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## Motivation and properties of the Black IV representation

Since the Black formula assumes lognormal dynamics for the underlying asset, re-parameterising with the normalised BS implied volatility seems appropriate when St is not only martingale (under the measure associated to Nt) but also exhibits ”close to lognormal” dynamics. In some practical instances, other simple dynamics such as the normal framework 3 can prove efficient, as will be discussed in Section 2.2.3. But, in most markets, the support of the asset marginal distribution is constrained (or assumed) to be asymmetric, typically bounded on the left. Therefore the (displaced) Black-Scholes implied volatility has proven to be a robust4 candidate for the re-parameterisation of the price map.

In a more general manner, it is in fact the ”implied parameter” approach, which consists in considering prices through a simple ”baseline” model, which allows us to compare ”raw” prices for different strikes and/or maturities. The normal and lognormal dynamics are merely instances of that approach, albeit very common and important ones (more on this in Section 2.2.3).

Another advantage of the IV map over the price map is its regularisation effect, which is ironically a consequence of its more limited definition domain.

Indeed, the Black Scholes formula (1.1.4) is only specified in the domain θ > 0, since at T = t the option price naturally equals the intrinsic payoff [ST − K]+. The latter, however, does not provide C1 regularity at-the-money. This is an issue since our method happens to be of an asymptotic nature. It uses expansions intensively and therefore requires/provides differentials of 3and therefore the Bachelier formula. 4There are other definitions and sources of robustness for Black and Scholes : see [KaSES98] for instance.

some transform of the price, taken precisely at that same IATM point (t, y = 0, θ = 0). Alternatively, if we assume that the implied volatility is well behaved for short maturities, typically if it admits a finite limit in T = t along with a sufficient number of its differentials, we can extend the IV map by continuity5. The Black-Scholes formula itself becomes valid in the full domain and allows to fall back effortlessly onto the intrinsic value. This re-parameterisation effectively contains the irregularity of the price functional to the Black formula itself, allowing the new functional (the implied volatility) to be infinitely smooth, if required.

In other words, the price always exhibit a singularity (a ”kink”) at the IATM point, while the implied volatility can be infinitely smooth. It then becomes clear that an additional and major attraction of re parameterising with the Implied Volatility is that it enables, at low cost, the local regularity that our methodology requires.

In the same vein, the expansion method that we use is necessarily less precise for strikes far from the money. Therefore, since the vega dies out in these regions, using volatility (as opposed to price) expansions artificially limits the resulting pricing error, which is most important tradingwise. In other words, in terms of magnitude the IV is usually a more uniform, precise albeit dangerous (c.f. validity issues) representation than the price itself.

To complement this point, it is also interesting to note that the implied volatility is in practice strongly linked to the delta, hence to the hedge, and especially so in the FX world. As a final word of caution, we stress that the argumentation above is valid for vanilla call and put options, but might not be so for other products, such as binaries : there will be more on this point in Section 2.2.3.1.1.2.3

### Motivation and properties of the sliding representation

We first put this technique in perspective and comment on its relevant mathematical properties. We then discuss the financial attractiveness of this simultaneous ”time and strike” slide. The general concept and the use of relative variables are certainly not new. In the rates environment for instance, it is common practice to denote a Libor rate either with fixed maturity or with fixed accrual6: each notation has its specific pros and cons (see [BGM97] or [Sch05], among others). In an option framework, sliding strikes are also frequently used in order to account for ”stickiness” : certain smiles are ”strike sticky” while most are ”delta sticky”.

