Pricing CPPI Capital Guarantees: A Lagrangian Framework

Abstract

A robust computational framework is presented for the risk-neutral valuation of capital guarantees written on discretely-reallocated portfolios following the Constant Proportion Portfolio Insurance (CPPI) strategy. Aiming to address the (arguably more realistic) cases where analytical results are unavailable, this framework accommodates risky-asset jumps, volatility surfaces, borrowing restrictions, nonuniform reallocation schedules and autonomous CPPI floor trajectories. The two-asset state space representation developed herein facilitates visualising the CPPI strategy, which in turn provides insight into grid design and interpolation. It is demonstrated that given a deterministic process for the risk-free rate, the pricing problem can be cast as solving cascading systems of 1D partial integro-differential equations (PIDEs). This formulation’s stability and monotonicity are studied. In addition to making more sense financially, the limited borrowing variant of the CPPI strategy is found to be better suited than the classical (unlimited borrowing) counterpart for bounded-domain calculations. Consequently, it is demonstrated how the unlimited borrowing problem can be approximated by imposing an artificial borrowing limit. For implementation validation, analytical solutions to special cases are derived. Numerical tests are presented to demonstrate the versatility of this framework.