Mapping Model Parameter Sensitivities to Market Quote Risk

# From parameter risk (sensitivities) to market risk (sensitivities)


# From parameter risk (sensitivities) to market risk (sensitivities)












In models where the underlying is not modeled directly - such as in the HJM framework or short rate models - how does one then compute the Greeks, i.e. sensitivites wrt. market variables.

As an example, let's say that I've used a calibrated short rate model and Monte Carlo simulation to find the value of an European Payer Swaption (i.e. a call option on a payer swap). I want to find it's delta and vega, which is the sensitivities of the swaption value with respect to changes in the price and volatility of the underlying, respectivly.

Formally, if we let $V_t$ denote the price of the swaption on the underlying swap $S$, then I am trying to find

$$\frac{\partial V_t}{\partial S} \quad \text{and} \quad \frac{\partial V_t}{\partial \sigma}.$$

In this simuation, I do not have an explicit formula for $V_t$ as a a function of the swap price $S$ and its volatility $\sigma$. However, we do have an explicit function for the swap price $S$ as a function of the model parameters or our state variables. And we also have an implicit function for the price of the swaption $V$ in the form of our Monte Carlo simulation (or if we are lucky some analytical or semi-analytical expression). Hence, we can find the risks / sensitivites wrt. to the models parameters or state varibles.

How can I go from the parameter sensitives to market sensitivites that we would use for risk report or hedging?

## Answer by Kermittfrog (score 3)

https://quant.stackexchange.com/a/77229

Formally, you have two ingredients:

- a pricing function for your specific instrument, $f$, that depends on some set of model parameters $\mathbf{r}$

- a parameterization $\mathbf{F}$ that consistently links model parameters $\mathbf{r}$ to observed quotes $\mathbf{q}$.

For example, $\mathbf{F}$ could represent your swap and short rate models that have been calibrated with respect to observed swap rates and quotes implied volatilities. You can think of $\mathbf{F}$ as a stacked vector of valuation functions (deposits, FRAs, swaps, caps/floors, swaptions). If calibrated correctly, the model must meet observed parameters $\mathbf{c}$ (...which could be traded prices or direct quotes):

$$ \mathbf{r}:\mathbf{F}(\mathbf{r},\mathbf{q})\stackrel{!}{=}\mathbf{c} $$

The subsequent derivation assumes that we have the same number of reference products in $\mathbf{F}$ as we have quotes $\mathbf{q}$ and parameters $\mathbf{r}$.

We are now interested in $\mathrm{d}f$ as a function of $d\mathbf{q}$. From the implicit function theorem

$$ \begin{align} \mathrm{d}\mathbf{F}&=\frac{\partial F}{\partial \mathbf{r}}\mathrm{d}\mathbf{r}+\frac{\partial F}{\partial \mathbf{q}}\mathrm{d}\mathbf{q}\stackrel{!}{=}0\\ \Rightarrow \mathrm{d}\mathbf{r}&=-\left(\frac{\partial F}{\partial \mathbf{r}}\right)^{-1}\frac{\partial F}{\partial \mathbf{q}}\mathrm{d}\mathbf{q} \end{align} $$

where $\frac{\partial F}{\partial \mathbf{r}}$ and $\frac{\partial F}{\partial \mathbf{q}}$ are to be understood as the Jacobians of $F$ w.r.t. rates and quotes. We can now write

$$ \mathrm{d}f=\frac{\partial f}{\partial \mathbf{r}}\mathrm{d}\mathbf{r}=-\frac{\partial f}{\partial \mathbf{r}}\left(\frac{\partial F}{\partial \mathbf{r}}\right)^{-1}\frac{\partial F}{\partial \mathbf{q}}\mathrm{d}\mathbf{q} $$