They review some of the weaknesses of the model, and concentrate on the problem of negative probabilities induced by the original approximation formula, especially at low strikes. To solve the problem they focus on the probably density function, similar to the approach taken by Andreason and Huge, ZABR — Expansions for the Masses. They find a solution which agrees closely with the original SABR formula but corrects the possibility of negative probabilities. The solution however is not an analytic formula, but a one-dimensional PDE which can be solved numerically using finite difference techniques. Crank-Nicolson can lead to spurious oscillation depending on the geometry of the finite difference grid specifically the Courant Number. A solution is to use alternative finite difference schemes, the TR-BDF2 and Lawson-Swayne schemes work well on this problem, and provide fast stable solutions.
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We'd like to understand how you use our websites in order to improve them. Register your interest. However these prices are only accurate for options with short maturities.
On the other-hand, discretely monitored barrier options cannot be priced using this approach and a numerical technique is required. A novel computational method based on a spectral discretisation of the pricing equation is proposed for the solution of these problems. The high accuracy of the method is first established for special cases of the SABR model where analytical solutions are available and the method is then applied to the pricing of discrete barriers under the arbitrage-free SABR model.
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Correspondence to Muddun Bhuruth. Reprints and Permissions. Comput Econ 54, — Download citation. Accepted : 24 October Published : 30 October
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