F. MPCA Vega Hedge
Data Decomposition
In curve PCA, we put together all daily curve movement (row) vectors from the past 3 months and form a data matrix \(X\). We then perform SVD to obtain
where \(T\) is the PC score matrix, and \(V^{\mathsf T}\) is a unitary matrix whose rows are the principal components \(\vec v_1^{\mathsf T}, \vec v_2^{\mathsf T}, \vec v_3^{\mathsf T}, \ldots\).
Similarly, in vol surface MPCA, we put together all daily vol surface movement matrices from the past 3 months to form a 3-dimensional data cube \(Z\). We then perform the higher-order singular value decomposition (HOSVD) which will give us two unitary matrices \(A\) and \(B\), and a 3-dimensional score cube \(U\), such that any specific layer of the score cube \(U^{(k)}\) and its corresponding vol surface movement matrix \(Z^{(k)}\) (from a specific date) satisfy the identity
The unitary matrices \(A\) and \(B\) together play the same role as \(V\) in SVD. Denote the column vectors of \(A\) by \(\vec a_1, \vec a_2, \ldots\), and the column vectors of \(B\) by \(\vec b_1, \vec b_2, \ldots\). The multilinear principal component surfaces are the outer product \(\vec a_i\otimes \vec b_j\).
Application in Portfolio Hedging
In curve PCA, we write tomorrow’s yield curve movement as
where \(t_1, t_2\) and \(t_3\) are random variables with variances estimated from the score matrix \(T\). We further write tomorrow’s PnL attribution as
where \(\vec d\) is today’s EOD portfolio delta ladder. The 3 terms on the right-hand side correspond to PnL contributed by PC1, PC2 and PC3, respectively. Now, to hedge out PC1 risk (that’s PnL contributed by parallel shift of the curve), we add a curve instrument to the portfolio, changing the PnL to
where \(\vec h\) is the delta ladder of the curve instrument added, and \(\beta\) the hedge ratio. To make the portfolio immune to parallel shift of the curve, we just have to find the hedge ratio \(\beta\) such that \(\langle \vec v_1, \vec d+\beta \vec h\rangle = 0\). The hedge ratio is
Similarly, in vol surface MPCA, we write tomorrow’s vol surface movement as
where \(u_{ij}\)’s are random variables with variances estimated from the score cube \(U\). Empirically, on the USD swaption vol surface, the 4 terms combined can often explain over \(90\%\) of the variances of the data. Given today’s EOD vega matrix \(D\), same size as \(Z\), tomorrow’s PnL is sum of elementwise products of \(D\) and \(Z\), which is conveniently given by the trace of \(ZD^{\mathsf T}\), which we denote by \(\text{tr}(ZD^{\mathsf T})\). It can be shown that \begin{align*} \text{tr}(ZD^{\mathsf T}) &\approx \text{tr}\left(\left(\sum_{i=1}^2\sum_{j=1}^2 u_{ij}(\vec a_i\otimes\vec b_j)\right)D^{\mathsf T}\right)\\ &= \sum_{i=1}^2\sum_{j=1}^2 u_{ij}(\vec a_i^{\mathsf T} D \vec b_j). \end{align*} This follows from the property of trace and outer product. To hedge out MPC1 risk, we add one instrument to the portfolio, say a 1y10y swaption with notional \(\beta\), whose vega matrix is \(\beta H\), changing the PnL to \begin{align*} \text{tr}(Z(D+\beta H)^{\mathsf T}) \approx \sum_{i=1}^2\sum_{j=1}^2 u_{ij}(\vec a_i^{\mathsf T} (D+\beta H) \vec b_j). \end{align*} To find the hedge ratio that makes the MPC1 term vanish, we set \(\vec a_1^{\mathsf T} (D+\beta H) \vec b_1 = 0\), which solves to