Topic 17 – Calibration

1. What calibration really is

Working definition:

“Choose model parameters so that model prices are consistent with selected market quotes.”

Inputs:

• A model with parameters θ (e.g. Black vol, Heston params, short-rate params…)
• A set of liquid instruments and their market prices:
  • Vanillas across strikes/tenors,
  • Swaps, caps/floors, swaptions,
  • In power/gas: strips, options on forwards, etc.

Output:

• Parameters $\theta^*$ solving “best fit” under some criterion.

Important:

• Calibration is not prediction. It is consistency with today’s tradable quotes.

2. Objective functions – what are we minimising?

Typical choices:

• Price error: $$\min_\theta \sum_i w_i \left(P_i^{\text{model}}(\theta) - P_i^{\text{mkt}}\right)^2$$
• Implied vol error (equity/FX-style): $$\min_\theta \sum_i w_i \left(\sigma_i^{\text{model}}(\theta) - \sigma_i^{\text{mkt}}\right)^2$$

Weights $w_i$:

• To reflect:
  • Liquidity (more weight on liquid points),
  • Bid–ask width (downweight noisy quotes),
  • Scaling (avoid big prices dominating small ones).

Alternatives:

• Relative errors: $(P^{\text{model}} - P^{\text{mkt}})/P^{\text{mkt}}$.
• Robust loss (e.g. Huber) to damp effect of outliers.

The key is to pick an objective that reflects:

• Desk priorities,
• Liquidity structure,
• Stability requirements.

3. Stability and overfitting

Signs of unstable calibration:

• Small changes in input quotes → huge swings in parameters.
• Errors jump around from day to day.
• Parameters hit hard bounds frequently.

Causes:

• Too many free parameters vs number/quality of quotes.
• Poor parameterisation (highly correlated parameters).
• Overly flexible model fitted to noisy data.

Mitigations:

• Regularisation / penalties:
  • Penalise parameters drifting too far from prior values.
• Parameter constraints:
  • Enforce positivity, bounds, ordering.
• Smoothing across time:
  • Fit today, but with penalty for deviating from yesterday.
• Simplify the model:
  • Fewer parameters with clearer interpretation.

Idea:

“A good calibration is not just a small error today; it is also stable over time and economically sensible.”

4. Sanity constraints and diagnostics

After calibration, you should ask:

1. Are parameters in plausible ranges?
   • Vol between 0 and, say, 300%.
   • Mean-reversion speeds not absurdly high or negative.
2. Does the implied surface behave well?
   • No arbitrage violations (monotonicity, convexity).
   • Smile / term-structure shapes that make economic sense.
3. How do residuals look?
   • Are misfits random-ish, or structured?
   • Are there regions (deep OTM, long-dated) with systematic bias?

Diagnostics:

• Plot model vs market implied vols or prices.
• Track time series of key parameters.
• Track fit error (RMSE, max error).

In power/commodities:

• Check that calibrated parameters / surfaces:
  • Are consistent with seasonality and shape,
  • Do not imply arbitrage across forwards / options.

5. Calibration in the system

From a systems point of view:

• Calibration is a job that produces a model artefact:
  • (model_type, snapshot_id, params, fit_metrics, date, version_id).
• Risk and pricing use model IDs, not raw parameters.
• You can:
  • Replay old calibrations.
  • Compare models across time.
  • Enforce “only approved model IDs” in production.

This ties into:

• Versioning, run provenance, model risk controls.

6. Interview soundbite

“Calibration picks model parameters to best fit a set of liquid instruments under a chosen objective – often least squares in price or implied vol with liquidity-based weights. A good calibration is not just low error; it is also stable and economically sane. We monitor parameter time series, fit residuals, and implied surface shape, and we impose bounds and regularisation to avoid overfitting. In the system, a calibration run produces a versioned model artefact tied to a specific market snapshot, so pricing and risk can be reproduced and audited later.”