Volatility dynamics for a single underlying : advanced methods 

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Pure asymptotic applications : qualitative approaches

Let us first turn to asymptotic applications, starting with what we shall call qualitative model design & analysis and which clearly refers to the direct problem. Most practitioners rely on stochastic instantaneous volatility models for pricing and hedging. It is therefore in their interest to better understand these models’ behaviour, in particular the influence or cross-interferences of the various parameters (vol of vol, correlation, mean-reversion, etc.) or of specific functional forms (local volatility, time-dependent parameters, etc.) on the volatility surface, on the joint dynamics of the underlying with its instantaneous volatility, or with the smile, etc.
Such a precise understanding enables the agent to deliver a better hedge, and the modeler to customise an existing model or even design a new one ex nihilo in order to fulfill given trading needs. However most SInsV models currently used22depend on numerical engines to price, whether it be Finite Differences or Monte-Carlo schemes, or even (Fast) Fourier Transform. Hence the difficulty with judging the impact of modeling choices on the smile, both in static and dynamic terms. Alternatively, the approach that we advocate consists in focusing on the IATM region and use a low-level differential approach, by manipulating the usual and meaningful smile descriptors which are level, skew, curvature, slope, etc. Clearly the Σ-(2,0) group generated by Theorem 1.2 provides most of these. We will show in Chapter 2 that, providing simple and realistic assumptions, all IATM differentials can be expressed. In particular Chapter 3 will give the most important ones, such as the twist and flattening (refer to the introduction of that Chapter for a typology of the smile). Very shortly, section 1.5.2 will be dedicated to an illustration of that approach. It will focus on the use of skew functions within stochastic volatility models, and in particular on the comparison of the Lognormal Displaced Diffusion and CEV instances.
Still in their pure asymptotic form, we can also use the formulas in an inverse manner. We can either exploit the recovery formulas provided by the Recovery Theorem 1.1, or try and invert the direct expressions of Theorem 1.2. The point is usually to re-parameterise, either totally or partially, a stochastic instantaneous volatility model by using meaningful, market-related quantities associated to the smile. As for model analysis, the rationale is the difficulty to appreciate the magnitude or impact of parameters in typical SinsV models, such as mean-reversion or vol of vol. Therefore that inverse method transforms the original model into an ”intuitive” version, which is parameterised via its most pertinent output, the smile. Section 1.5.3 will illustrate that method on several simple examples.

Whole smile extrapolations and the pertinence of polynomials

When considering the direct problem, the temptation is great to extrapolate the IATM differ-entials that have been computed (in particular those given by Theorem 1.2) to the whole smile, i.e. for all strikes and all maturities. The instinctive method to exploit these results is to develop Taylor/MacLaurin series, both in strike and in maturity. This is indeed the approach taken by [Dur06] to describe the absolute surface Σ(t S t K T ), and we propose now to mimic that effort. Within our framework, we denote Σ∗(t y θ ) the polynomial approximation of the sliding im-plied volatility Σ(t y θ ), matched at the IATM point (t 0 0). Let us invoke the IATM Identity (1.2.36) along with the static results of Theorem 1.2 : (1.4.51), (1.4.52) and (1.4.53). We then write the MacLaurin series as as the dynamic coefficients b, ν and n also need to ensure the validity of the surface in the future, as well as satisfy the ZDC everywhere. Finally, as approximations of the dynamics, they must tally the dynamics of the (static) approximations : in a nutshell, both groups have to be consistent. At this stage of the study, we choose not to dwell into the various flavours and difficulties of those extrapolations. Some basic considerations will later be discussed in section 4.1 and a practical application will be presented in section 4.5. For now, it suffices to say that Taylor/McLaurin series on the implied volatility and w.r.t. strike are certainly not the only option, and unfortu-nately not as straightforward as we might hope. Also, we can already anticipate that the two hurdles that any approximation will have to tackle are on one hand the validity of the associ-ated price surface (both statically and dynamically), and on the other hand the precision of that proxy w.r.t. the real model. The fact that we are dealing with such approximations argues in itself against pure pricing ap-plications : the risk of inconsistency and of arbitrage could indeed be significant. But on the other hand, these extrapolations can provide good initial guesses that can be used in static or dynamic calibration procedures : indeed, most global calibration procedures use an optimiser, which itself invokes heavily the actual pricer.
More precisely, having selected a (large) collection of options [K T ] for which market/target prices are available, then for a given set of model parameters the market error is defined as a metric24 between the model option prices and the corresponding targets. The optimisation engine is tasked with minimising the market error, as a function of the model parameters, which are our (usually constrained) variables. Any such engine will therefore make numerous calls to the pricer, which itself for complex models usually consists in another numerical engine : FFT, PDE solver (finite differences or finite elements) or Monte-Carlo. That pricer will therefore be much slower than an extrapolation formula based on our asymptotic results, coming in closed form.

