On VIX Futures in the Rough Bergomi Model

Jacquier, Antoine; Martini, Claude; Muguruza, Aitor

doi:10.2139/ssrn.2900248

Cited by 17 publications

(17 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then, recalling that ρ ≤ 0, the proof of [31, Theorem 1], using a localisation argument with the increasing sequence of stopping times τ n := inf{t > 0 : V t = n}, remains the same, and the stock price is a true martingale. This in particular means that a local volatility version of the rough Bergomi model [8,51,52] falls within our current setup.…”

Section: 2mentioning

confidence: 79%

“…Not only is this feature consistent with historical time series [34], but it further allows to capture the notoriously steep at-the-money skew of Equity options, as highlighted in [3,9,8,29,30]. Since then, a lot of effort has been devoted to advocating this new class of models and showing the full extent of their capabilities, in particular as good estimators for a large class of assets [11], and for consistent pricing of volatility indices [46,51]. Nothing comes for free though, and the flip side of this new paradigm is the computational cost.…”

Section: Introductionmentioning

confidence: 86%

See 1 more Smart Citation

Deep PPDEs for Rough Local Stochastic Volatility

Jacquier

Oumgari²

2019

SSRN Journal

Self Cite

View full text Add to dashboard Cite

Section: 2mentioning

confidence: 79%

Section: Introductionmentioning

confidence: 86%

Deep PPDEs for Rough Local Stochastic Volatility

Jacquier

Oumgari²

2019

SSRN Journal

Self Cite

View full text Add to dashboard Cite

“…As noted by many authors (e.g. [16,21]), there might exist an inconsistency between the volatility-of-volatility inferred from SPX and VIX.…”

Section: Introductionmentioning

confidence: 98%

Joint Modelling and Calibration of SPX and VIX by Optimal Transport

et al. 2020

View full text Add to dashboard Cite

This paper addresses the joint calibration problem of SPX options and VIX options or futures. We show that the problem can be formulated as a semimartingale optimal transport problem under a finite number of discrete constraints, in the spirit of [12]. We introduce a PDE formulation along with its dual counterpart. The optimal processes can then be represented via the solutions of Hamilton-Jacobi-Bellman equations arising from the dual formulation. A numerical example shows that the model can be accurately calibrated to the SPX European options and the VIX futures simultaneously.Recently, the theory of optimal transport has proved to be successful for robust hedging and volatility calibration. The discrete-time martingale optimal transport has been applied to derive model-independent bounds of VIX derivatives by De Marco and Henry-Labordere [5]. The theory has been further used to the calibrate the non-parametric discrete-time model proposed by Guyon [15]. In terms of continuous-time optimal transport, the authors of this paper have applied the theory to the calibration of local volatility [13] and local-stochastic volatility models [12] to European options. Furthermore, in [10], the first two authors have extended the semimartingale optimal transport problem [22] to a more general path-dependent setting. Their work expands the available calibration instruments from European options to path-dependent options, such as Asian options, barrier options and lookback options.In this paper, we introduce a time continuous formulation of the joint calibration problem. Identifying suitable state variables, we formulate the joint calibration problem as a semimartingale optimal transport problem under a finite number of discrete constraints, as studied in [12]. Instead of directly modelling the volatility or VIX process, we consider a semimartingale X whose first element X 1 is the logarithm of the SPX price and second element is defined as X 2 t := E(X 1 T | F t ). The filtration F t represents the market information available at time t. Then, the payoff of VIX futures can be expressed in the form of E( X 1 t − X 2 t ). By applying the localisation result of [12], we show that the solutions can be found among a set of Markov processes that the drift and diffusion are functions of t and X t . We then introduce a PDE formulation along with its dual counterpart. Furthermore, we show that the solutions can be represented in terms of solutions of Hamilton-Jacobi-Bellman (HJB) equations arising from the dual formulation.There are two motivations for identifying the conditional expectation as a state variable. Firstly, it makes the joint calibration problem fall into the class of the problems studied in [12]. Secondly, taking X 2 as a state variable, we can use conventional Monte Carlo methods or PDE methods to calculate the prices of VIX derivatives. If we only have X 1 , due to the nonlinearity of the square root function, the evaluation of the conditional expectation cannot be computed by direct Monte Carlo methods. It often req...

show abstract

“…Perhaps, options on volatility itself are the most natural object to first analyse within the class of rough volatility models. In this direction, Jacquier, Martini, and Muguruza [JMM18] provide algorithms for pricing VIX options and futures. Horvath, Jacquier and Tankov [HFT18] further study VIX smiles in the presence of vol-of-vol combined with rough volatility.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics for Volatility Derivatives in Multi-Factor Rough Volatility Models

2019

Self Cite

View full text Add to dashboard Cite

We present small-time implied volatility asymptotics for Realised Variance (RV) and VIX options for a number of (rough) stochastic volatility models via large deviations principle. We provide numerical results along with efficient and robust numerical recipes to compute the rate function; the backbone of our theoretical framework. Based on our results, we further develop approximation schemes for the density of RV, which in turn allows to express the volatility swap in close-form. Lastly, we investigate different constructions of multi-factor models and how each of them affects the convexity of the implied volatility smile. Interestingly, we identify the class of models that generate non-linear smiles around-the-money.Date: March 25, 2019.2010 Mathematics Subject Classification. Primary 60F10, 60G22; Secondary 91G20, 60G15, 91G60. ε > 0, P ε is the joint law and, for each δ > 0, the set ω :and lim sup ε↓0 h ε log P ε (Γ δ ) = −∞, where Γ δ := {(ỹ, y) : d(ỹ, y) > δ} ⊂ Y × Y. Theorem D.2. Let X and Y be topological spaces and f : X → Y a continuous function. Consider a good rate function I : X → [0, ∞]. For each y ∈ Y, define I (y) := inf{I(x) : x ∈ X , y = f (x)}. Then, if I controls the LDP associated with a family of probability measures {µ ε } on X , the I controls the LDP associated with the family of probability measures {µ ε • f −1 } on Y and I is a good rate function on Y.

show abstract

On VIX Futures in the Rough Bergomi Model

Cited by 17 publications

References 22 publications

Deep PPDEs for Rough Local Stochastic Volatility

Deep PPDEs for Rough Local Stochastic Volatility

Joint Modelling and Calibration of SPX and VIX by Optimal Transport

Asymptotics for Volatility Derivatives in Multi-Factor Rough Volatility Models

Contact Info

Product

Resources

About