Pricing CPPI Capital Guarantees: A Lagrangian Framework
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Date
2011-09-28T19:10:54Z
Authors
Morley, Christopher Stephen Band
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
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.
Description
Keywords
Computational Finance, CPPI, Constant Proportion Portfolio Insurance, Capital Guarantees, Numerical methods, Partial integro-differential equations, Valuation, Borrowing limit, Exotic options, Jump diffusion