Implicit Differentiation for Calibration and Market-Rate Sensitivities

# Implicit function theorem and sensitivities to market risk for Nelson-Siegel-Svensson model


# Implicit function theorem and sensitivities to market risk for Nelson-Siegel-Svensson model












I’m calibrating the Nelson-Siegel-Svensson model to market rates and I’m trying to compute the sensitivities of the NSS parameters to those said rates:

$$r\left(T\right)=\beta_{0}+\beta_{1}{\frac{\left[{1-\exp\left({-T/\lambda}\right)}\right]}{T/\lambda}}+\beta_{2}{\left({\frac{\left[{1-\exp\left({-T/\lambda}\right)}\right]}{T/\lambda}}-\exp\left({-T/\lambda}\right)\right)}+\beta_{3}{\left({\frac{\left[{1-\exp\left({-T/\kappa}\right)}\right]}{T/\kappa}}-\exp\left({-T/\kappa}\right)\right)}.$$

I'm calibrating the model using Levenberg-Marquardt algorithm.

Let's call the parameters $b=\{\beta_0^*,\beta_1^*,\beta_2^*,\beta_3^*,\lambda^*,\kappa^*\}\in\mathbb{R}^{n=6}$, such that, for $m$ observed rates in the market for $m$ maturities ($m>n$), we have:

$$\forall i = 1,...,m, \, f_{i}(r_{i},T_{i};b)=r(T_i;b) - r_i=0.$$

What I want to compute is the sensitivities of $b$ w.r.t ${r_1,...,r_m}$:

$$\frac{\partial b}{\partial r}={\begin{pmatrix}{\dfrac {\partial \beta_0^*}{\partial r_{1}}}&\cdots &{\dfrac {\partial \beta_0^*}{\partial r_{m}}}\\\vdots &\ddots &\vdots \\{\dfrac {\partial \kappa^*}{\partial r_{1}}}&\cdots &{\dfrac {\partial \kappa^*}{\partial r_{m}}}\end{pmatrix}}\in\mathbb{R}^{n\times m}.$$

I found this paper which is the closest to possibly have the answer to my problem:

- From model to market risks: The Implicit Function Theorem (IFT) demystified

But I'm still missing some things. It says the following (page. 3): using the implicit function theorem, we have

$$\underbrace{\frac{\partial b}{\partial r}}_{n\times m}=\underbrace{\left(\frac{\partial I}{\partial b}\right)^{-1}}_{m\times i}\left[\underbrace{\frac{\partial I}{\partial r}}_{i\times n}-\underbrace{\frac{\partial I}{\partial c}}_{i\times p}\underbrace{\frac{\partial c}{\partial r}}_{p\times n}\right]$$

with:

- $I$ being a "set of $i$ calibrated instruments" (?)

- $c$ being a "set of $p$ additional parameters" (?)

That's where I fail to go further. In my specific case, what are my set $I$ and $c$?

## Answer by Kermittfrog (score 4, accepted)

https://quant.stackexchange.com/a/79866

To my understanding, there exist two ways to invoke the IFT in quantitative finance applications in a calibration / bootstrapping context.

- When minimizing some error function during bootstrapping.

- When bootstrapping a set of parameters (e.g. a zero curve) so that all reference instruments are priced exactly to par (e.g. zero PV for a swap),

### IFT in optimization problems

I understand that you tackle the first question, where you minimize a scalar error function. To simplify notation, collect all the parameters into the $K \times 1$ vector $\theta$ and write

$$ \min_{\theta} f(\theta,y) = \sum_i\left(y(T_i)-r(T_i,\theta)\right)^2 $$

Once the minimization of $f$ has been performed, we have obtained some optimal $\theta^*$. From optimization theory, we know that the following optimization condition must hold at the optimum:

$$ g\equiv\left.\frac{\mathrm{d}f(\theta,y)}{\mathrm{d}\theta}\right|_{\theta=\theta^*}=0 $$

This can be used as the starting point for finding the relationship between changes in input data and the calibrated parameters. In layman's terms: When the inputs $y$ change (by a little bit), the optimization routine will still induce that $g=0$ at the optimum, i.e. it stays at zero:

$$ \begin{align} \frac{\mathrm{d}g}{\mathrm{d}y}&= \frac{\mathrm{d}g}{\mathrm{d}\theta}\frac{\mathrm{d}\theta}{\mathrm{d}y}+\frac{\mathrm{d}g}{\mathrm{d}y}=0 \\ \Rightarrow \frac{\mathrm{d}\theta}{\mathrm{d}y}&=-\left(\frac{\mathrm{d}g}{\mathrm{d}\theta}\right)^{-1}\frac{\mathrm{d}g}{\mathrm{d}y} \end{align} $$

Hence, given our scalar optimization problem $f$, the relationship between (small) changes to the inputs and the calibrated parameters are found using derivatives of the first-order-condition.

