ARBITRAGE-FREE SABR PDF

The code is not meant for production purpose and does not cater for corner cases. It is only meant to illustrate the main techniques described in the paper. This function will solve the arbitrage-free PDE on a grid using the specified number of space steps N and time steps timesteps, and spanning nd standard deviations below and above. The last parameter specifies the interpolation in between grid nodes. The original Hagan et al. In the future, we may add a quadratic interpolation.

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The code is not meant for production purpose and does not cater for corner cases. It is only meant to illustrate the main techniques described in the paper. This function will solve the arbitrage-free PDE on a grid using the specified number of space steps N and time steps timesteps, and spanning nd standard deviations below and above.

The last parameter specifies the interpolation in between grid nodes. The original Hagan et al. In the future, we may add a quadratic interpolation. Examples Price from the paper: Hagan example We use the same parameters as the example of negative density with the standard SABR formula in Hagan et al. As we used only 50 space steps, we can see clearly the staircase. We use the same parameters as Antonov et al. Because the model allows for negative strikes, we will plot the volatility in the Bachelier model the b.

This spike stems from the model, and not from any artificial numerical error. Convergence table Here, we compute the at-the-money implied volatility for a sequence of doubling time steps and space steps, as well as the ratio of the differences between consecutive results.

A ratio of 4 corresponds to second-order convergence.

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Arbitrage free SABR and near negative rates

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. When implementing one will encounter two problems which can be resolved; Problem 1 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. This problem is not new and can be remedied by a change variable, as suggested in the well-known Anderson-Piterbarg book p.

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