Model Calibration & Parameter Estimation

Calibration usually emphasizes consistency with current market prices or implied volatilities. Parameter estimation is broader: it can use history as well as the live chain. They are related. They are not interchangeable.

For Heston, the knobs include v0, kappa, theta, xi, and rho. Calibration tries to reproduce a chosen set of quotes. A classroom example: the model prices a $200 call at $7.25 and the quote is $8.10. The model is not on the market until the knobs move. Those dollars are teaching figures, not a live print.

A simple price-error objective sums weighted squared gaps between model price and market price. A volatility-error objective compares implied vols instead. Price errors and vol errors do not treat a cheap short-dated wing the same way as an expensive long-dated option.

Weights can be equal, tilted to liquidity, scaled by the bid–ask, or aimed at a risk region. No scheme is automatically correct. Write down why you chose it.

Clean stale and crossed quotes. Check conventions, rates, and dividends. Select the model and the objective. Optimize. Run diagnostics. Then approve, document, and monitor. The optimizer is the middle of the process, not the end.

• Sparse or noisy quotes in the wings and long expiries.
• Parameters that move together, so many sets fit almost the same prices.
• Local minima, numerical error, and a model that cannot fit every feature.
• Overfitting today and behaving poorly between quotes or under stress.

A low calibration error does not prove the model is economically reasonable or suitable for every product.

In-sample fit asks how closely the model reproduces the quotes you used. Out-of-sample validation asks how it behaves on quotes or periods you held out, or under a small perturbation.

Useful tests: residuals by strike and maturity, recalibrate after a tiny quote bump, parameter stability over time, price an instrument you did not fit, and compare with a benchmark. Fitness-for-purpose is the decision, not the headline error.

Local volatility typically aims to reproduce the observed vanilla surface and infer a local variance function. Surface quality and arbitrage consistency dominate.

Heston seeks five (or so) stochastic-variance parameters. It may not fit every surface, and the knobs can be hard to identify. Judge both on stability, intended products, and hedge behavior — not only on the error number.

A calibration report should answer:

1. What data and timestamp were used?
2. Which instruments were included or excluded, and why?
3. Which model, objective, and weights?
4. Did the optimizer converge? Were bounds hit?
5. How large are residuals by strike and maturity?
6. How sensitive are parameters to a small quote change?
7. What changed since the last run, and who approved it?

Triggers for a review: error above a stated threshold, a sharp jump in a key parameter, thin quote coverage, an optimizer that pins a bound, or a material gap versus an independent benchmark. Thresholds belong to the book and the use — they are not copied from another desk.

Next: hedge and surface risk. Educational only. Not a production model policy.