3.11.1 Calibration from Zero

Basic Prerequisites: Black–Scholes–Merton Hub. No prior derivatives knowledge needed.

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1. Intuition & Practical Objective

Start with the dumbest possible question: if Black–Scholes says an option has one price, why does a desk need a “calibration” step at all? Because the model has free parameters (volatility, and in fancier models several of them), and the market has already decided the actual prices. Calibration is the act of choosing the parameters so the model agrees with the market. Everything else on this page is unpacking that sentence.

Think of a model as a machine with dials. You feed it a strike and a maturity and it prints a price. The dials are $\theta$ (for BSM just $\sigma$; for Heston $v_0,\bar{v},\lambda ,\eta ,\rho$; for SABR $\alpha ,\beta ,\rho ,\nu$). Market quotes are the answer key. Calibration turns the dials until the machine’s answers match the key.

Three things to un-learn first:

1. Calibrating is not fitting history, and it is not forecasting. You are not averaging past prices and you are not predicting the future. You are solving: “what parameter values make this model’s today quotes equal the market’s today quotes?” The parameters are a static snapshot of the market’s view, not a law of motion.
2. The market is the authority, not the model. When model and market disagree, it is the model that is wrong. Calibration makes the model bow to the market. The moment you change the parameters tomorrow (recalibration), you are admitting the previous parameters were only ever a best-fit to yesterday.
3. A good fit is necessary, not sufficient. Fitting the smile perfectly does not mean the model’s dynamics are right. Two models can agree on every vanilla price today and disagree violently on an exotic or a hedge ratio tomorrow (Bergomi Ch 2: this is the local-vol vs stochastic-vol distinction, and Ch 8: the volatility-of-volatility expansion).

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2. Mathematical Ground Truth & Derivations

The calibration problem, written down. We observe $N$ market quotes $\{O_i^{\text{mkt}}\}$ (option prices, or equivalently implied vols ${\hat{\sigma }}_i$). We pick a pricing function $P_i(\theta )$ (model price for quote $i$ under parameters $\theta$). Calibration is

$$\hat{\theta }=\arg \underset{\theta }{\min }\mathcal{L}(P_1(\theta ),\dots ,P_N(\theta );O_1^{\text{mkt}},\dots ,O_N^{\text{mkt}}),$$

where $\mathcal{L}$ is an objective function (a distance; the lookup table is on the Index Hub). For BSM with a single free parameter $\sigma$, this is a one-dimensional root-finding/minimization: find $\sigma$ so the model call price equals the market call price.

The single-parameter case is already instructive. Given one market option, $\sigma$ is chosen so $P(\sigma )=O^{\text{mkt}}$. Invert BSM: this $\sigma$ is called the implied volatility - it is literally the BSM parameter calibrated to that one quote. With several quotes at once (a smile), no single $\sigma$ fits all, so you must trade off which strikes matter most. That trade-off is exactly the choice of objective function.

Two dangers that appear even here.

• Objective mismatch: the price-RMSE optimum and the vol-RMSE optimum differ, because price is a nonlinear function of vol (dollar-weighted toward ATM).
• Over-determination: with more quotes than parameters, the fit is a compromise; the residuals tell you where the model is structurally wrong (the smile BSM cannot reach).

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3. Computational Implementation - one dial, two objectives, two answers

Fit a single BSM vol $\sigma$ to the whole market smile, first minimizing price RMSE, then implied-vol RMSE. The two answers differ - proof that the objective function is a real choice. Stdlib only.

Price-RMSE picks $\sigma =0.239$ (biased to ATM where prices are largest); vol-RMSE picks $\sigma =0.26$ (spreads error evenly across the smile). Same market, same model, one free parameter - two calibrations. That is the seed of the whole folder: calibration is a decision, not a computation.

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4. Failure Modes & First-Principles Breakdowns

1. Treating calibration as “the answer.” A best-fit $\theta$ is conditional on today’s quotes. Recalibrating tomorrow changes it - so any derivative priced on it inherits recalibration drift (see 05 · Failure Modes).
2. The residual you can’t fit is information. If a one-$\sigma$ BSM fit leaves a systematic smile-shaped error, that is not noise to shrug off - it is the model telling you it is structurally wrong (constant vol cannot reach the wings). This is the motivation for local/stochastic vol.
3. Wrong objective ⇒ wrong parameter. Fitting in price space quietly ignores the wings; fitting in vol space can leave large dollar mispricings at ATM. Choose deliberately and report both.
4. More parameters ≠ better. A richer model (SABR, Heston) fits better but invites non-identifiability and overfitting - the machine has too many dials and many dial-settings print the same prices.

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5. References

• Bergomi, Stochastic Volatility Modeling
• Gatheral, The Volatility Surface
• Hull, Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Base: Black–Scholes–Merton Hub · Stochastic Calculus & Itô
• Continue: 02 · The Calibration Problem · Index Hub
• Sibling: Implied Volatility Surfaces