quantitative analysis in finance | ||
Current research My current research in quantitative finance is mainly about nonlinear option pricing, the associated numerical probabilistic methods, and volatility and correlation modeling. My main contributions include the following. The VIX Future in Bergomi Models: Analytic Expansions and Joint Calibration with S&P 500 Skew We derive the expansion of the price of a VIX future in various Bergomi models at order 6 in small volatility-of-volatility. We introduce the notion of volatility of the VIX squared implied by the VIX future, which we call "VIX2 implied volatility", expand this quantity at order 5, and show that the implied volatility expansion converges much faster than the price expansion. We cover the one-factor, two-factor, and skewed two-factor Bergomi models and allow for maturity-dependent and/or time-dependent parameters. The expansions allow us to precisely pinpoint the roles of all the model parameters (volatility-of-volatility, mean reversions, correlations, mixing fraction) in the formation of the prices of VIX futures in Bergomi models. The derivation of the expansion naturally involves the (classical or dual bivariate) Hermite polynomials and exploits their orthogonality properties. When the initial term-structure of variance swaps is flat, the expansion is a closed-form expression; otherwise, it involves one-dimensional integrals which are extremely fast to compute. The VIX2 implied volatility expansion is extremely precise for both the one-factor model and the two-factor model with independent factors, even for the very large values of volatility-of-volatility that are usual in equity derivatives markets, and can virtually be considered an exact formula in those cases. We use the new expansion together with the Bergomi-Guyon expansion of the S&P 500 smile to (instantaneously) calibrate the two-factor Bergomi model jointly to the term-structures of S&P 500 at-the-money skew and VIX2 implied volatility. Our tests and the new expansion shed more light on the inability of traditional stochastic volatility models to jointly fit S&P 500 and VIX market data. The (imperfect but decent) joint fit requires much larger values of volatility-of-volatility and fast mean reversion than the ones previously reported in Bergomi (2005, 2016). This is explained in the following article: - The VIX Future in Bergomi Models: Analytic Expansions and Joint Calibration with S&P 500 Skew, submitted, 2020. Preprint available online here. Joint calibration of SPX and VIX options using a dispersion-constrained martingale transport approach Since VIX options started trading in 2006, many researchers have tried to build a model that jointly and exactly calibrates to the prices of S&P 500 (SPX) options, VIX futures and VIX options. So far the best attempts, which used parametric continuous-time jump-diffusion models on the SPX, could only produce an approximate fit. In this talk we solve this puzzle using a nonparametric discrete-time model. Given a VIX future maturity T1, we build a joint probability measure on the SPX at T1, the VIX at T1, and the SPX at T2 = T1 + 30 days which is perfectly calibrated to the SPX smiles at T1 and T2, and the VIX future and VIX smile at T1. Our model satisfies the martingality constraint on the SPX as well as the requirement that the VIX at T1 is the implied volatility of the 30-day log-contract on the SPX. We prove by duality that the existence of such model means that the SPX and VIX markets are jointly arbitrage-free. The discrete-time model is cast as a dispersion-constrained martingale transport problem and solved using the Sinkhorn algorithm, in the spirit of De March and Henry-Labordere (2019). The algorithm identifies joint SPX/VIX arbitrages should they arise. Our numerical experiments show that the algorithm performs very well in both low and high volatility environments. Finally we explain how to handle the fact that the VIX future and SPX option monthly maturities do not perfectly coincide, and how to extend the two-maturity model to include all available monthly maturities. This is explained in the following article: - The Joint S&P 500/VIX Smile Calibration Puzzle Solved, Risk Magazine, April 2020. Available online here. Extended version available online here. Inversion of convex ordering in the VIX market: a remarkable feature of the SPX and VIX markets We investigate conditions for the existence of a continuous model on the S&P 500 index (SPX) that jointly calibrates to a full surface of SPX implied volatilities and to the VIX smiles. We present a novel approach based on the SPX smile calibration condition (the fact that the conditional expectation of the instantaneous variance given the spot equals the market local variance). In the limiting case of instantaneous VIX, a novel application of martingale transport to finance shows that such model exists if and only if, for each time t, the local variance is smaller than the instantaneous variance in convex order. The real case of a 30 day VIX is more involved, as averaging over 30 days and projecting onto a filtration can undo convex ordering. We show that in usual market conditions, and for reasonable smile