Besides, we emphasize that the nature of the benefit brought by this sliding convention, in our specific framework, is more style than substance. Indeed it does not lead to fundamental or technical results which an absolute setting could not reach. Which is a positive feature, since our choice of a strike representation (log-moneyness) is partly subjective and certainly no panacea. It is therefore comforting that our results can practically be transferred to another convention :

the practicalities of this transfer are discussed in section 1.1.3.2. It remains that in principle there are many such ways to define the slide, especially in strike. An obvious candidate is proportional moneyness (K/St), but any other adequate function of K and St can be considered : such adequacy obviously requires a bijectivity in K and also a sufficient regularity, especially at the money. In [Haf04] one can find a general definition for the strike slide, called simply moneyness. But it is stressed therein that the choice should be made on an ad hoc basis, an assertion that we support. Indeed, for a given market, a good parameterisation should provide a smile dynamically as stable and stationary as possible. The overall principle consists in conditioning the smile w.r.t. our only observable state variables, i.e. t and St. For a complementary discussion on this subject, refer to section 2.2.1.1 [p.103]. 5It is for this reason that, in the sequel, any value of the implied volatility taken in θ = 0, typically in (t, y, 0) or (t, 0, 0), must be as a notation abuse, in fact a limit. 6L(t, T,U) vs Lδ(t, T ) △= L(t, T, T + δ) We believe however that our specific choice of a sliding convention is justified, for reasons that we expose now. First of all, and on a mathematical level, we elected to use lognormal dynamics to define an implied parameter : this is in no way mandatory and simply the most common market practice. It leads however to Black’s formula, which itself clearly makes of the log-moneyness y the natural variable to consider.

Also, the results that we present herein are structurally complex, hence any approach that clarifies the interpretation and the role of the various terms is a priori welcome. In particular we find that the sliding representation usually allows a better understanding of stationarity and time-homogeneity issues, especially in the second Part devoted to term structure models. Furthermore, and as will be covered in the various application sections, the practical efficiency of our methodology depends as much on the pure asymptotic results as on the chosen representation of the variables. This is generally true for most extrapolation methods, but also for numerical reasons as well as for the analysis of model behaviour. For the latter in particular, when the model and/or smile specification themselves are (pseudo-) sliding or time-shift homogeneous, we find that better efficiency is attained by using the sliding versions of our results. In more formal terms, the main attraction towards sliding versions of the price and implied volatility mappings lies in the dynamic and stochastic properties brought by this change of coordinates. Indeed, reducing the number of arguments from an absolute representation (four arguments : t, St, T and K) to a sliding one (three arguments only : t, y and θ) effectively ”transfers” the underlying St, and therefore the driver Wt into the functional (here eΣ or e C). Let us quickly illustrate this point with two simple examples.

#### The stochastic volatility models

From now on, we will focus on the Sliding Implied Volatility Surface eΣ(t, y, θ) associated to a given model : we are interested in its shape and also in its joint dynamics with the underlying. In our framework, we specifically want this map to exhibit stochastic dynamics, which should be driven by two orthogonal Wiener processes :

◮ the endogenous driver of the underlying, denoted Wt. In all generality the dimension of this driver will noted as nw, but initially it will be taken as mono-dimensional since our underlying itself has been defined as a scalar. The consequences of relaxing this assumption will be exposed in Section

◮ the exogenous driver −→Z t, which enables movements of the implied volatility surface independently of the underlying dynamics. It is taken as multi-dimensional (with finite dimension nz) to allow for the complex deformation modes observed in practice. By convention, we will take Wt and −→Z t to be independent, and all multidimensional Brownian motions (including −→Z t) to be uncorrelated (i.e. to exhibit a unit covariance matrix). One might question why we chose to express our dynamics along two uncorrelated Wiener processes. Indeed other authors have opted for a single one, unified driver : this is the case for instance in [Dur06].

Clearly this is mathematically insignificant, an purely a matter of presentation. Our view is that it brings two main advantages, for only one drawback.

The first advantage is technical, and is analog to manipulating independent (as opposed to correlated) gaussian vectors. The volatility and correlation structures are then combined in (products of) tensorial coefficients, which simplifies the computation of brackets hd·, d·i. The second advantage is linked to modeling and interpretation. Indeed, we are attached to the incompleteness, endogenous/exogenous interpretation detailed above. Also, for option pricing purposes and certainly in numerical terms, it makes sense to orthogonalise the drivers.

**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

**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**