Calibration Theory

This document provides the theoretical foundation for model calibration in quantitative finance.

Table of Contents

1. Calibration Problem Formulation
2. Optimization Methods
3. Loss Functions
4. Parameter Constraints
5. Regularization
6. Calibration Diagnostics
7. Model-Specific Calibration

---

Calibration Problem Formulation

Inverse Problem

Model calibration is an inverse problem: Given observed market prices \(\{P_i^{\text{market}}\}_{i=1}^N\), find model parameters \(\boldsymbol{\theta}\) such that model prices \(P_i^{\text{model}}(\boldsymbol{\theta})\) match the market.

Objective:

\[ \boldsymbol{\theta}^* = \arg\min_{\boldsymbol{\theta} \in \Theta} \mathcal{L}(\boldsymbol{\theta}) \]

where \(\mathcal{L}(\boldsymbol{\theta})\) is a loss function measuring model-market discrepancy.

Reference: [Cont & Tankov, 2009]; [Guyon & Henry-Labordère, 2014]

Ill-Posedness

Calibration problems are often ill-posed in the sense of Hadamard:

1. Existence: Solution may not exist (model cannot fit all prices)
2. Uniqueness: Multiple parameter sets may fit equally well
3. Stability: Small changes in market data → large changes in \(\boldsymbol{\theta}\)

Consequences: - Need regularization for stability - Overfitting risk (fitting noise rather than signal) - Prior information helps (parameter bounds, market conventions)

Reference: [Cont & Tankov, 2009]; [Engl et al., 1996]

Well-Posed Calibration

To ensure well-posedness:

1. Parameter bounds: \(\boldsymbol{\theta} \in [\boldsymbol{\theta}_{\min}, \boldsymbol{\theta}_{\max}]\)
2. Regularization: Add penalty term \(R(\boldsymbol{\theta})\)
3. Prior selection: Choose liquid instruments with tight spreads
4. Model selection: Use parsimonious models (fewer parameters)

Reference: [Guyon & Henry-Labordère, 2014], Chapter 4

---

Optimization Methods

Gradient-Based Methods

Gradient Descent

Update rule:

\[ \boldsymbol{\theta}_{k+1} = \boldsymbol{\theta}_k - \alpha_k \nabla \mathcal{L}(\boldsymbol{\theta}_k) \]

where \(\alpha_k > 0\) is the learning rate.

Convergence: \(O(1/k)\) for convex \(\mathcal{L}\)

Variants: - Momentum: Adds inertia to escape local minima - Nesterov: Accelerated gradient (look-ahead)

Reference: [Nocedal & Wright, 2006], Chapter 3

Adam Optimizer

Adaptive Moment Estimation (Adam) [Kingma & Ba, 2014] is the de facto standard in machine learning.

Update:

\[ \begin{align} m_k &= \beta_1 m_{k-1} + (1 - \beta_1) g_k \quad \text{(first moment)} \\ v_k &= \beta_2 v_{k-1} + (1 - \beta_2) g_k^2 \quad \text{(second moment)} \\ \hat{m}_k &= m_k / (1 - \beta_1^k), \quad \hat{v}_k = v_k / (1 - \beta_2^k) \quad \text{(bias correction)} \\ \boldsymbol{\theta}_{k+1} &= \boldsymbol{\theta}_k - \alpha \frac{\hat{m}_k}{\sqrt{\hat{v}_k} + \varepsilon} \end{align} \]

Hyperparameters: \(\beta_1 = 0.9\), \(\beta_2 = 0.999\), \(\varepsilon = 10^{-8}\) (defaults)

Advantages: - Adaptive learning rates per parameter - Robust to noisy gradients - Little tuning required

Reference: [Kingma & Ba, 2014]

Implementation: src/neutryx/calibration/ uses Adam via optax library

L-BFGS-B

Limited-memory BFGS with Box constraints is a quasi-Newton method.

Idea: Approximate Hessian \(H \approx \nabla^2 \mathcal{L}\) using gradient history (last \(m\) iterations).

Update:

\[ \boldsymbol{\theta}_{k+1} = \boldsymbol{\theta}_k - \alpha_k H_k^{-1} \nabla \mathcal{L}(\boldsymbol{\theta}_k) \]

Advantages: - Superlinear convergence (near minimum) - Handles box constraints naturally - Memory-efficient (\(O(md)\) storage)

Disadvantages: - Requires smooth \(\mathcal{L}\) (not robust to noise) - Can get stuck in local minima

Reference: [Nocedal & Wright, 2006], Chapter 7; [Byrd et al., 1995]

Gradient-Free Methods

Differential Evolution

Differential Evolution (DE) [Storn & Price, 1997] is a global optimization algorithm.

