Large-Step Monte Carlo Simulation for SABR-LMM

Summary

The document describes a simulation question for a SABR-LMM model under a selected numeraire. Forward LIBOR rates have shifted SABR dynamics, stochastic lognormal volatilities, and correlated Brownian drivers. The author already uses Euler discretization with predictor-corrector drift estimates and reports that it works reasonably for six-month steps when pricing swaptions, but seeks a fast and accurate approach for larger time steps.

It presents no proposed scheme, comparison, or additional numerical evidence; it is a request for guidance rather than a worked method. The stated experience only supports the current discretization at the shorter step size. Performance and accuracy for larger steps remain unresolved, and any recommended method would need to account for the model’s correlations, drift structure, and swaption-pricing objective.

Key ideas

• The model combines shifted SABR dynamics for forward LIBOR rates with stochastic volatility.
• The rate and volatility Brownian drivers have specified cross-correlations.
• Euler discretization with predictor-corrector drift approximation is reported to work reasonably at six-month steps for swaption pricing.
• The document asks for a fast, reliable Monte Carlo scheme at larger time steps but does not provide one.

Tags

# SABR-LMM: best way to perform a MC simulation


# SABR-LMM: best way to perform a MC simulation












I am working on a SABR-LMM model with the following system of SDEs under a numeraire $N$:

$$ \begin{align} &\mathrm{d} F_i(t) = \sigma_i (t) (F_i(t) + s)^{\beta} \Big( \mu^f_i (t) \mathrm{d}t + \mathrm{d} W^{N}_i(t) \Big) \ , \notag \\ &\mathrm{d} \sigma_i(t) = \nu_i \sigma_i(t) \Big( \mu^\sigma_i (t) \mathrm{d}t + \mathrm{d} Z^{N}_i(t) \Big) \ , \notag \\ &E \big\{\mathrm{d} W_i(t) \mathrm{d} Z_j(t)\big\} = \rho_{ij} \mathrm{d} t \ , \label{Eq. SABR-LMM} \\ &E \big\{\mathrm{d} W_i(t) \mathrm{d} W_j(t) \big\} = \gamma_{ij} \mathrm{d}t \ , \notag \\ &E \big\{\mathrm{d} Z_i(t) \mathrm{d} Z_j(t) \big\} = \phi_{ij} \mathrm{d}t \ , \notag \end{align} $$

where $F_i (t)$ is the forward LIBOR rate fixing at $T_{i-1}$. Assuming that I already have a way to calibrate the model and that I have chosen a numeraire, what is the best way to perform a large-step Monte-Carlo simulation (timestep > 6M)? Until now I have performed a simple Euler-discretization with predictor-corrector approach for approximating the drifts. This seems to work relatively well for 6M steps when I price swaptions, however I fail to find a good and fast simulation scheme for steps larger than 6 months.

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