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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?

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  • $\begingroup$ How does the number of market factors ro which you need sensitivities compare to the number of model parameters to which which you have sensitivities? Would you be able to calculate a matrix of sensitivities of market factors to model parameters and invert this matrix? $\endgroup$ Oct 29, 2023 at 16:58
  • $\begingroup$ I guess that would depend on how many market factors I would like to get the risk for. In the example of the swaption one choice could be the underlying swaption itself - hence one market variable. Or it could be all of the underlying (LIBOR) forward rates in the swap contract. Depending on the model we might have very few or many parameters. In the simple case of a Vasicek model it would be current level of the short rate $r0$, speed of mean reversion $a$, long term level $b$ and volatility of the short rate $\sigma_r$ (not be confused with $\sigma$ as in the volatility of the swap). $\endgroup$
    – Landscape
    Oct 29, 2023 at 17:05
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    $\begingroup$ OK, consider an extreme example: suppose that a model assumes that an interest rate curve can only change so rigidly that just one model parameter determines the swap rates for all tenors. It would be challenging to find the impact of petrutbung a swap rate at just one tenor ceteris paribus. $\endgroup$ Oct 29, 2023 at 17:26
  • $\begingroup$ I can see that. But does it not matter that we also know the sensitivity of the swap rate wrt. model parameters? I am tempted to use a huristic chain rule like so: $\frac{dV}{dS} = \frac{dV}{dx} \frac{dx}{dS} = \frac{dV}{dx} [\frac{dS}{dx}]^{-1}$ $\endgroup$
    – Landscape
    Oct 29, 2023 at 17:53
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    $\begingroup$ The market spot rates are the par swap rates, for different tenors which are used to build a discount curve, yield curve, and forward curve. Usually, initial value(s) of the internal rate(s) are deduced from the market yield curve (“stripped” from the par swap curve). Insofar, delta is really the market sensitivity to interest rates and generally means the change in price due to a parallel bump in the par swap curve. No need to recalibrate the parameters. In terms of HW1F (which is part of the HJM framework), only change the mean reversion to match the bumped forward. $\endgroup$
    – AKdemy
    Oct 29, 2023 at 19:58

1 Answer 1

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Formally, you have two ingredients:

  1. a pricing function for your specific instrument, $f$, that depends on some set of model parameters $\mathbf{r}$
  2. 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

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

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