Bibliography of the history of technology

Neu, John

doi:10.1016/0016-0032(70)90276-0

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Appendices a Derivation Of The Effective Forward Equation1

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2014

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“…We will successively transform this equation order by order until all the derivatives are negligibly small [17,18]. Instead of constructing explicit asymptotic solutions to the resulting problem, as was done in the original SABR paper [1], here we seek to write Q (2) in terms of Q (0) .…”

Section: Appendices a Derivation Of The Effective Forward Equationmentioning

confidence: 99%

Arbitrage-Free SABR

Hagan¹,

Kumar²,

Lesniewski³

et al. 2014

Wilmott

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Smile risk is often managed using the explicit implied volatility formulas developed for the SABR model [1]. These asymptotic formulas are not exact, and this can lead to arbitrage for low strike options. Here we provide an alternate method for pricing options under the SABR model: We use asymptotic techniques to reduce the SABR model from two dimensions to one dimension. This leads to an effective one-dimensional forward equation for the probability density which has the same asymptotic order of accuracy as the explicit implied volatility formulas. We obtain arbitrage-free option prices by numerically solving this PDE. The implied volatilities obtained from the numerical solutions closely match the explicit implied volatility curves, apart from a boundary layer at very low rates. For very low-rate environments, or for very low strikes, the implied absolute (normal) volatility dips downward, closely matching market observations. We also show how negative rates can be accommodated by replacing the F factor with (F + a) .

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Section: Appendices a Derivation Of The Effective Forward Equationmentioning

confidence: 99%