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Approximation methods for inhomogeneous geometric Brownian motion. (English) Zbl 1411.91549

Summary: 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.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
30B10 Power series (including lacunary series) in one complex variable
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