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Suppose I have a volatility model of the form $\sigma=f(X(x,y), Y(x, y))$ where $f$ is some function of the variables $X, Y$ which are calibrated using some calibration procedure with market implied volatilities $x,y$. My question is about the PnL expansion in terms of the Greeks, mainly, vega.

Is the vega calculated by taking the derivative with respect to all parameters $X,Y,x,y$ such that the vega PnL for an option is $$Vega_{PnL}=\frac{\partial V}{\partial X}dX+ \frac{\partial V}{\partial Y}dY+ \frac{\partial V}{\partial x}dx+ \frac{\partial V}{\partial y}dy?$$ Or is it in fact the partial with respect to the parameters $x,y$ I.e., $Vega_{PnL}= \frac{\partial V}{\partial x}dx+ \frac{\partial V}{\partial y}dy$?

I would have thought it would be the partial with respect to $x, y$ only.

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To get to P&L we first need sensitivities (partial derivatives). For simplicity, let's assume that $V$ only consumes volatility via $g$, that is P&L and Vega P&L are the same thing:

$$ V(x,y) = g(\sigma(x,y)) = g(f(X(x,y), Y(x,y))) = (g\circ f)(X(x,y), Y(x,y))= U(X(x,y), Y(x,y))$$

Note that $$ V = U \circ (X, Y)$$

$$ \frac{\partial \sigma}{\partial x} = \frac{\partial f}{dX}\frac{\partial X}{\partial x} + \frac{\partial f}{\partial Y}\frac{\partial Y}{\partial x} $$

$$ \frac{\partial \sigma}{\partial y} = \frac{\partial f}{\partial X}\frac{\partial X}{\partial y} + \frac{\partial f}{\partial Y}\frac{\partial Y}{\partial y} $$

$$ \frac{\partial V}{\partial x} = \frac{\partial g}{d\sigma}\frac{\partial \sigma}{\partial x} $$ $$ \frac{\partial V}{\partial y} = \frac{\partial g}{d\sigma}\frac{\partial \sigma}{\partial y} $$

$$ \frac{\partial V}{\partial x} = \frac{\partial U}{dX}\frac{\partial X}{\partial x} + \frac{\partial U}{\partial Y}\frac{\partial Y}{\partial x} $$

$$ \frac{\partial V}{\partial y} = \frac{\partial U}{dX}\frac{\partial X}{\partial y} + \frac{\partial U}{\partial Y}\frac{\partial Y}{\partial y} $$

In the P&L context, the Taylor expansion/approximation is used:

$$ V(x,y) - V(x_0,y_0) = \frac{\partial V}{\partial x}(x_0,y_0) (x-x_0) + \frac{\partial V}{\partial y}(x_0,y_0) (y-y_0) $$ $$ + 0.5 \frac{\partial^2 V}{\partial x^2}(x_0,y_0) (x-x_0)^2 + 0.5 \frac{\partial^2 V}{\partial y^2}(x_0,y_0) (y-y_0)^2 $$ $$+ 0.5 \frac{\partial^2 V}{\partial x \partial y}(x_0,y_0) (x-x_0)(y-y_0) + 0.5 \frac{\partial^2 V}{\partial y \partial x}(x_0,y_0) (x-x_0)(y-y_0)$$ $$+ ..., $$ which can be notationally abbreviated as

$$ dV = \frac{\partial V}{\partial x}dx + \frac{\partial V}{\partial y}dy + 0.5 \frac{\partial^2 V}{\partial x^2}(dx)^2 + 0.5 \frac{\partial^2 V}{\partial y^2}(dy)^2 + 0.5 \frac{\partial^2 V}{\partial x \partial y}dxdy + 0.5 \frac{\partial^2 V}{\partial y \partial x}dx dy + ...$$

If one drops the second-order terms, we get the usual:

$$ dV = \frac{\partial V}{\partial x}dx + \frac{\partial V}{\partial y}dy $$

Note that $$ \frac{\partial V}{\partial X}, \; \frac{\partial V}{\partial Y} $$ can be ambiguous notations.