Procedure: 1. Initialize population of \(P\) candidates 2. For each candidate \(\boldsymbol{\theta}_i\): - Mutation: \(\boldsymbol{\theta}'_i = \boldsymbol{\theta}_a + F(\boldsymbol{\theta}_b - \boldsymbol{\theta}_c)\) (random triplet) - Crossover: Mix \(\boldsymbol{\theta}_i\) and \(\boldsymbol{\theta}'_i\) with probability \(CR\) - Selection: Keep better candidate

Advantages: - Global search (avoids local minima) - No gradient required - Robust to noisy objectives

Disadvantages: - Slow convergence (many function evaluations) - Not suitable for high-dimensional problems (\(d > 20\))

Reference: [Storn & Price, 1997]; [Price et al., 2005]

Nelder-Mead Simplex

Nelder-Mead [Nelder & Mead, 1965] uses a simplex (\((d+1)\) points in \(d\) dimensions) that reflects, expands, and contracts.

Operations: Reflect, expand, contract, shrink

Advantages: - Derivative-free - Simple implementation

Disadvantages: - Slow for \(d > 10\) - Can stagnate without converging

Reference: [Nelder & Mead, 1965]

Gradient Computation

Finite Differences:

\[ \frac{\partial \mathcal{L}}{\partial \theta_j} \approx \frac{\mathcal{L}(\boldsymbol{\theta} + h \mathbf{e}_j) - \mathcal{L}(\boldsymbol{\theta} - h \mathbf{e}_j)}{2h} \]

Cost: \(O(d)\) function evaluations

Automatic Differentiation (JAX):

\[ \nabla \mathcal{L}(\boldsymbol{\theta}) = \texttt{jax.grad}(\mathcal{L})(\boldsymbol{\theta}) \]

Cost: \(O(1)\) (same as function evaluation, up to small constant)

Advantage: Exact gradients, enabling efficient gradient-based optimization.

Reference: [Bradbury et al., 2018]; [Griewank & Walther, 2008]

Implementation: All calibration in src/neutryx/calibration/ uses JAX autodiff

---

Loss Functions

Mean Squared Error (MSE)

Definition:

\[ \mathcal{L}_{\text{MSE}}(\boldsymbol{\theta}) = \frac{1}{N} \sum_{i=1}^N \left(P_i^{\text{market}} - P_i^{\text{model}}(\boldsymbol{\theta})\right)^2 \]

Properties: - Quadratic: Penalizes large errors heavily - Differentiable: Suitable for gradient methods - Scale-dependent: Sensitive to price magnitudes

Use case: General-purpose calibration

Reference: Standard

Root Mean Squared Error (RMSE)

\[ \mathcal{L}_{\text{RMSE}}(\boldsymbol{\theta}) = \sqrt{\frac{1}{N} \sum_{i=1}^N \left(P_i^{\text{market}} - P_i^{\text{model}}(\boldsymbol{\theta})\right)^2} \]

Advantage: Same units as prices (interpretable)

Relative Error

\[ \mathcal{L}_{\text{rel}}(\boldsymbol{\theta}) = \frac{1}{N} \sum_{i=1}^N \left(\frac{P_i^{\text{market}} - P_i^{\text{model}}(\boldsymbol{\theta})}{P_i^{\text{market}}}\right)^2 \]

Advantage: Scale-invariant (treats all prices equally in percentage terms)

Use case: When prices span multiple orders of magnitude

Implied Volatility Error

\[ \mathcal{L}_{\text{IV}}(\boldsymbol{\theta}) = \sum_{i=1}^N w_i \left(\sigma_i^{\text{market}} - \sigma_i^{\text{model}}(\boldsymbol{\theta})\right)^2 \]

where \(\sigma_i\) is the implied volatility (Black-Scholes).

