Calibrating the LFM to Caps and Swaptions

A calibrated LFM must simultaneously price caps and swaptions traded in the market. These two sets of instruments identify different parameters: caps identify the instantaneous vol functions $\sigma_k(t)$ (which set the marginal distribution of each $F_k$, see the term structure of volatility), while swaptions identify the instantaneous correlation structure $\rho_{k,j}$ (which governs the joint distribution, see instantaneous and terminal correlation). The standard calibration procedure is therefore a two-step decoupled optimisation: (1) strip caplet vols from cap quotes and set vol parameters $\Phi_k$ or $\eta_{k-j}$; (2) minimise the sum-squared-error between model and market swaption vols (using the Rebonato approximation) by optimising the correlation angles $\theta_k$. This lesson works through both steps with the May 2000 calibration example from IRM Chapter 7, identifies where the TABLE 5 vol parameterisation succeeds and where it fails, and explains the stability-vs-accuracy trade-off when the swaption matrix is large.