A static replication approach for callable interest rate derivatives: mathematical foundations and efficient estimation of SIMM–MVA (Q6576883)

The authors present an algorithm to construct an approximate static replicating portfolio for Bermudan swaptions under an affine term-structure model, and use it to compute Greeks efficiently for the purpose of initial margin (IM) calculation along Monte Carlo paths in the standard initial margin model (SIMM).\N\NA Bermudan swaption gives the holder the right to enter, at any of the dates \(T_{0},\dots, T_{M-1}\), into a fixed-for-floating interest rate swap with maturity at \(T_{M}\) at a pre-specified fixed rate \(K\). The authors assume the short rate is of the form \(r(t) = g(x(t))\) where \(g \colon \mathbb{R}^{d}\rightarrow \mathbb{R}\) and \(dx(t) = \mu (t, x(t)) dt + \eta (t) dW(t)\) with \(\mu \colon [0,T] \times \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}\), \(\eta \colon [0, T ] \rightarrow \mathbb{R}^{d\times d}\), and \(W(t)\) denoting a \(d\)-dimensional Brownian motion.\N\NIM is a protection against exposure changes during the margin period, typically formalized as 99\% value-at-risk of the portfolio change over a \(10\)-day time interval. Under the SIMM, IM, a stochastic process, is to be calculated as the sum of three components: Delta risk, measuring the sensitivity of the portfolio to changes of the risk-free rate at a pre-specified set of tenors; Vega risk, measuring the sensitivity of the portfolio to changes of volatilities at the same set of tenors; and Curvature risk, calculated from Vega risk.\N\NThe static replicating portfolio for a Bermudan swaption is found using the regress-later technique as in [\textit{S. Jain} and \textit{C. W. Oosterlee}, Appl. Math. Comput. 269, 412--431 (2015; Zbl 1410.91486)]. Using backward induction, a static portfolio of options is found which approximately replicates the Bermudan swaption value until it matures or is exercised. By appropriately constraining the structure of the shallow neural network used in the regression, the options in the replicating portfolio are obtained to be European swaptions, for which the Greeks needed in the IM calculation can be computed efficiently.\N\NThe authors illustrate the accuracy of their static replication approach by comparing the resulting time-zero and future sensitivities, IM components, expected positive exposure, and credit value adjustment to those obtained by Monte Carlo methods.