Abstract
In this chapter we turn to even more practical considerations, by applying ACE results to some popular stochastic (instantaneous) volatility models, namely the SABR and FL-SV classes. We start by discussing the financial, practical and numerical issues involved. We then derive the chaos dynamics of each model, up to the third layer, stressing the technical benefits of staying model-generic and of exploiting induction. We can then express the desired static IATM differentials, which we subsequently use in either direct or inverse mode. In inverse fashion, we use those quantities to illustrate an “intuitive” model re-parametrisation of the generic SABR class. In direct mode, we test the flexibility and quality of static smile approximations provided by ACE for the CEV-SABR model, compared to Hagan et al’s benchmark.
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- 1.
In order to improve Monte-Carlo speed or the exercise boundary definition.
- 2.
Often these methods will use singular perturbation and/or expansions on the vol of vol parameter, which creates the deviation from the Normal or Lognormal model, which itself provides either a manageable infinitesimal generator or the closed-form formula. Pure chaos as in [2] is less frequently found.
- 3.
As described initially in [7] and then in this study.
- 4.
If \(\kappa \) was null then we would have \(a_{12}=0\), which is not the case.
- 5.
See Sect. 3.3 [p. 146] for a more in-depth discussion of the various extensions and alternatives to the mainstream methodology.
- 6.
The backbone is defined as the function \(S_t \rightarrow \varSigma (K=S_t,T)\). In the absence of parameter \(T\) this function will refer to a very short time-to-maturity.
- 7.
Although the IATM SImpV constraints ensure a large amount of redundancy between static and dynamics: see Proposition 2.2 [p. 54].
- 8.
Which is not to say that it has not been published or that this avenue should be abandoned.
- 9.
On EUR markets for instance, quotes for zero-strike floors on nominal rates can be obtained from the brokers.
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Nicolay, D. (2014). Practical Applications and Testing. In: Asymptotic Chaos Expansions in Finance. Springer Finance(). Springer, London. https://doi.org/10.1007/978-1-4471-6506-4_4
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