ~ microseconds.
  • Greeks “for free”:
    • Differentiable expressions → analytic delta, gamma, vega, etc.
  • Stability / smoothness:
    • No Monte Carlo noise; smooth curves for calibration and risk.
  • Debuggability:
    • Easier to test against known limits and asymptotics.

Limitations:

  • Need strong assumptions:
    • Simple dynamics (GBM, constant vol, affine models).
    • Simple payoff (no path-dependence, no complex constraints).
  • Hard to handle:
    • Early exercise (American),
    • Multi-dimensional path-dependence (barriers with path constraints, storage, swing),
    • Local/stochastic vol with complex features.

Think of analytic pricers as reference engines and sanity checks, not universal tools.

2. Numerical methods – why we need them

When payoffs or dynamics get richer:

  • Monte Carlo (MC):
    • Path-dependent payoffs (Asian, barrier, storage, swing).
    • High-dimensional risk (many factors, early exercise via LSMC).
  • PDE / finite differences:
    • Early exercise in low dimensions (American options, some callable structures).
    • Local/stochastic volatility in 1–2 dimensions.
  • Trees / lattices:
    • Simple early-exercise structures, pedagogy, quick approximations.

Advantages:

  • Much more flexible in payoff and model structure.
  • Can directly handle path dependence, constraints, exercise rules.

Costs:

  • Performance:
    • Need many paths / grid points.
  • Noise / error control:
    • MC sampling error, PDE time/space discretisation error.
  • Greeks:
    • Often approximated via finite differences or AAD → more complexity.
  • Implementation complexity:
    • Seeds, variance reduction, stability conditions, boundary conditions.

3. How a practitioner thinks about the trade-off

For a given product:

  1. Is there a trusted analytic or semi-analytic benchmark?
    • Yes → use it as primary pricer if within assumptions, or as a test target.
    • No → numerical is unavoidable.
  2. Dimensionality and path structure:
    • 1D, European-like → analytic / PDE / lattice viable.
    • Many risk factors, strong path-dependence → MC / LSMC.
  3. Use-case:
    • Calibration:
      • Need fast, smooth pricing → analytic / semi-analytic or very fast PDE.
    • Exotic trade pricing:
      • Can accept heavier numerical cost per trade, but need robust Greeks.
    • Risk / intraday:
      • Need speed and determinism; may use approximations or surrogate models.

Typical pattern:

“Use analytic pricing wherever it is structurally appropriate, and numerical engines for complex payoffs. Use analytic prices as regression tests and calibration anchors for numerical methods.”

4. Interview soundbite

“Analytic pricers are fast, smooth, and give clean Greeks, but require strong assumptions on dynamics and payoff. Numerical methods – Monte Carlo, PDEs, trees – buy flexibility for path-dependence, early exercise, and richer models at the cost of performance and numerical noise. In practice you mix them: analytic formulas as benchmarks and calibration anchors; numerical pricers for more complex structures, backed by regression tests against the analytic cases where they overlap.”