extrapolations, the distribution of the VIX squared in the market local volatility model is larger than the market-implied distribution of the VIX squared in convex order for short maturities T, and that the two distributions are not rankable in convex order for intermediate maturities. In particular, a necessary condition for continuous models to jointly calibrate to the SPX and VIX markets is the inversion of convex ordering property: the fact that, even though associated local variances are smaller than instantaneous variances in convex order, the VIX squared is larger in convex order in the associated local volatility model than in the original model for short maturities. We argue and numerically demonstrate that, when the (typically negative) spot-vol correlation is large enough in absolute value, (a) traditional stochastic volatility models with large mean reversion, and (b) rough volatility models with small Hurst exponent, satisfy the inversion of convex ordering property, and more generally can reproduce the market term-structure of convex ordering of the local and stochastic squared VIX. This is explained in the following article: - Inversion of Convex Ordering in the VIX Market, to appear in Quantitative Finance, 2019. Available online here. Local volatility does not maximize the price of VIX futures: a consequence of the inversion of convex ordering It has often been stated that, within the class of continuous stochastic volatility models calibrated to vanillas, the price of a VIX future is maximized by the Dupire local volatility model. In this article we prove that this statement is incorrect: we build a continuous stochastic volatility model in which a VIX future is strictly more expensive than in its associated local volatility model. More generally, in this model, strictly convex payoffs on a squared VIX are strictly cheaper than in the associated local volatility model. This corresponds to an inversion of convex ordering between local and stochastic variances, when moving from instantaneous variances to squared VIX, as convex payoffs on instantaneous variances are always cheaper in the local volatility model. We thus prove that this inversion of convex ordering, which is observed in the SPX market for short VIX maturities, can be produced by a continuous stochastic volatility model. We also prove that the model can be extended so that, as suggested by market data, the convex ordering is preserved for long maturities. This is explained in the following article: - Inversion of Convex Ordering: Local Volatility Does Not Maximize the Price of VIX Futures, SIAM Journal on Financial Mathematics 11(1):SC1-SC13, 2020 (with B. Acciaio). Available online here or here. The VIX future in Bergomi models: closed form expansions at order 6 in volatility of volatility We derive the expansion of the price of a VIX future in various Bergomi models at order 6 in small volatility-of-volatility. We introduce the notion of volatility of the VIX squared implied by the VIX future price, expand this quantity at order 5, and show that the implied volatility expansion converges much more quickly than the price expansion. The expansions allow us to precisely pinpoint the roles of volatility-of-volatility and mean reversion in the formation of the prices of VIX futures in Bergomi models. They also shed light on the (in)ability of Bergomi models to jointly calibrate to S&P 500 options, VIX futures, and VIX options. We cover the one-factor, two-factor, and skewed two-factor Bergomi models and allow for maturity-dependent and/or time-dependent parameters. The derivation of the expansion naturally involves the (classical or dual bivariate) Hermite polynomials and exploits their orthogonality properties. When the initial term-structure of variance swaps is flat, the expansion is a closed form expression. Otherwise, it involves one-dimensional integrals which are extremely fast to compute. Often the implied volatility expansion is extremely precise, even for the very large values of volatility-of-volatility that are usual in equity derivatives markets. This is explained in the following article: - The VIX Future in Bergomi Models, preprint, 2020. [Slides] Model-free bounds for VIX futures given S&P 500 smiles using optimal transport We derive sharp bounds for the prices of VIX futures using the full information of S&P 500 smiles. To that end, we formulate the model-free sub/superreplication of the VIX by trading in the S&P 500 and its vanilla options as well as the forward-starting log-contracts. A dual problem of minimizing/maximizing certain risk-neutral expectations is introduced and shown to yield the same value. The classical bounds for VIX futures given the smiles only use a calendar spread of log-contracts on the S&P 500. We analyze for which smiles the classical bounds are sharp and how they can be improved when they are not. In particular, we introduce a family of functionally generated portfolios which often improves the classical bounds while still being tractable; more precisely, determined by a single concave/convex function on the line. Numerical experiments on market data and SABR smiles show that the classical lower bound can be improved dramatically, whereas