Calibration and Parameter Estimation in Models › Objective Functions for Calibration

The loss function defines what "good calibration" means. Different objectives can lead to different calibrated parameters, and the choice should reflect how the model will be used. A model calibrated for pricing exotic derivatives may need different parameter values than one calibrated for risk management, even if based on the same underlying framework.

The most direct approach minimizes differences between model and market prices. Matching vanilla quotes aligns the model prices of those hedging instruments with the market. Exotic valuations still depend on dynamics and assumptions that the vanilla calibration may not identify.

For options, a common choice is to calibrate in volatility space rather than price space. Instead of minimizing price errors, we minimize implied volatility errors:

$$L(\theta) = \sum_{i=1}^{n} w_i \left( \sigma_i^{model}(\theta) - \sigma_i^{mkt} \right)^2$$

where:

• $\sigma_i^{model}(\theta)$: implied volatility derived from the model price for instrument $i$
• $\sigma_i^{mkt}$: observed market implied volatility for instrument $i$
• $w_i$: nonnegative weight for instrument $i$, with at least one positive weight
• $n$: number of calibration instruments

Volatility-space calibration offers several useful interpretations:

• Quote units are comparable: a one-percentage-point volatility error has the same volatility units across options. Its dollar impact still depends on Vega, the sensitivity of price to volatility.
• Numerical behavior depends on the instruments and model. Low Vega can amplify price errors during IV inversion, so volatility loss need not have smoother gradients or better conditioning.
• Errors are convenient to interpret when the market quotes the instruments in implied volatility.

Model and market prices must admit an implied volatility under the same option conventions. For the no-dividend European calls used below, prices strictly between discounted intrinsic value and spot have a finite positive Black-Scholes IV. Boundary and invalid prices require explicit handling.

Market prices are not single values; they come with bid-ask spreads that reflect the cost of transacting and the uncertainty about fair value. A sophisticated calibration might seek parameters such that model prices fall within observed bid-ask bounds rather than matching midpoint prices exactly:

$$\text{minimize } L(\theta) \quad \text{subject to} \quad P_i^{bid} \leq M_i(\theta) \leq P_i^{ask} \quad \text{for all } i$$

where:

• $P_i^{bid}$: market bid price, the highest price at which someone is willing to buy
• $P_i^{ask}$: market ask price, the lowest price at which someone is willing to sell
• $M_i(\theta)$: model price
• $L(\theta)$: loss function to minimize
• $\theta$: model parameters

A model price inside an executable bid-ask interval is compatible with that individual quote, but it does not establish joint no-arbitrage or model adequacy. The constraints are usable only when they are jointly feasible: a restricted model may not intersect all quote bands. If no feasible fit exists, investigate the quotes and model restrictions or use explicit slack variables or outside-spread penalties.

For time-series models like GARCH or short-rate models estimated from historical data, maximum likelihood estimation (MLE) provides a principled framework rooted in statistical theory. Given observations $y_1, y_2, \ldots, y_T$ and a model specifying the conditional density $f(y_t | y_{t-1}, \ldots, y_1; \theta)$, the log-likelihood is:

$$\ell(\theta) = \sum_{t=1}^{T} \log f(y_t | y_{t-1}, \ldots, y_1; \theta)$$

where:

• $\ell(\theta)$: log of the observed data's joint density under the model, viewed as a function of the parameters
• $\theta$: vector of model parameters
• $T$: number of time series observations
• $f(\cdot)$: conditional probability density specified by the model
• $y_t$: observation at time $t$

For continuous observations, likelihood compares joint densities as a function of $\theta$; the probability of one exact observed vector is zero. For discrete observations, use probability masses instead. Higher likelihood means a better fit by this criterion under the assumed model, rather than a higher posterior probability that the parameter is true.

Maximizing $\ell(\theta)$ fits the specified joint or conditional likelihood to the observed sample. The displayed sum uses the conditional-density factorization; the first term requires an initial-density or conditioning convention. This likelihood approach is commonly used for fitting GARCH models to return data.

Under appropriate regularity, identification and correct-specification conditions, MLE is consistent and asymptotically normal. It can attain the asymptotic information bound within the regular estimator class; this is not a minimum-variance guarantee among every consistent estimator. A computable likelihood alone does not ensure a finite maximizer, correct specification or reliable finite-sample recovery.

Calibration may need to preserve an initial yield curve while fitting option prices. Exact curve matching must be built into the parameterization, imposed as equality constraints or preserved by a hierarchical procedure. A weighted composite objective generally trades discrepancies unless a separate mechanism enforces the exact curve fit.