Advantages: - Market convention: Traders quote volatility, not price - Comparable: Volatilities are dimensionless percentages - Vega-weighted: Implicitly weights by vega (sensitivity)

Disadvantage: Requires Black-Scholes inversion (may not exist for all prices)

Reference: [Gatheral, 2006], Chapter 2; [Cont & Tankov, 2009]

Use case: Standard for volatility surface calibration (Heston, SABR)

Implementation: src/neutryx/calibration/losses.py

Vega-Weighted Loss

\[ \mathcal{L}_{\text{vega}}(\boldsymbol{\theta}) = \sum_{i=1}^N \mathcal{V}_i \left(P_i^{\text{market}} - P_i^{\text{model}}(\boldsymbol{\theta})\right)^2 \]

where \(\mathcal{V}_i = \frac{\partial P_i}{\partial \sigma}\) is the vega.

Motivation: ATM options have high vega (most sensitive to vol) → weight them more.

Effect: Prioritizes fitting liquid, vega-rich options.

Reference: [Cont & Tankov, 2009]

Bid-Ask Spread Weighting

\[ \mathcal{L}_{\text{spread}}(\boldsymbol{\theta}) = \sum_{i=1}^N \frac{1}{s_i^2} \left(P_i^{\text{mid}} - P_i^{\text{model}}(\boldsymbol{\theta})\right)^2 \]

where \(s_i = P_i^{\text{ask}} - P_i^{\text{bid}}\) is the bid-ask spread.

Motivation: Tight spreads → high liquidity → more reliable prices → higher weight.

Effect: Ignores illiquid options with wide spreads.

Reference: Market practice

---

Parameter Constraints

Box Constraints

Simple bounds:

\[ \theta_j^{\min} \leq \theta_j \leq \theta_j^{\max}, \quad j = 1, \ldots, d \]

Example (Heston): - \(\kappa > 0\) (mean-reversion) - \(\theta > 0\) (long-term variance) - \(\sigma_v > 0\) (vol-of-vol) - \(\rho \in [-1, 1]\) (correlation) - \(v_0 > 0\) (initial variance)

Enforcement: - Clipping: \(\theta_j \gets \max(\theta_j^{\min}, \min(\theta_j, \theta_j^{\max}))\) - Projected gradient descent - Barrier methods (interior-point)

Reference: [Nocedal & Wright, 2006], Chapter 16

Implementation: src/neutryx/calibration/constraints.py

Parameter Transformations

Unconstrained optimization is easier. Transform constrained \(\theta \in [a, b]\) to unconstrained \(\phi \in \mathbb{R}\):

Log transform (positive constraints \(\theta > 0\)):

\[ \phi = \log \theta, \quad \theta = e^\phi \]

Logit transform (bounded \(\theta \in [a, b]\)):

\[ \phi = \log\left(\frac{\theta - a}{b - \theta}\right), \quad \theta = a + \frac{b - a}{1 + e^{-\phi}} \]

Tanh transform (bounded \(\theta \in [-1, 1]\), e.g., correlation):

\[ \phi = \text{arctanh}(\theta), \quad \theta = \tanh(\phi) \]

Advantage: Optimize over \(\phi\) without constraints (simpler).

Disadvantage: Jacobian adjustment required for gradients.

Reference: [Nocedal & Wright, 2006]; [Guyon & Henry-Labordère, 2014]

Implementation: src/neutryx/calibration/transforms.py

Feller Condition

For CIR and Heston models, the Feller condition ensures positivity:

\[ 2\kappa\theta > \sigma^2 \quad \text{(CIR: } \sigma_v^2 \text{ for Heston)} \]

Enforcement: - Add penalty: \(\mathcal{L}_{\text{total}} = \mathcal{L} + \lambda \max(0, \sigma_v^2 - 2\kappa\theta)\) - Hard constraint: Project parameters after each update

Reference: [Heston, 1993]; [Cox et al., 1985]

No-Arbitrage Constraints

For local volatility and implied volatility surfaces:

1. Calendar spread arbitrage: \(\frac{\partial C}{\partial T} \geq 0\)
2. Butterfly arbitrage: \(\frac{\partial^2 C}{\partial K^2} \geq 0\) (density non-negative)

Enforcement: - Regularization (penalize violations) - Constrained optimization - Post-processing (arbitrage removal)

Reference: [Gatheral, 2006], Section 5.2; [Fengler, 2009]

---

Regularization

Tikhonov Regularization

Penalize large parameter values:

\[ \mathcal{L}_{\text{reg}}(\boldsymbol{\theta}) = \mathcal{L}(\boldsymbol{\theta}) + \lambda \|\boldsymbol{\theta} - \boldsymbol{\theta}_0\|^2 \]

where: - \(\boldsymbol{\theta}_0\): prior/initial guess - \(\lambda > 0\): regularization strength

Effect: Shrinks \(\boldsymbol{\theta}\) toward prior (prevents overfitting).

