3.11 Calibration & Market Practice

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma, Implied Volatility Surfaces, and Heston & SABR. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

A model is not a fact - it is a choice of dynamics whose free parameters must be set so that the model reproduces the prices the market already trades. That act - finding the parameters that make the model’s quotes match observed market quotes - is calibration, and it is the daily craft of every options desk.

The objective is blunt: fit the model to the smile without believing the fit. Every parameter you change to match today’s vanilla quotes is a static repair; it says nothing about the future dynamics the model is actually used for (exotic pricing, hedging, risk). This folder is the calibration-and-market-practice hub for Pillar 3: it is a lookup for the two things that matter most (the objective function you minimize, and the workflow you run each morning), and a router into six sub-pages that take you from the one-line idea to professional failure-mode depth.

The one-sentence essence. “Calibration = solve ${\min }_{\theta }\text{distance}(\text{model prices/vols}(\theta ),\text{market quotes})$, then distrust $\theta$: it is a static best-fit, not a law of motion.”

2. Mathematical Ground Truth - the objective functions (job #1)

The core choice is what distance to minimize. Three families dominate practice (Gatheral Ch 3; Bergomi Ch 7 §7.5):

The three disagree. In 01 · From Zero a single-parameter fit shows the price-RMSE optimum lands at a different vol than the implied-vol optimum. In practice desks fit in implied vol (it is what is quoted, so a fit that misses it is untradeable) and often in relative price for exotic books, but they always report fit quality in both.

Regularization. Non-uniqueness and instability force a penalty. A ridge term $+\lambda \Vert \theta -{\theta }_0{\Vert }^2$ (or $+\lambda \Vert \theta {\Vert }^2$) keeps parameters near a prior and stabilizes the inverse problem (see 02 · The Calibration Problem, 04 · Stochastic Vol). It is the price you pay for trading off fit (bias) against stability (variance) - the standard bias–variance decomposition.

Why calibration is hard - the three structural obstacles:

1. Non-identifiability - different parameters give the same smile (SABR’s $\beta$/$\rho$ ridge; the $\kappa$/$\eta$ collinearity of Heston). The smile pins the combination, not the parameters.
2. Ill-posedness - the local-vol inversion (Dupire/Gatheral) differentiates noisy data, so tiny bid/ask noise explodes into huge local-vol spikes (03 · Local Vol, 05 · Failure Modes).
3. Overfitting - a rich parametric surface fits today’s quotes perfectly and is useless tomorrow (or even mid-curve).

3. Computational Implementation - the objective-function engine

Two objective functions on a real smile, fit with a single free parameter $\sigma$. Stdlib only.

The price-RMSE optimum ($\sigma =0.239$) sits below the vol-RMSE optimum ($\sigma =0.26$): dollar weighting pulls the fit toward the ATM strike, where prices are largest. Same data, same model, different objective → different calibrated parameter. This is job #1 of the hub: choose the objective deliberately.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - full analysis lives in 05 · Failure Modes & Practice. In one line each:

1. Objective mismatch - fitting in price space leaves the wings wrong (vega-weighted risk is mis-hedged); fitting in vol space can leave dollar P&L large at ATM.
2. Non-identifiability - the smile fixes combinations of parameters, not the parameters themselves; two fits with identical RMSE carry different hedge ratios.
3. Instability - differentiation of noisy quotes (the local-vol inversion) amplifies noise by $\sim 1/d{y}^2$; overfit surfaces are static, not predictive.
4. Recalibration drift - a model “meant to be recalibrated daily” (Bergomi on local vol) has forward/future skews that change with recalibration - unhedgeable carry.

5. References

• Gatheral, Jim: The Volatility Surface: A Practitioner’s Guide (Wiley 2006)
• Bergomi, Lorenzo: Stochastic Volatility Modeling (CRC 2016)
• Brigo–Mercurio: *Interest Rate Models
• Duffy: Finite Difference Methods in Financial Engineering (Wiley 2006)

6. Connected Graph Bridges

• Foundational base: Stochastic Calculus & Itô · Implied Volatility Surfaces
• Sibling models: Heston & SABR · Black–Scholes–Merton Hub · Interest Rate Models
• Sub-pages (in-folder): 01 From Zero · 02 The Calibration Problem · 03 Calibrating Local Vol · 04 Calibrating Stochastic Vol · 05 Failure Modes · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05