Approximations of Bond and Swaption Prices in a Black–karasiński Model

Abstract

We derive semi-analytic approximation formulae for bond and swaption prices in a Black-Karasiński interest rate model. Approximations are obtained using a novel technique based on the Karhunen-Loève expansion. Formulas are easily computable and prove to be very accurate in numerical tests. This makes them useful for numerically efficient calibration of the model.

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

“…Here we observe that the GTFK approximation is hardly distinguishable from the PDE result up to T = 5, and remains very accurate even for large time horizons. This is also confirmed by the results for zero-coupon bonds (6) reported in Table I illustrating how the GTFK method compares favorably with the results obtained with recently proposed semi-analytical approximations, namely the Exponent Expansion (EE) (Stehlíková and Capriotti, 2014), and the Karhunen-Loéve (KL) expansions (Daniluk and Muchorski, 2016) when benchmarked agains a numerical solution of the associated PDE. In particular, for short time horizons, the GTFK approximation has comparable accuracy with the EE.…”

“…In this region, both α(x) and ω 2 (x) display T EE KL(1) KL(2) GTFK PDE 0.1 0.9939 (0.00%) 0.9939 (0.00%) 0.9939 (0.00%) 0.9939 (0.00%) 0.9939 0.5 0.9681 (0.00%) 0.9681 (0.00%) 0.9681 (0.00%) 0.9681 (0.00%) 0.9681 1.0 0.9331 (0.00%) 0.9331 (0.00%) 0.9331 (0.00%) 0.9331 (0.00%) 0.9331 2.0 0.8581 (0.01%) 0.8580 (0.02%) 0.8581 (0.01%) 0.8582 (0.00%) 0.8582 3.0 0.7845 (0.01%) 0.7842 (0.05%) 0.7844 (0.02%) 0.7847 (0.01%) 0.7846 5.0 0.6595 (0.04%) 0.6582 (0.24%) 0.6593 (0.08%) 0.6602 (0.06%) 0.6598 10.0 -0.4545 (1.69%) 0.4601 (0.48%) 0.4628 (0.10%) 0.4623 20.0 -0.2440 (9.06%) 0.2592 (3.38%) 0.2672 (0.41%) 0.2683 (Stehlíková and Capriotti, 2014), the Karhunen-Loéve (KL) expansion of Ref. (Daniluk and Muchorski, 2016) to first and second order, and by solving numerically the associated PDE. The parameters of the BK process are: mean-reversion speed a = 0.1, level b = ln 0.04, volatility σ = 0.85, and initial rate r0 = 0.06.…”

“…(Stehlíková and Capriotti, 2014), the Karhunen-Loéve (KL) expansion of Ref. (Daniluk and Muchorski, 2016) to first and second order, and by solving numerically the associated PDE. The parameters of the BK process are: mean-reversion speed a = 0.1, level b = ln 0.04, volatility σ = 0.85, and initial rate r0 = 0.06.…”

Small-Time Asymptotics for Gaussian Self-Similar Stochastic Volatility Models

“…For Brownian motion, Brownian bridge, and OU processes, such formulas can be found in [16]. For OU bridges, one can consult [17,15], and for the Gaussian process introduced in [18], the Karhunen-Loève decomposition can be found in the same paper. Unfortunately, even for classical fractional Gaussian processes, e.g., fBm or fOU, the Karhunen-Loève characteristics are not known.…”

The Black‐Karasinski Model: Thirty Years On

“…Another method again is that proposed by Daniluk and Muchorski (2016), based on the Karhunen-Loève representation of the Ornstein-Uhlenbeck process, which gives rise to a small term variance expansion in terms of Hermite polynomials. This they use to infer expressions for zero-coupon bond and swaption prices.…”