Reference: [Engl et al., 1996]; [Guyon & Henry-Labordère, 2014]

Total Variation Regularization

For local volatility surfaces \(\sigma_{\text{loc}}(K, T)\):

\[ \mathcal{L}_{\text{TV}}(\sigma) = \mathcal{L}(\sigma) + \lambda \int \|\nabla \sigma(K, T)\| \, dK \, dT \]

Effect: Penalizes rapid changes (produces smooth surfaces).

Reference: [Fengler, 2009]; [Andersen & Brotherton-Ratcliffe, 1998]

Early Stopping

In iterative optimization, stop before full convergence:

\[ k^* = \arg\min_k \mathcal{L}_{\text{validation}}(\boldsymbol{\theta}_k) \]

Effect: Prevents overfitting to training data.

Reference: [Goodfellow et al., 2016], Section 7.8

---

Calibration Diagnostics

Residual Analysis

Residuals:

\[ r_i = P_i^{\text{market}} - P_i^{\text{model}}(\boldsymbol{\theta}^*) \]

Check: 1. Mean: \(\bar{r} \approx 0\) (unbiased) 2. Pattern: No systematic structure (e.g., smile across strikes) 3. Outliers: Flag instruments with \(|r_i| > 3\hat{\sigma}_r\)

Visual: Plot residuals vs. strike, maturity, moneyness.

Reference: Standard statistical practice

Implementation: src/neutryx/calibration/diagnostics.py:residual_analysis()

Goodness of Fit

R-squared:

\[ R^2 = 1 - \frac{\sum_i r_i^2}{\sum_i (P_i^{\text{market}} - \bar{P}^{\text{market}})^2} \]

Interpretation: Proportion of variance explained (\(R^2 \approx 1\) is good).

Mean Absolute Error (MAE):

\[ \text{MAE} = \frac{1}{N} \sum_{i=1}^N |r_i| \]

Reference: Standard

Parameter Uncertainty

Covariance matrix (Cramér-Rao bound):

\[ \text{Cov}(\boldsymbol{\theta}^*) \approx \left(\mathbf{J}^T \mathbf{J}\right)^{-1} \]

where \(\mathbf{J}\) is the Jacobian: \(J_{ij} = \frac{\partial P_i}{\partial \theta_j}\).

Standard errors:

\[ \text{SE}(\theta_j) = \sqrt{[\text{Cov}(\boldsymbol{\theta}^*)]_{jj}} \]

Confidence intervals (95%):

\[ \theta_j^* \pm 1.96 \cdot \text{SE}(\theta_j) \]

Reference: [Nocedal & Wright, 2006], Section 10.4

Implementation: src/neutryx/calibration/diagnostics.py:parameter_uncertainty()

Identifiability

Correlation matrix of parameters:

\[ \text{Corr}(\theta_i, \theta_j) = \frac{\text{Cov}(\theta_i, \theta_j)}{\text{SE}(\theta_i) \text{SE}(\theta_j)} \]

High correlation (e.g., \(|\text{Corr}| > 0.9\)) indicates weak identifiability: Parameters compensate for each other.

Example (Heston): \(\kappa\) and \(\theta\) often highly correlated (both control long-term behavior).

Solution: Fix one parameter or add prior information.

Reference: [Cont & Tankov, 2009]; [Guyon & Henry-Labordère, 2014]

Implementation: src/neutryx/calibration/diagnostics.py:identifiability()

---

Model-Specific Calibration

Heston Model

Parameters: \(\boldsymbol{\theta} = \{v_0, \kappa, \theta, \sigma_v, \rho\}\)

Instruments: European options across strikes and maturities.

Loss: Implied volatility error (vega-weighted).

\[ \mathcal{L}(\boldsymbol{\theta}) = \sum_{i,j} w_{ij} \left(\sigma_{ij}^{\text{market}} - \sigma_{ij}^{\text{Heston}}(\boldsymbol{\theta})\right)^2 \]

Pricing: FFT (Carr-Madan) or semi-analytical (Heston formula).