Let's try this for the NSS example you provide, assuming we want to minimize the squared errors. Further, let us introduce simplifying notation:

- $g_i$ is the the $k\times 1$ gradient of the NSS model at $T_i$, and

- $H_i$ is the $k\times k$ matrix of second derivatives of the NSS model at $T_i$.

Then:

$$ \begin{align} f&=\sum_i(y_i-r(T_i))^2\\ g&\equiv \frac{\mathrm{d}f}{\mathrm{d}\theta}=-2\sum_i(y_i-r(T_i))g_i\}\stackrel{!}{=}0 \end{align} $$ Note that I have used the "$\stackrel{!}{=}0$" rather sloppily to indicate that, at $\theta=\theta^*$, the gradient of the problem must be zero, else we could improve the optimization further. Now we repeat the tools from above, i.e. invoke the notion that, $\mathrm{d}g$ must be zero when changing to another $y$. To this end, we need

$$ \frac{\mathrm{d}g}{\mathrm{d}y}=-2\begin{pmatrix} \frac{\mathrm{d}r(T_1)}{\mathrm{d}\theta} & \frac{\mathrm{d}r(T_2)}{\mathrm{d}\theta} & \ldots & \frac{\mathrm{d}r(T_n)}{\mathrm{d}\theta} \end{pmatrix}\equiv J $$ which is of dimension $k \times n$ and of course $$ \frac{\mathrm{d}g}{\mathrm{d}\theta}=-2\sum_iy_iH_i-g_ig_i^T- r(T_i)H_i\equiv G $$ which is of dimension $k \times k$

Finally, we have

$$ \frac{\mathrm{d}\theta}{\mathrm{d}y}=-G^{-1}J $$ which is of dimension $k \times n$ and to be calculated at $\theta=\theta^*$.

NB: Most optimizers will offer both Hessian and Jacobian of the model as a by-product, AFAIK.

### IFT in standard bootstrapping

A common bootstrapping problem is to find a set of $K\leq N$ parameters $\theta$ (e.g. zero rate nodes in a interest rate curve) so that we can calibrate $N$ reference instruments to their respective quotes, stored in vector $Q$, dimension $N \times 1$.

Let $f$ be a $N\times 1$ vector of reference instruments that (conveniently) have to be priced at par (i.e. zero). Assume that $\theta^*$ has been calibrated such. Any update of $Q$ shall result in an update of $\theta$ such that, again, $f=0$. Then

$$ \frac{\mathrm{d}f}{\mathrm{d}y}=\frac{\mathrm{d}f}{\mathrm{d}\theta}\frac{\mathrm{d}\theta}{\mathrm{d}y}+\frac{\mathrm{d}f}{\mathrm{d}y}\stackrel{!}{=}0 $$

and again

$$ \frac{\mathrm{d}\theta}{\mathrm{d}y}=-\left(\frac{\mathrm{d}f}{\mathrm{d}\theta}\right)^{-1}\frac{\mathrm{d}f}{\mathrm{d}y} $$

If $K<N$, we have to come up with a pseudeo inverse, i.e.

$$ \frac{\mathrm{d}\theta}{\mathrm{d}y}=-\left[\left(\frac{\mathrm{d}f}{\mathrm{d}\theta}\right)^T\frac{\mathrm{d}f}{\mathrm{d}\theta}\right]^{-1}\left(\frac{\mathrm{d}f}{\mathrm{d}\theta}\right)^T\frac{\mathrm{d}f}{\mathrm{d}y} $$

A typical application, then, is to find sensitivites of some instrument $h$ in our portfolio with respect to market observed swap rates. Let $ \theta$ be parameters of the (zero-root) bootstrapped curve, commonly zero rates. Then:

$$ \frac{\mathrm{d}h}{\mathrm{d}q}=\frac{\mathrm{d}h}{\mathrm{d}\theta}\frac{\mathrm{d}\theta}{\mathrm{d}r}=-\frac{\mathrm{d}h}{\mathrm{d}\theta}\left(\frac{\mathrm{d}f}{\mathrm{d}\theta}\right)^{-1}\frac{\mathrm{d}f}{\mathrm{d}y} $$

where we use the "zero sensitivities" of our instrument and the (precomputed) sensitivities of the zero rates w.r.t. the quotes.