Analytic Swaption Pricing in the Black-Karasinski Model

Turfus, Colin

doi:10.2139/ssrn.3253866

Cited by 6 publications

(3 citation statements)

References 7 publications

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“…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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Section: Solution Of the Heat Equationmentioning

confidence: 99%

Multilayer heat equations: Application to finance

Itkin¹,

Lipton²,

Muravey³

2022

FMF

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Section: Pricing Zero-coupon Bonds and Barrier Options For The Black-...mentioning

confidence: 99%

Multilayer heat equations: application to finance

Itkin¹,

Lipton²,

Muravey³

2021

Preprint

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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.

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“…However, in the BK model, the price F (t, r) of a ZCB is not known in the 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., (Antonov and Spector, 2011;Capriotti and Stehlikova, 2014;Horvath et al, 2017), and also survey in (Turfus, 2020).…”

Section: T Yearsmentioning

confidence: 99%

From the Black-Karasinski to the Verhulst model to accommodate the unconventional Fed's policy

Itkin,

Lipton,

Muravey

2020

Preprint

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In this paper, we argue that some of the most popular short-term interest models have to be revisited and modified to reflect current market conditions better. In particular, we propose a modification of the popular Black-Karasinski model, which is widely used by practitioners for modeling interest rates, credit, and commodities. Our adjustment gives rise to the stochastic Verhulst model, which is well-known in the population dynamics and epidemiology as a logistic model. We demonstrate that the Verhulst model's dynamics are well suited to the current economic environment and the Fed's actions. Besides, we derive new integral equations for the zero-coupon bond prices for both the BK and Verhulst models. For the BK model for small maturities up to 2 years, we solve the corresponding integral equation by using the reduced differential transform method. For the Verhulst integral equation, under some mild assumptions, we find the closed-form solution. Numerical examples show that computationally our approach is significantly more efficient than the standard finite difference method.1. The diffusion rt is recurrent if and only if q ≤ 0, where q = 1 2 − κ(t) θ(t) σ 2 (t) . 2. If q < 0, the diffusion rt converges in law towards the unique stationary Gamma probability distribution γ −2q, σ 2 (t)/(2κ(t) t→∞ . 3. If q > 0, the diffusion goes a.s. to zero when time goes to infinity.Proof. The proof can be obtained by applying the Itô's lemma to Eq. ( 5) and using Eq. ( 6). The second part follows from Proposition 3.3 in (Giet et al., 2015). It is interesting to note, that the condition q < 1/2 is precisely the Feller condition for the famous CIR model, (Andersen and Piterbarg, 2010).

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Analytic Swaption Pricing in the Black-Karasinski Model

Cited by 6 publications

References 7 publications

Multilayer heat equations: Application to finance

Multilayer heat equations: Application to finance

Multilayer heat equations: application to finance

From the Black-Karasinski to the Verhulst model to accommodate the unconventional Fed's policy

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