Pricing function $U$ (different from $V$) whose 'inputs' are $X$ and $Y$, as directly observed numbers, has its own P&L: $$ dU = \frac{\partial U}{\partial X}dX + \frac{\partial U}{\partial Y}dY, $$ that is $$ U(X,Y) - U(X_0,Y_0) = \frac{\partial U}{\partial X}(X_0,Y_0) (X-X_0) + \frac{\partial U}{\partial Y}(X_0,Y_0) (Y-Y_0) $$

If we want to involve $U$'s sensitivities in $V$'s P&L calculation, we use (from above list of chain rule applications):

$$ V(x,y) - V(x_0, y_0) = \left(\frac{\partial U}{dX}(X_0,Y_0)\frac{\partial X}{\partial x}(x_0, y_0) + \frac{\partial U}{\partial Y}(X_0,Y_0)\frac{\partial Y}{\partial x}(x_0, y_0)\right) (x-x_0) + \left(\frac{\partial U}{dX}(X_0,Y_0)\frac{\partial X}{\partial y}(x_0, y_0) + \frac{\partial U}{\partial Y}(X_0,Y_0)\frac{\partial Y}{\partial y}(x_0, y_0)\right)(y-y_0), $$ where $X_0 = X(x_0,y_0), Y_0=Y(x_0,y_0)$.

If we want to involve $V$'s sensitivities in $U$'s P&L calculation, we use (assuming invertibility): $$ \begin{bmatrix} \frac{\partial U}{dX} \\ \frac{\partial U}{dY} \end{bmatrix}(X_0,Y_0) = \begin{bmatrix} \frac{\partial X}{dx} & \frac{\partial X}{dy} \\ \frac{\partial Y}{dx} & \frac{\partial Y}{dy} \end{bmatrix}^{-1}(x_0,y_0) \cdot \begin{bmatrix} \frac{\partial V}{dx} \\ \frac{\partial V}{dy} \end{bmatrix}(x_0,y_0) $$

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You observe/mark implied volatilities $x$ and $y$. Your motivation to use a parametric model for vol is to calculate fair price $V$ of some instrument that needs implied volatility $\sigma=z$ that's not directly observable - perhaps some other expiry or moneyness or underlying tenor. (I'll continue using $x,y,z$, but you can have lots more observable and unobservable vols.) You interpolate (or even extrapolate) $z$ from $x$ and $y$, using your model, and making some assumptions about the shape of your volatility surface/cube. (It may seem that you're using $X,Y$ to get $V$ directly and then a little extra work is needed to extract the $z$.)

If your book has the convenient property that the sum of sensitivities of $V$ to each observable vol (assuming we bump one observable vol at a time and don't change others) $\approx$ the sum of sensitivities of $V$ to each vol (observable or not, assuming we bump one vol point at a time and don't change others) $\approx$ the sensitivity of $V$ of all vols being bumped together in parallel, then my suggestion is to report $z$, the sensitivity of $V$ to $z$, the utilization of the market risk limit to the sensitivity of $V$ to $z$, and the P&L attributable to the change in $z$ as the product of the change in $z$ from prior day $\times$ the prior sensitivity of $V$ to $z$. (Nothing so simple may work well for more complicated products, where the vega is materially non-linear, and you need 3rd order risks, cross gammas between vol points, etc.)

You can also supplement these numbers by reporting, as you proposed, the sensitivities of $V$ and of $z$ to each of the observables, assuming other observable are unchanged; and also to your internal model parameters $X,Y$. Someone seeking to understand what caused the P&L is very likely to prefer $z$-based explanation to the observables, but this additional information may help further.

Also, as a practical P&L explain aside/advice, many products have surprisingly material cross gammas between vol, underlying, and time. If you experiment and decide that including these cross gammas in the P&L explain would reduce the unexplained P&L enough to be worth the effort, then for cross gammas you can assume that all vols move in parallel, and not drill down into the structure of the vol surface/cube.

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