Calibration and Parameter Estimation in Models › The Calibration Problem

Calibration begins with a model that produces prices or other observable quantities as a function of parameters. To understand this formally, let $M(\theta)$ denote model prices for a vector of parameters $\theta = (\theta_1, \theta_2, \ldots, \theta_k)$. The market provides observed prices $P^{mkt} = (P_1^{mkt}, P_2^{mkt}, \ldots, P_n^{mkt})$ for $n$ instruments. The calibration problem seeks parameters $\hat{\theta}$ such that model prices $M(\hat{\theta})$ are as close as possible to market prices $P^{mkt}$.

This simple description masks several important choices. What does "as close as possible" mean? How do we measure discrepancy? How do we search for optimal parameters? And what happens when perfect agreement is impossible because the model cannot reproduce all market prices simultaneously? These questions form the core of calibration theory and drive the methodological choices we explore throughout this chapter.

Consider calibrating the Black-Scholes model to a set of option prices. Black-Scholes assumes constant volatility $\sigma$ for all options on the same underlying. But as we discussed in the chapter on implied volatility and the volatility smile, the market assigns different implied volatilities to options with different strikes and maturities. No single value of $\sigma$ can reproduce all observed option prices.

This fundamental tension arises because models simplify reality. The Black-Scholes model ignores jumps, stochastic volatility, and many other market features. When we calibrate, we're finding the best compromise: the parameter values that make the model least wrong across the calibration instruments. This perspective is essential for understanding calibration not as a search for "truth" but as a practical optimization that acknowledges model limitations.

More flexible models can reduce these discrepancies, and a finite set of quotes can sometimes be fitted exactly. The five-parameter Heston model captures some smile dynamics but generally cannot reproduce an arbitrary set of market quotes. For nested model families fitted to the same data under the same objective, the exact optimal loss cannot worsen when parameters are added, although it need not improve strictly. A finite optimizer may not reach that optimum. Additional flexibility can also complicate optimization, weaken identification and increase the risk of fitting noise.

Let $M_i(\theta)$ denote the model price for instrument $i$ given parameters $\theta$. The calibration problem is typically formulated as an optimization problem. We seek parameters that minimize some measure of discrepancy between what the model predicts and what the market shows us. Mathematically, this takes the form:

where:

• $\hat{\theta}$: the optimal set of parameters that we seek
• $\theta$: a vector of model parameters over which we optimize
• $L(\theta)$: a loss function measuring the discrepancy between model and market prices
• $\Theta$: the set of feasible parameters, typically constrained by economic requirements such as positivity of volatility or valid correlation bounds

This formulation captures the essence of calibration: we have a loss function that quantifies how poorly the model matches the market, and we search over the space of allowable parameters to find those that minimize this loss. The choice of loss function determines the calibration outcome because different functions emphasize different aspects of fit quality.

The most common loss functions include several variations that each serve distinct purposes:

Sum of squared errors (SSE):

where:

• $n$: number of calibration instruments
• $M_i(\theta)$: model price for instrument $i$ given parameters $\theta$
• $P_i^{mkt}$: observed market price for instrument $i$

This is the simplest and most intuitive loss function. It sums the squared differences between model and market prices across all instruments. Squaring ensures that positive and negative errors don't cancel, and larger errors receive disproportionately more penalty than smaller ones. This makes SSE particularly sensitive to outliers, which can be either an advantage (outliers might indicate important mispricings) or a disadvantage (outliers might represent data errors).

Weighted sum of squared errors:

where:

• $w_i$: nonnegative weight assigned to instrument $i$, with at least one positive weight
• $n$: number of calibration instruments
• $M_i(\theta)$: model price for instrument $i$ given parameters $\theta$
• $P_i^{mkt}$: observed market price for instrument $i$

Weighted SSE introduces flexibility by allowing different instruments to contribute differently to the objective. You might assign higher weights to more liquid instruments whose prices are more reliable, or to instruments that are most relevant to your trading book. The weights $w_i$ encode your beliefs about which prices are most important to match accurately.

Relative errors:

where:

• $n$: number of calibration instruments
• $M_i(\theta)$: model price for instrument $i$
• $P_i^{mkt}$: observed market price for instrument $i$

Relative errors normalize by a nonzero target price, so a $1 error on a $100 instrument counts the same as a $0.10 error on a $10 instrument. This removes emphasis caused solely by absolute price scale; an instrument can still dominate if its proportional error is large. Zero prices make this loss undefined, and very small prices can receive excessive influence. A spread-based or noise-based scale may be more appropriate for such quotes.

The loss function determines which errors receive the most attention. Weights can emphasize liquid near-the-money options or maturities relevant to a trading book, but actual liquidity depends on the market, maturity and liquidity measure. Relative errors instead compare proportional discrepancies. These are choices about the purpose and reliability of the calibration targets.