The Free Boundary SABR: Natural Extension to Negative Rates

Antonov, Alexandre; Konikov, Michael; Spector, Michael

doi:10.2139/ssrn.2557046

Cited by 42 publications

(41 citation statements)

References 7 publications

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“…The shifted-SABR model was introduced in Antonov et al (2015) to account for possible negative interest rates in the current low interest environment. Its dynamics are given by…”

Section: Sabr and Shifted-sabrmentioning

confidence: 99%

“…where > 0, ∈ (0, 1), and [dW (1) t dW (2) t ] = dt with −1 < < 1, and for some shift s ≥ 0, so that S t ∈ [−s, ∞). The volatility process follows a geometric Brownian motion (GBM) with volatility , and the local volatility component is given by the constant elasticity variance (CEV) form.…”

Section: Sabr and Shifted-sabrmentioning

confidence: 99%

“…The rate matrix captures the dynamics of t , in a similar manner as one would design a recombining trinomial tree to match transition probabilities at each node (here we will match the local moments of the dynamics d (t) to those of d t ). 2 More specifically, the volatility process is solved by t = 0 exp(− 2 t∕2 + W (2) t ), with moments (t) ∶= [ t ] = 0 and 2 (t) ∶= Var( t ) = 2 0 (exp( 2 t) − 1). With = 4 ∼ 6 fixed, we choose the variance boundary grid points so that the grid is ± standard deviations about (t) with t ∶= √ T∕2: 3…”

Section: Dimension Reductionmentioning

confidence: 99%

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Full‐fledged SABR Through Markov Chains

Cui¹,

Kirkby²,

Nguyen³

2019

Wilmott

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We present a general purpose technique for the efficient and accurate valuation of options in the shifted stochastic alpha, beta, rho (shifted‐SABR) model which includes SABR as a special case. The method is based on a novel double‐layer continuous‐time Markov chain from which closed form matrix expressions for European options are derived. We also propose a recursive risk‐neutral valuation technique for pricing discretely monitored path‐dependent options, and use it to price Bermudian and barrier options. In addition, we provide single Laplace transform formulae for discretely monitored arithmetic Asian options. Numerical experiments confirm the accuracy and efficiency of the proposed method, which is suitable for practical use.

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“…The shifted-SABR model was introduced in Antonov et al (2015) to account for possible negative interest rates in the current low interest environment. Its dynamics are given by…”

Section: Sabr and Shifted-sabrmentioning

confidence: 99%

Section: Sabr and Shifted-sabrmentioning

confidence: 99%

Section: Dimension Reductionmentioning

confidence: 99%

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Full‐fledged SABR Through Markov Chains

Cui¹,

Kirkby²,

Nguyen³

2019

Wilmott

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“…Several variations of this formula are used in practice [4][5][6][7][8][9][10][11][12][13], but they all agree to within O ( 2 ) . The SABR model usually fits the observed smile N ( T ex , K ) quite well at any given expiry T ex , but fitting smiles at different expiries T ex usually requires a different set of SABR parameters ( , , ) for each expiry.…”

Section: Echnical Papermentioning

confidence: 99%

Effective Media Analysis for Stochastic Volatility Models

Hagan¹,

Lesniewski²,

Woodward³

2018

Wilmott

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Analysis of the standard SABR model leads to an effective forward equation which has time-independent coefficients, and analysis of this reduced-dimensionality equation leads to explicit asymptotic formulas for the implied normal volatilities of European options. These formulas are accurate to within O( 2 ), and are used extensively in practice for pricing and managing the risks of European options. A recent analysis of the dynamic SABR model leads to an identical effective forward equation, except that the coefficients are time-dependent. Here we use singular perturbation methods to analyze this new equation. For each exercise date T ex , we derive a set of constant coefficients. Replacing the time-dependent coefficients with the constant coefficients, and solving the effective forward equation, yields the correct probability density function -and thus the correct European option values -to within O ( 2 ) . These constant coefficients now let us apply the existing analysis to obtain explicit asymptotic formulas for the implied volatilities of European options, accurate to within O ( 2 ) , for the dynamic SABR model. Current analyses of several other stochastic volatility models, including the -SABR and generalized Heston models, also yield identical effective forward equations with time-dependent coefficients. Therefore, the implied volatilities for these models are also given by the SABR implied volatility formulas.

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“…Furthermore, the distribution shift had to be recalibrated each time there were significant negative rate moves in order to guarantee non-negativity. It was only in 2015 that an elegant new solution was introduced (Antonov, Konikov, and Spector, 2015). In The free boundary SABR, the authors proposed an extension of the SABR model in which rates can go negative, with no need to decide in advance how negative.…”

Section: Finding Sigmamentioning

confidence: 99%

A brief history of quantitative finance

Cesa¹

2017

Probab Uncertain Quant Risk

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which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.Abstract In this introductory paper to the issue, I will travel through the history of how quantitative finance has developed and reached its current status, what problems it is called to address, and how they differ from those of the pre-crisis world.

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The Free Boundary SABR: Natural Extension to Negative Rates

Cited by 42 publications

References 7 publications

Full‐fledged SABR Through Markov Chains

Full‐fledged SABR Through Markov Chains

Effective Media Analysis for Stochastic Volatility Models

A brief history of quantitative finance

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