the upper bound is often close to optimal. This is explained in the following article: - Bounds for VIX Futures given S&P 500 Smiles, Finance and Stochastics, 21(3):593-630, 2017 (with R. Menegaux and M. Nutz). Available online here. Preprint available online here. Cross-dependent volatility Cross-dependent volatility models are a multidimensional generalization of path-dependent volatility models. The idea is very simple and can be summarized as follows: the volatility of asset 1 also depends on the price (or returns) of asset 2. We propose a general framework for pricing and hedging derivatives in cross-dependent volatility (CDV) models, i.e., multi-asset models in which the volatility of each asset is a function of not only its current or past levels, but also those of the other assets. For instance, CDV models can capture that stock volatilities are driven by an index level, or recent index returns. We explain how to build all the CDV models that are calibrated to all the asset smiles, solving in particular the longstanding smiles calibration problem for the 'cross-aware' multidimensional local volatility model. CDV models are rich enough to be simultaneously calibrated to other instruments, such as basket smiles, and we show that the model can fit a basket smile either by means of a correlation skew, like in the classical 'cross-blind' multi-asset local volatility model, or using only the cross-dependency of volatilities itself, in a correlation-skew-free model, thus proving that steep basket skews are not necessarily a sign of correlation skew. We can even calibrate CDV models to basket smiles using correlation skews that are opposite to the ones generated by the classical cross-blind models, e.g., calibrate to large negative index skews while requiring that stocks are less correlated when the market is down. All the calibration procedures use the particle method; the calibration of the implied 'local in basket' CDV uses a novel fixed-point particle method. Numerical results in the case of the FX smile triangle problem illustrate our results and the capabilities of CDV models. This is explained in the following article: - Cross-Dependent Volatility, Risk Magazine, April 2016. Available online here. Extended version available online here. Path-dependent volatility ath-dependent volatility models are a very natural class of models that relates the two quantitities that have a natural scale: asset returns and volatility levels. So far, path-dependent volatility models have drawn little attention from both practitioners and academics compared to local volatility and stochastic volatility models. This is unfair: in this article we show that they combine benefits from both. Like the local volatility model, they are complete and can fit exactly the market smile; like stochastic volatility models, they can produce rich implied volatility dynamics. Not only that: given their huge flexibility, they can actually generate a much broader range of spot-vol dynamics, thus possibly preventing large mispricings, and they can also capture prominent historical patterns of volatility. We give many examples to showcase their capabilities. This is explained in the following article: - Path-dependent volatility, Risk Magazine, October 2014. Available online here. Extended version available online here. Local correlation families We described all the local correlation structures that are consistent with the smile of a basket and the smiles of its constituents, thanks to a novel method that we have coined the "affine transform trick". Allowing correlation to be local, i.e., state-dependent, in multi-asset models allows better hedging by incorporating correlation moves in the delta. When options on a basket, be it a stock index, a cross FX rate, or an interest rate spread, are liquidly traded, one may calibrate a local correlation to these option prices. In this article we introduce a new family of local correlation models from which one can pick a model that not only calibrates to the basket smile but also has extra desirable properties, like fitting a view on correlation skew, mimicking historical correlation, or matching prices of exotic options. The family is built using the particle method and the procedure is easily adapted to calibrate path-dependent volatility models, path-dependent correlation models, and to include stochastic rates, stochastic dividend yield, and stochastic volatility. This is explained in the following article: - Local correlation families, Risk Magazine, February 2014. Available online here. Extended version available online here. The smile of stochastic volatility models: the Bergomi-G. expansion We consider general stochastic volatility models with no local volatility component and derive the general expression of the volatility smile at order two in volatility-of-volatility. We show how, at this order, the smile only depends on three dimensionless numbers whose precise expressions as functionals of the model's spot/variance and variance/variance covariance functions we provide. This is explained in the following article: - Stochastic volatility's orderly smiles, Risk Magazine, May 2012 (with L. Bergomi). Available online