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Sensitivities and hedge ratios : Delta, Gamma, Vega & co.

In both application fields mentioned above, i.e. pure asymptotics and extrapolations, the IATM differential results in general, and the first layer in particular, grant access to more than just the smile’s shape and dynamics. Indeed they also provide hedging and risk information in the form of many hedge ratios, including Delta, Gamma, Vega, Volga and Vanna (see p. 30 for definitions). As will shortly be discussed, in a market completed with options it is ultimately the hedge strategy that is the real determinant in terms of pricing and overall risk management. And in the context of unobservable and/or non-tradeable state variables, in particular with SV models (both instantaneous and implied), the very notion of hedge and its relationship to various price differentials becomes ambiguous. With these features in mind, we start by comparing the SImpV class to the dominant type of SInsV models, i.e. parametric diffusions. The ultimate purpose of this model class is to describe the dynamics of the (single) underlying St and a priori of no other financial quantity, especially not of options on St. To express those dynamics, the model is equipped with a finite number of auxiliary state variables (typically some stochastic volatility) gathered under the notation −→ Xt. Restricting ourselves to Itˆo processes, this dynamic system is then driven by the following Markovian parametric diffusion : t →− t −→t −→ ⊥ −→t (1.5.77) dS = f t S X ; β dB.

Table of contents :

Introduction
I Single Underlying 
1 Volatility dynamics for a single underlying : foundations 
1.1 Framework and objectives
1.2 Derivation of the Zero-Drift Conditions
1.3 Recovering the instantaneous volatility : the first layer
1.4 Generating the implied volatility : the first layer
1.5 Illustrations and applications
1.6 Conclusion and overture
2 Volatility dynamics for a single underlying : advanced methods 
2.1 Higher-order expansions : methodology and automation
2.2 Framework extensions and generalisation
2.3 Multi-dimensional extensions, or the limitations of recovery
2.4 Illustration of the vectorial framework : the basket case
3 Meaningful differentials through further layers 
3.1 Some minor but useful results
3.2 Computation of the hyperskew
3.3 Computation of the hypercurve
3.4 Computation of the twist
3.5 Computation of the flattening
3.6 Computation of the arch
3.7 Illustration of the maturity effect
4 Practical applications and testing 
4.1 General considerations on practical applications
4.2 The generic SABR class
4.3 The CEV-SABR model
4.4 The FL-SV class
4.5 Numerical Implementation
II Term Structures 
5 Volatility dynamics in a term structure 
5.1 Framework and objectives
5.2 Derivation of the zero-drift conditions
5.3 Recovering the instantaneous volatility
5.4 Generating the SIV surface : the first layer
5.5 Extensions, further questions and conclusion
6 Implied Dynamics in the SV-HJM framework 
6.1 Definitions, notations and objectives
6.2 Dynamics of rebased bonds
6.3 Bond Options
6.4 Caplets
6.5 Swaptions
6.6 Indirect approaches : assets vs rates
7 Implied Dynamics in the SV-LMM framework 
7.1 Definitions, notations and objectives
7.2 Chaos dynamics of the Zeros in an LMM framework
7.3 Bond options
7.4 Caplets
7.5 Swaptions
7.6 Approximating the swap rate volatility
Conclusion

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