Approximation Methods for Inhomogeneous Geometric Brownian Motion

Abstract

We present an accurate and easy-to-compute approximation of the transition probabilities and the associated Arrow-Debreu (AD) prices for the inhomogeneous geometric Brownian motion (IGBM) model for interest rates, default intensities or volatilities. Through this procedure, dubbed exponent expansion, transition probabilities and AD prices are obtained as a power series in time to maturity. This provides remarkably accurate results — for time horizons up to several years — even when truncated after the first few terms. For farther time horizons, the exponent expansion can be combined with a fast numerical convolution to obtain high-precision results.

Physics and Derivatives: Effective-Potential Path-Integral Approximations of Arrow-Debreu Densities

“…In this section we illustrate the effectiveness of the GFTK approach by discussing its application to a few diffusions processes of the form (1), starting from two cases in which the method gives exact results, namely the Vasicek and the so-called quadratic short-rate model. We then discuss the Black-Karasinski (BK) (Black and Karasinski, 1991) and GARCH linear SDE model (Capriotti et al, 2019;Li et al, 2018) -for which the AD density (2) or zero-coupon bonds (6) are not know analyticallyby presenting the comparison of the GTFK results with those obtained by solving numerically the relevant PDEs and by employing other approximations.…”

“…ing how the GTFK method compares favorably with the results obtained with recently proposed semi-analytical approximations, namely the EE (Capriotti et al, 2019), when benchmarked agains a numerical solution of the associated PDE. In general, although less accurate than in the BK case, due to the more complex form of the drift potential (65), the approximation produces satisfactory results for maturities up to several years even in regimes of high volatility.…”

“…Accounting for this , the WK and the GTFK effective potentials do agree (Cuccoli et al, 1992;Vaia and Tognetti, 1990). Similarly, GTFK is distinct from the exponential power series expansion of (Makri and Miller, 1989), previously applied successfully in the financial context (Capriotti, 2006;Capriotti et al, 2019;Stehlíková and Capriotti, 2014), and which we will use as one of the benchmarks when discussing our numerical results. With respect to these approaches, the GTFK method has a strong advantage: it still gives a meaningful representation of the thermodynamics down to zero temperature, where it is equivalent to the so-called self-consistent harmonic approximation (Koehler, 1966a,b), that was initially applied to quantum crystal lattices.…”

“…The accuracy of the GTFK method for the GARCH linear SDE is also illustrated for zero-coupon bonds (6) in Table II and III for two (Capriotti et al, 2019), and by solving numerically the associated PDE. The parameters of the process are: mean-reversion level a = 0.1, level b = 0.04, volatility σ = 0.6, and initial rate y0 = 0.06.…”

“…Remarkably, the GTFK method, yielding exact results in the limit of zero volatility and time to maturity as any semi-classical approximation, is also exact whenever the drift potential is quadratic, which means it is exact, as we will recall, for the Vasicek (Vasicek, 1977) and quadratic model (Kakushadze, 2015). We finally illustrate the remarkable accuracy of the GTFK method for models for which an analytical solution is not available via the application to the so-called Black-Karasinski (BK) model (Black and Karasinski, 1991) and the so-called GARCH linear stochastic differential equation (SDE) (Capriotti et al, 2019;Li et al, 2018), both of particular relevance for the valuation of credit derivatives.…”