here. Preprint ``The smile in stochastic volatility models'' available here. Particle method: The Smile Calibration Problem Solved Following previous work on calibration of multi-factor local stochastic volatility models to market smiles, we show how to calibrate exactly any such models. Our approach, based on McKean's particle method, extends to hybrid models, for which we provide a Malliavin representation of the effective local volatility. We illustrate the efficiency of our algorithm on hybrid local stochastic volatility models. This is explained in the following article: - Being particular about calibration, Risk Magazine, January 2012 (with P. Henry-Labordere). Available online here. Long version ``The smile calibration problem solved'' available here. From spot volatilities to implied volatilities: a heat kernel expansion approach The link between spot volatilities and implied volatilities has been actively investigated in the last two decades. Since the pioneering work of Dupire (1994), one knows how to infer the local volatility function from the implied volatility surface. Inverting this formula, i.e., computing implied volatilities from local volatilities, is not an easy task. The classical solution consists in solving numerically the forward equation for call prices. In this article, we suggest new methods for computing implied volatilities, based on a very general result which expresses the square of the implied volatility as an average over time and space of the square of the spot volatilities. This is explained in the following article: - From spot volatilities to implied volatilities, Risk Magazine, June 2011 (with P. Henry-Labordere). Available online here. Long version available here. Monte Carlo pricing in the Uncertain Volatility Model The uncertain volatility model has long ago attracted the attention of practitioners as it provides worst-case pricing scenario for the sell-side. The valuation of a financial derivative based on this model requires solving a fully non-linear PDE. One can rely on finite difference schemes only when the number of variables (that is, underlyings and path-dependent variables) is small - in practice no more than three. In all other cases, numerical valuation seems out of reach. In this paper, we outline two accurate, easy-to-implement Monte-Carlo-like methods which hardly depend on dimensionality. The first method requires a parameterization of the optimal covariance matrix and consists in a series of backward low-dimensional optimizations. The second method relies heavily on a recently established connection between second-order backward stochastic differential equations and non-linear second-order parabolic PDEs. Both methods are illustrated by numerical experiments. This is explained in the following article: - Uncertain volatility model: a Monte-Carlo approach, Journal of Computational Finance, 14(3), 2011 (with P. Henry-Labordere). Available online here. Preprint available here. Other topics addressed in the book "Nonlinear Option Pricing": - Monte Carlo methods for chooser options; lower and upper bounds (optimal stopping problems, dual formulation) - Pricing of reinsurance deals with uncertainty on lapse and mortality (PDEs, Monte Carlo, backward SDEs) This is explained in Chapters 6 and 8 of the book Nonlinear Option Pricing. Past research Past research topics in quantitative finance include: - Stochastic volatility models: static and dynamic properties - FX options vanilla smile - Constant Maturity Swaps and the smile of swaptions - Time-extrapolations for numerical schemes for SDEs PhD research - Approximation in density of stochastic processes. Application to option pricing and hedging. I have derived functional expansions for the density of the Euler scheme approximating the solution of a regular stochastic differential equation. Such expansions allow infinite differentiation with respect to both space variables, the starting point x and the current point y. They involve functional coefficients and remainders that have gaussian tails. I have also studied the way these coefficients and remainders explode when time tends to zero. I have given applications to option pricing and hedging, proving numerical convergence rates for prices, deltas and gammas on a lognormal stock market. I have proved that applying the Euler scheme of order n to the logarithm of the underlying leads to approximations of the price, the deltas and the gammas which converge to the true price, deltas and gammas with speed 1/n. I have also showed that the principal part of the error explodes as the maturity t tends to 0 as t^{-1/2} for the prices, t^{-1} for the deltas and t^{-3/2} for the gammas. As for the O(1/n^2) remainder, the rates are respectively t^{-2}, t^{-5/2} and t^{-3}. - Stochastic volatility models. I have studied ergodic stochastic volatility models, as presented in the book "Derivatives in Financial Markets with Stochastic Volatility" by J.-P. Fouque, G. Papanicolaou and K. R. Sircar (Cambridge University Press). Besides the theoretical study of such models, I have performed simulations in the case when the mean reversion parameter is not large. Such models generate a wide range of volatility smiles. |