Abstract
In the low or negative interest rates environment, extending option models to negative rates becomes important. This chapter describes two such extensions of the SABR model: free SABR and mixture SABR. For free SABR, an exact formula is derived for option prices in the case of zero correlation between the rate and its volatility. For nonzero correlation, a mapping procedure onto a mimicking zero-correlation model is applied. Mixture SABR always has a closed-form solution for option prices, and has additional degrees of freedom allowing it to calibrate to a broader set of trades, e.g a set of swaptions and a CMS payment. Analytical results for free and mixture SABR models are compared with the Monte Carlo simulation ones.
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Notes
- 1.
Note that the SABR approximation [39] based on the Heat-Kernel expansion cannot be applied to the free SABR because it does not take into account the boundary conditions.
- 2.
To avoid potential problems related with a non-smooth behavior around \(F_0=10\, K\) we suggest \( \max (K, 0.1F) \approx 0.1F + \frac{1}{2}\left( K - 0.1F + \sqrt{(K-0.1F)^2+\varepsilon ^2} \right) \) for small parameter \(\varepsilon \) around 1bp.
- 3.
We will write here \(\alpha \) instead of \(v_0\).
- 4.
The normal model component can be obviously written via zero beta Free SABR form, \(\mathscr {O}_{N}(T,\, K;\,\alpha _2,\rho _2, \gamma _2)= \mathscr {O}_{F}(T,\, K;\,\alpha _2, 0, \rho _2, \gamma _2)\).
- 5.
As far as the SABR is mainly the strike interpolation (not a term-structure model) its process-level dynamic properties, i.e. conditional expectation \(\mathbb {E}{[(F_T-K)^+\;|\;F_t=f]}\), are irrelevant.
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Antonov, A., Konikov, M., Spector, M. (2019). Extending SABR Model to Negative Rates. In: Modern SABR Analytics . SpringerBriefs in Quantitative Finance. Springer, Cham. https://doi.org/10.1007/978-3-030-10656-0_5
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DOI: https://doi.org/10.1007/978-3-030-10656-0_5
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