Calibration inverse problems and identifiability

Definition A model with parameter vector $\theta$ in an admissible set $\Theta \subseteq \mathbb{R}^p$ assigns to each of $m$ quoted instruments a model value $g_i(\theta)$. The forward map

contains everything that turns parameters into quotes, namely the contract terms and dates, the valuation time, the settlement and discounting conventions, and the conversion into the quotation convention (a price, an implied volatility or a spread), so that $g_i(\theta)$ and the observed quote $q_i$ are in the same units. Calibration chooses, when a minimizer exists,

for a loss $\mathcal{L}$ that measures the misfit between $g(\theta)$ and the quote vector $q = (q_1, \dots, q_m)$. Recovering inputs from outputs in this way is an inverse problem. In the sense of Hadamard it is well posed when, for each admissible data vector, a solution exists, is unique and depends continuously on the data. Calibration can fail each of the three conditions. Noisy, stale or mutually inconsistent quotes, or a misspecified model, leave no exact fit. Too few independent quotes leave many parameter vectors with the same fit. Nearly dependent quotes make the solution move far when a quote moves a little.

Definition Write $r_i(\theta) = g_i(\theta) - q_i$ for the residual of quote $i$ and let $W = \operatorname{diag}(w_1, \dots, w_m)$ with every $w_i > 0$. The weighted calibration loss, with an optional penalty, is

with a penalty weight $\eta \ge 0$, a prior parameter vector $\theta_0$ and a penalty matrix $\Gamma$. This card takes $\eta = 0$. For a linear forward map $g(\theta) = A\theta$ with $A \in \mathbb{R}^{m \times p}$, the loss is $(A\theta - q)^{\top} W (A\theta - q)$ and its gradient is $2A^{\top}W(A\theta - q)$. Setting the gradient to zero gives the weighted normal equations

The Hessian $2A^{\top}WA$ is positive semidefinite, so the loss is convex and every solution of the normal equations is a global minimizer. Solutions always exist, because $A^{\top}WA$ and $A^{\top}$ have the same range, which contains $A^{\top}Wq$. For every direction $\delta$ the quadratic form $\delta^{\top}A^{\top}WA\,\delta = \sum_i w_i (A\delta)_i^2$ is positive exactly when $A\delta \ne 0$. So $A^{\top}WA$ is positive definite, and the minimizer is unique, if and only if $A$ has full column rank $p$. Multiplying every weight by the same positive constant does not change the minimizer when $\eta = 0$. If the quote errors are independent with variances $v_i$, the weights $w_i = 1/v_i$ give generalized least squares, which is maximum likelihood when the errors are also Gaussian. Weights can instead express how much each instrument matters for the intended use. A bid-ask spread is a range of trading prices, not automatically the standard deviation of an independent Gaussian error, so turning spreads into weights is a modeling choice.

Definition The parameters of a forward map $g \colon \Theta \to \mathbb{R}^m$ are identifiable from the quoted instruments if $g(\theta) = g(\theta')$ with $\theta, \theta' \in \Theta$ implies $\theta = \theta'$, that is, if $g$ is injective on $\Theta$. They are locally identifiable at $\theta^*$ if $g$ is injective on $\Theta \cap U$ for some neighborhood $U$ of $\theta^*$. Identifiability is a property of the model, the instrument set and the admissible set together, stated for noise-free outputs, and no optimizer can create it. For a linear map $g(\theta) = A\theta$ on $\Theta = \mathbb{R}^p$, the parameters are identifiable exactly when the null space of $A$ is $\{0\}$, that is, when $A$ has rank $p$, which needs at least $p$ quotes whose rows of $A$ are linearly independent. If $A\delta = 0$ for some $\delta \ne 0$, then $\theta$ and $\theta + s\delta$ give the same quotes for every real $s$. A combination $c^{\top}\theta$ is still identified exactly when $c$ lies in the row space of $A$, because then it is a fixed combination of model quotes. For nonlinear maps the rank condition is local. If $g$ is continuously differentiable near an interior point $\theta^*$ and its Jacobian at $\theta^*$ has rank $p$, then the parameters are locally identifiable at $\theta^*$. Global identifiability can still fail, and a singular Jacobian does not by itself rule identifiability out.

