Analytic Option Prices for the Black-Karasinski Short Rate Model

Horvath, Blanka; Jacquier, Antoine; Turfus, Colin

doi:10.2139/ssrn.3253833

Cited by 15 publications

(14 citation statements)

References 10 publications

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“…The calculation for a risk-free cash flow in our model is very similar to that performed by Horvath et al (2017) and essentially corresponds to taking the distinguished limit as λ → 0 then r → 0. The same result is naturally obtained, namely that f T t = X T (x, t), where, with the convention that…”

Section: Appendix a Green's Functionmentioning

confidence: 94%

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Quantifying Correlation Uncertainty Risk in Credit Derivatives Pricing

Turfus

2018

IJFS

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We propose a simple but practical methodology for the quantification of correlation risk in the context of credit derivatives pricing and credit valuation adjustment (CVA), where the correlation between rates and credit is often uncertain or unmodelled. We take the rates model to be Hull-White (normal) and the credit model to be Black-Karasinski (lognormal). We summarise recent work furnishing highly accurate analytic pricing formulae for credit default swaps (CDS) including with defaultable Libor flows, extending this to the situation where they are capped and/or floored. We also consider the pricing of contingent CDS with an interest rate swap underlying. We derive therefrom explicit expressions showing how the dependence of model prices on the uncertain parameter(s) can be captured in analytic formulae that are readily amenable to computation without recourse to Monte Carlo or lattice-based computation. In so doing, we crucially take into account the impact on model calibration of the uncertain (or unmodelled) parameters.

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Section: Appendix a Green's Functionmentioning

confidence: 94%

“…with in general lim t→T − f T t = P(x T ) for some payoff function P(x). 3 In the absence of closed form solutions to (19) and guided by the work of Hagan et al (2005), Pagliarani and Pascucci (2011) and Horvath et al (2017), we propose a perturbation expansion approach as follows.…”

Section: Derivation Of Governing Pdementioning

confidence: 99%

Quantifying Correlation Uncertainty Risk in Credit Derivatives Pricing

Turfus

2018

IJFS

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“…(28) reduce the number of unknown variables to 2N − 2, because χ + i (τ ), Θ i (τ ) can be expressed via χ − i (τ ), Ω i (τ ) and substituted into Eq. (20). Thus, the GIT method provides a significant simplification of the system of Volterra equations as compared with the HP method.…”

Section: Solution Of the Heat Equationmentioning

confidence: 99%

“…However, in the BK model, the price P (t, T ) of a (ZCB) with the maturity T is not known in closed form since this model is not affine. Multiple good approximations have been developed in the literature using asymptotic expansions of various flavors, see, e.g., [3,44,20], and also survey in [46].…”

Section: Solution Of the Heat Equationmentioning

confidence: 99%

Multilayer heat equations: Application to finance

Itkin¹,

Lipton²,

Muravey³

2022

FMF

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<p style='text-indent:20px;'>In this paper, we develop a Multilayer (ML) method for solving one-factor parabolic equations. Our approach provides a powerful alternative to the well-known finite difference and Monte Carlo methods. We discuss various advantages of this approach, which judiciously combines semi-analytical and numerical techniques and provides a fast and accurate way of finding solutions to the corresponding equations. To introduce the core of the method, we consider multilayer heat equations, known in physics for a relatively long time but never used when solving financial problems. Thus, we expand the analytic machinery of quantitative finance by augmenting it with the ML method. We demonstrate how one can solve various problems of mathematical finance by using our approach. Specifically, we develop efficient algorithms for pricing barrier options for time-dependent one-factor short-rate models, such as Black-Karasinski and Verhulst. Besides, we show how to solve the well-known Dupire equation quickly and accurately. Numerical examples confirm that our approach is considerably more efficient for solving the corresponding partial differential equations than the conventional finite difference method by being much faster and more accurate than the known alternatives.</p>

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“…This work was extended to address swaption prices under the B-K model by Turfus (2018a). The B-K expansion of Horvath et al (2017) was subsequently extended to an explicit expression for the pricing kernel to infinite order by Turfus (2018b). From this result can easily be inferred zero-coupon bond prices to any required degree of accuracy; option prices require some extra effort.…”

Section: Low-rate Approximationmentioning

confidence: 99%

The Black‐Karasinski Model: Thirty Years On

Turfus¹

2021

Wilmott

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This year marks the 30th anniversary of the publication of the seminal short-rate model of Black and Karasinski (1991). Colin Turfus looks back over the early career of its co-originator Piotr Karasinski and records his story of how the model came to be developed, going on to review the impact it had subsequently, and the reasons behind the revival of interest which has been evident in recent years. 1

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Analytic Option Prices for the Black-Karasinski Short Rate Model

Cited by 15 publications

References 10 publications

Quantifying Correlation Uncertainty Risk in Credit Derivatives Pricing

Quantifying Correlation Uncertainty Risk in Credit Derivatives Pricing

Multilayer heat equations: Application to finance

The Black‐Karasinski Model: Thirty Years On

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