Constraints: - \(v_0, \kappa, \theta, \sigma_v > 0\) - \(\rho \in [-1, 1]\) - Feller: \(2\kappa\theta > \sigma_v^2\) (optional)

Typical values (equity): - \(v_0 \approx 0.04\) (20% vol) - \(\kappa \approx 2\) (mean-reversion) - \(\theta \approx 0.04\) (long-term vol) - \(\sigma_v \approx 0.4\) (vol-of-vol) - \(\rho \approx -0.7\) (negative correlation, leverage effect)

Reference: [Heston, 1993]; [Gatheral, 2006], Chapter 3

Implementation: src/neutryx/calibration/heston.py

SABR Model

Parameters: \(\boldsymbol{\theta} = \{\alpha, \beta, \rho, \nu\}\) (typically fix \(\beta\)).

Instruments: Swaption or cap volatilities (interest rate markets).

Loss: Implied volatility error.

Pricing: Hagan's approximation (fast, analytical).

Constraints: - \(\alpha, \nu > 0\) - \(\beta \in [0, 1]\) (often fixed: \(\beta = 0.5\) for shifted lognormal) - \(\rho \in [-1, 1]\)

Typical values (interest rates): - \(\alpha \approx 0.02\) (ATM vol) - \(\beta = 0.5\) (fixed) - \(\rho \approx -0.3\) (negative correlation) - \(\nu \approx 0.3\) (vol-of-vol)

Reference: [Hagan et al., 2002]; [Gatheral, 2006], Chapter 4

Implementation: src/neutryx/calibration/sabr.py

Jump-Diffusion (Merton)

Parameters: \(\boldsymbol{\theta} = \{\sigma, \lambda, \mu_J, \sigma_J\}\)

Instruments: Short-dated options (capture jump risk).

Loss: Price or implied volatility error.

Pricing: Analytical series or FFT.

Constraints: - \(\sigma, \lambda, \sigma_J > 0\) - \(\mu_J \in \mathbb{R}\)

Typical values: - \(\lambda \approx 0.5\) (1 jump every 2 years) - \(\mu_J \approx -0.1\) (10% downward jump on average) - \(\sigma_J \approx 0.2\) (20% jump volatility)

Reference: [Merton, 1976]; [Cont & Tankov, 2004], Chapter 9

Implementation: src/neutryx/calibration/jump_diffusion.py

Local Volatility (Dupire)

Parameters: \(\sigma_{\text{loc}}(K, T)\) (function, not finite-dimensional)

Instruments: Complete implied volatility surface.

Method: Dupire's formula (analytical, no optimization).

\[ \sigma_{\text{loc}}^2(K, T) = \frac{\frac{\partial C}{\partial T}}{\frac{1}{2} K^2 \frac{\partial^2 C}{\partial K^2}} \]

Challenges: - Noisy derivatives → regularization required - Arbitrage-free interpolation - Extrapolation beyond liquid strikes

Reference: [Dupire, 1994]; [Gatheral, 2006], Chapter 5

Implementation: src/neutryx/calibration/local_vol.py

---

Practical Considerations

Multi-Start Optimization

Problem: Non-convex loss landscapes have many local minima.

Solution: Run optimization from \(M\) random initializations, select best result.

Initialization strategies: 1. Random sampling within bounds 2. Latin hypercube sampling (space-filling) 3. Prior-based (perturb market-calibrated parameters)

Reference: Market practice

Incremental Calibration

Daily recalibration: Use previous day's parameters as initial guess.

Advantage: Warm start (faster convergence, parameter stability).

Disadvantage: Can get stuck in local minima if market regime shifts.

Solution: Periodic global search (e.g., weekly).

Reference: Market practice

Cross-Validation

Procedure: 1. Split data: Training (80%), validation (20%) 2. Calibrate on training set 3. Evaluate loss on validation set 4. Select hyperparameters (\(\lambda\), model complexity) minimizing validation loss

Prevents: Overfitting to calibration data.

Reference: [Goodfellow et al., 2016], Chapter 5

---

Summary

This document covered:

1. Problem formulation: Inverse problem, ill-posedness
2. Optimization: Gradient-based (Adam, L-BFGS-B) and gradient-free (DE, Nelder-Mead)
3. Loss functions: MSE, RMSE, implied vol, vega-weighted, spread-weighted
4. Constraints: Box constraints, transformations, Feller condition, no-arbitrage
5. Regularization: Tikhonov, total variation, early stopping
6. Diagnostics: Residuals, R-squared, parameter uncertainty, identifiability
7. Model-specific: Heston, SABR, jump-diffusion, local volatility

All calibration methods are implemented in src/neutryx/calibration/ with extensive diagnostics and validation.

See also: Pricing Models Theory | Numerical Methods

---

References: See Bibliography for complete citations.