Exact calibration of structural models via time-change

Battle Count: 56

The paper is primarily relevant to credit risk modeling, CDS pricing, and counterparty risk management rather than direct quantitative trading strategies. However, it has indirect relevance for: (1) credit portfolio risk assessment, (2) CVA calculations affecting trading desk P&L, (3) structural model calibration for credit derivatives pricing, and (4) understanding the AT1P model's volatility interpretation. The time-change technique could potentially be adapted for other calibration problems in quantitative finance.

The core calibration (inverting the latent survival curve) is straightforward for the Brownian case (closed-form formula 7). For drifted Brownian motion and AT1P, numerical inversion is required but computationally efficient. The main complexity lies in: (1) understanding the stochastic calculus framework (running infimum, time-change, Dambis-Dubins-Schwarz theorem), (2) handling the singularity at the origin in practice, (3) implementing Monte Carlo verification with proper discretization, and (4) extending to multi-factor/joint models with different clocks. The mathematical prerequisites are substantial (stochastic processes, first-passage theory).

The paper provides fully analytical closed-form expressions (Theorem 1, Corollary 1) and detailed proofs. Monte Carlo verification is described with specific parameters (N=1000 trajectories, δt=0.01, T=20). The implied clock formula (7) and diffusion coefficient (9) are explicit. However, no code repository is provided. The mathematical derivations are self-contained and verifiable.