Definition If $g$ is differentiable at $\theta$, the price Jacobian is the $m \times p$ matrix $J(\theta)$ with entries

in units of quote $i$ per unit of parameter $j$. It gives the first-order response of every quote to a small parameter move $\delta$,

A direction $\delta \ne 0$ with $J(\theta)\delta = 0$ is a null direction, which leaves every quote unchanged to first order. With weights $W$, the singular values $s_1 \ge \dots \ge s_p \ge 0$ of $W^{1/2}J(\theta)$, the square roots of the eigenvalues of $J^{\top}WJ$, measure how strongly parameter directions move the weighted quotes, since $\lVert W^{1/2}J\delta \rVert \ge s_p \lVert \delta \rVert$ for every $\delta$, with equality when $\delta$ is an eigenvector of $J^{\top}WJ$ for its smallest eigenvalue. A small $s_p$ marks a weakly identified direction, and a large condition number $s_1/s_p$ marks a calibration that turns small quote changes into large parameter changes. For a linear map, $J = A$ at every point. For a nonlinear map, a damped Gauss-Newton step linearizes the residual $r(\theta) = g(\theta) - q$ at the current iterate $\theta_k$ and solves

with $J_k = J(\theta_k)$ and a damping $\mu \ge 0$, where $\mu = 0$ gives the Gauss-Newton step itself. A standard local result, stated here without proof, links the Jacobian to parameter uncertainty. Suppose $q = g(\theta^*) + \varepsilon$ with $\mathbb{E}[\varepsilon] = 0$ and $\operatorname{Cov}(\varepsilon) = \Sigma$ positive definite, take $W = \Sigma^{-1}$, and suppose that $g$ is twice continuously differentiable, that $J = J(\theta^*)$ has rank $p$ at an interior point $\theta^*$, and that the errors are small. Then, to first order,

and an unquoted price $f(\theta)$ has variance about $\nabla f^{\top}\bigl(J^{\top}\Sigma^{-1}J\bigr)^{-1}\nabla f$, with the gradient taken at $\theta^*$. For a linear map with full column rank and a linear unquoted price, both formulas hold exactly, without normality. Neither accounts for model misspecification.

Definition Take a linear map, a penalty weight $\eta > 0$, a prior $\theta_0$ and the identity penalty $\Gamma = I$. Ridge calibration minimizes

Its gradient is $2A^{\top}W(A\theta - q) + 2\eta(\theta - \theta_0)$, so the minimizer solves

For $\eta > 0$ the matrix $A^{\top}WA + \eta I$ is positive definite even when $A$ is rank deficient, so the minimizer $\theta_\eta$ exists and is unique. It satisfies

which lies in the row space of $A$. Hence the component of $\theta_\eta$ along every null direction of $A$ equals that of $\theta_0$, for every $\eta > 0$. As $\eta \to 0^{+}$, the solution tends to the minimizer of the unpenalized weighted loss that is closest to $\theta_0$ in the Euclidean norm, and as $\eta \to \infty$ it tends to $\theta_0$. With a general penalty matrix $\Gamma$ the Hessian is $2(A^{\top}WA + \eta\,\Gamma^{\top}\Gamma)$, and the minimizer is unique exactly when the null spaces of $A$ and $\Gamma$ meet only in $0$. The value of $\eta$ is a further choice, and so is $\theta_0$.

Definition Let $\hat\theta$ be calibrated on a data set $D$, and count every choice made while fitting, including the model, the instruments, the weights, the prior and $\eta$, as part of the fit. A position with value $V$ is hedged by selling $h(\hat\theta)$ units of hedge instruments with prices $H$, where $h$ is sized from the fitted model, often as a vector of model sensitivities. An out-of-sample hedge test measures the hedged profit and loss

over moves or instruments that were not used to make any choice in the fit, with consistent accounting of timestamps, cash flows, funding and costs. The calibration residual measures agreement on $D$ only. Parameters that agree on $D$ but differ along a null or weak direction can give different hedges, and only data outside $D$ can separate them. A held-out test is evidence about the period and the moves it covers, and it bears on future use only to the extent that those moves are relevant. It is not a forecast of hedge performance.