Chapter 2 Volatility Dynamics for a Single Underlying : Foundations

In this first and fundamental chapter we lay out the core principles of the Asymptotic Chaos Expansion (ACE) methodology. We investigate the relationship between stochastic instantaneous volatility (SInsV) and stochastic implied volatility (SImpV)models, in the simple case of a single underlying, and when the endogenous driver is scalar. We discuss both the inverse (or recovery) and the direct problem, initially limiting the asymptotic expansion to its lowest order, which we call the first layer. We illustrate these asymptotic results first with the local volatility (LV) class, and then with a comprehensive extension to stochastic volatility (SV) dynamics. In Sect. 2.1, we define the market environment: the underlying, the numeraire and the liquid European options. We define and justify the re-parametrisation of the option price surface via a sliding implied volatility map. We can then introduce both stochastic volatility models (SInsV and SImpV) as well as some sufficient regularity assumptions. Finally we state our objectives, which we split into a direct and an inverse problem. In Sect. 2.2, we establish the fundamental result of this chapter, and of the ACE methodology. This is the Zero Drift Condition (ZDC), a PDE constraining the shape and dynamics of the stochastic implied volatility model in the whole strike/expiry domain, in order to respect the no-arbitrage assumption (NAA). We then specialise that result to the immediate or zero-expiry sub-domain, which leads us to a pair of Immediate ZDCs. Finally we specialise again these results to the Immediate At-TheMoney (IATM) point, which is our most limited but fertile asymptotic, and quote the IATM Identity linking the implied and instantaneous volatilities. In Sect. 2.3, we solve part of the inverse problem, which is to recover the instantaneous model from the implied one. First we establish arbitrage constraints between the coefficients of the SImpV model at the IATM point, which emphasises the structural over-specification of that class. Then we show that at a given level of precision (the first layer, which involves a group of low-order IATM differentials of the smile) the implied model injects itself into the instantaneous class. In Sect. 2.4, we tackle the more popular direct problem, which consists in generating the smile, and more generally the implied model associated to a given instantaneous class. For the first layer, we establish the opposite connection from before, which confirms a full correspondence between the two classes (at that level of © Springer-Verlag London 2014 D. Nicolay, Asymptotic Chaos Expansions in Finance, Springer Finance Lecture Notes, DOI 10.1007/978-1-4471-6506-4_2 23 24 2 Volatility Dynamics for a Single Underlying: Foundations precision). We comment, illustrate and contrast these results against the available literature, within the simple class of local volatility models, thereby exposing some shortcomings in a popular heuristic. In Sect. 2.5 we turn to some practical applications of these results, which we classify as either pure asymptotic, whole-smile or sensitivity-oriented. In terms of pure asymptotics, we define a stochastic instantaneous volatility model class, covering most popular SV models, for which we provide the first layer differentials. In particular we analyse the respective merits and properties of the Lognormal displaced diffusion (LDD) and CEV specifications as skew functions. We also use the pure asymptotic results to re-parametrise such instantaneous volatility models into more intuitive versions, based on the smile that they generate. Then we briefly discuss the caveats involved in extrapolating the whole smile in the naive way, via Taylor expansions. Eventually, in Sect. 2.6 we conclude this chapter and open onto the more specialised subjects covered thereafter. We also provide a diagram gathering the main proofs of this chapter, and hence capturing the main body of the ACE approach, as a basis and a comparison tool for the more advanced results coming thereafter. Note that a general roadmap for this chapter is provided by Fig. 2.4 [p. 113]. 2.1 Framework and Objectives 2.1. 1 Market and Underlyings We consider a market equipped with the usual filtered objective probability space (Ω,F ,P,Ft). Unless explicitly specified otherwise, all processes mentioned thereafter will be assumed to be continuous and Ft-adapted. Although we impose the No-Arbitrage Assumption (NAA), we do not, however, demand market completeness. This choice is obviously motivated by the stochastic volatility specification, and therefore in the sequel the term “risk-neutral measure” should be understood as “chosen risk-neutral measure” with respect to the volatility risk premium. Some considerable literature has been devoted to the economic significance and modelling of this risk premium. We have chosen not to dwell on this interesting subject, since it is of less relevance in our “completed” framework (which includes vanilla options), and also because it presents less interest in the prospect of pure hedging (using dynamically those options). Those readers interested in the subjects of market equilibrium, and market price of volatility risk, can refer to [1] or [2], for instance. As for the theoretical market, we start by considering a single, scalar asset, with continuous price process ̂ St . We also select a numeraire asset Nt , so that under the risk-neutral measure Q and using Lognormal conventions, their dynamics come as d̂ St/ St = rtdt + σtdW t and dNt/Nt = rtdt + λtdW t 2.1 Framework and Objectives 25 with W t a scalar Q-Wiener process, while both σt and λt are undefined but continuous scalar volatility processes. We then define the deflated or rebased asset St and its martingale measure QN by writing St = ̂ St/Nt and dW N t = dW t − λtdt and specifying that W N t be a Q N -Wiener process. But now the process St is not a priori a tradeable asset any more. For all intents and purposes, it should be seen as an index, a reference defining the payoffs of our soon-to-come vanilla options. For that reason, it will be called an underlying, with dynamics coming driftless as dSt St = σtdW N t . (2.1) Nevertheless, as amatter of convention St will often be called the “spot” process in the sequel. Also, regarding notation, we will forgo the QN identifier for the relevant Wiener process and simply denote it by Wt . 2.1. 2 Vanilla Options Market and Sliding Implied Volatility 2.1. 2.1 Definitions and Notations On top of the underlying St and of the numeraire Nt , we now assume a market continuum of pricesC(t, St, T , K) forCall optionswritten on St . Their payoff is either defined or equivalent (using NAA arguments) to the following cashflow, transferred at time T : C(T , ST , T , K) = NT (ST − K)+ . (2.2) The continuum is assumed both in maturity (until a finite horizon Tmax) and in strike (for all K ∈ ] − ∞,∞[). In fact we could consider Puts or Straddles instead of Calls: a smooth continuum in strike is indeed equivalent to assuming that the full marginal distribution is given. Providing it is valid,1 this surface of option prices is associated to an implied volatility mapping Σ(t, St, K, T) via the classical Lognormal re-parametrisation: C(t, St, T , K) = NtC ( St, K,Σ(t, St, K, T). √ T − t )