Columbia HomeSchedule for Fall 2024
Seminars are on Thursdays
Time: 4:10 pm - 5:25 pm
Location: Room 903, 1255 Amsterdam Ave.
Building access currently requires CUID or advance notice. Please contact the organizers if you need to be added to the guest list.
Organizers: Steven Campbell, Ioannis Karatzas, Marcel Nutz, Philip Protter
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9/12/2024
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Florian Bourgey (Bloomberg) Title: Smile Dynamics and Rough Volatility Abstract: We investigate the dynamic properties of various stochastic and notably rough volatility models, focusing on the dynamics of implied volatilities. While recent literature has extensively analyzed static properties, such as a model's calibration power or the term structure of ATM skews, dynamic features have received less attention. We focus on the Skew-Stickiness Ratio (SSR), an industry-standard indicator of joint spot price and implied volatility dynamics, pursuing the analysis of [Bergomi, Smile dynamics IV, Risk 2009] and extending it to rough volatility models. Using different numerical estimators, we compare the behavior of the model SSR for several models (not limited to the affine framework) with the empirical market SSR for the SPX Index; this comparison sheds light on the suitability of certain modeling choices. Notably, we observe that Bergomi's original intuition—that a forward variance model with a power-law kernel should generate an SSR with a constant term structure—turns out to be accurate, but only for small volatilities of volatilities. On the contrary, the typical parameter sets required for the calibration of fractional models to the SPX options surface (with high levels of volatilities of volatilities) generate a term structure of the SSR that displays important deviations with respect to the market, leading to preliminary conclusions not in favor of models such as rough Bergomi or rough Heston. |
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9/19/2024
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Ruixun Zhang (Peking University) Title: On Consistency of Signature Using Lasso Abstract: Signatures are iterated path integrals of continuous and discrete-time processes, and their universal nonlinearity linearizes the problem of feature selection in time series data analysis. This paper studies the consistency of signature using Lasso regression, both theoretically and numerically. We establish conditions under which the Lasso regression is consistent both asymptotically and in finite sample. Furthermore, we show that the Lasso regression is more consistent with the Itô signature for time series and processes that are closer to the Brownian motion and with weaker inter-dimensional correlations, while it is more consistent with the Stratonovich signature for mean-reverting time series and processes. We demonstrate that signature can be applied to learn nonlinear functions and option prices with high accuracy, and the performance depends on properties of the underlying process and the choice of the signature. This is joint work with Xin Guo, Binnan Wang, and Chaoyi Zhao.
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9/26/2024
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NO SEMINAR EASTERN CONFERENCE ON MATHEMATICAL FINANCE |
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10/3/2024
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Eduardo Abi Jaber (Ecole Polytechnique) Title: From the Quintic model to signature volatility models Abstract: We will introduce the Quintic Ornstein-Uhlenbeck model that jointly calibrates SPX-VIX options with a particular focus on its mathematical tractability namely for fast pricing SPX options using Fourier techniques. Then, we will consider the more general class of stochastic volatility models where the dynamics of the volatility are given by a possibly infinite linear combination of the elements of the time extended signature of a Brownian motion. First, we show that the model is remarkably universal, as it includes, but is not limited to, the celebrated Stein-Stein, Bergomi, and Heston models, together with some path-dependent variants. Second, we derive the joint characteristic functional of the log-price and integrated variance provided that some infinite-dimensional extended tensor algebra valued Riccati equation admits a solution. This allows us to price and (quadratically) hedge certain European and path-dependent options using Fourier inversion techniques. We highlight the efficiency and accuracy of these Fourier techniques in a comprehensive numerical study |
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10/10/2024
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Zoltan Eisler (Gardening) VIRTUAL-ONLY TALK Enhanced measurement of broker performance via explicit models of execution cost Join Zoom Meeting
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10/17/2024
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Jaehyuk Choi (Peking U Shenzhen) Title: Efficient simulation of the SABR model Abstract: We propose an efficient and reliable simulation scheme for the stochastic-alpha-beta-rho (SABR) model. The two challenges of the SABR simulation lie in sampling (i) the integrated variance conditional on terminal volatility and (ii) the terminal price conditional on terminal volatility and integrated variance. For the first sampling procedure, we analytically derive the first four moments of the conditional average variance, and sample it from the moment-matched shifted lognormal approximation. For the second sampling procedure, we approximate the conditional terminal price as a constant-elasticity-of-variance (CEV) distribution. Our CEV approximation preserves the martingale condition and precludes arbitrage, which is a key advantage over Islah's approximation used in most SABR simulation schemes in the literature. Then, we adopt the exact sampling method of the CEV distribution based on the shifted-Poisson-mixture Gamma random variable. Our enhanced procedures avoid the tedious Laplace inversion algorithm for sampling integrated variance and non-efficient inverse transform sampling of the forward price in some of the earlier simulation schemes. Numerical results demonstrate our simulation scheme to be highly efficient, accurate, and reliable. |
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10/24/2024
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Alexander Schied (Waterloo) Title: Exploring Roughness in Stochastic Processes: From Weierstrass Bridges to Volatility Estimation Abstract: Motivated by the recent success of rough volatility models, we introduce the notion of a roughness exponent to quantify the roughness of trajectories. It can be computed for many stochastic processes and fractal functions and also inspired the introduction of a new class of stochastic processes, the so-called Weierstrass bridges. After taking a look at Weierstrass bridges and their sample path properties, we discuss the relations between the roughness exponent and other roughness measures, such as weighted quadratic variation and Besov regularity. We show furthermore that the roughness exponent can be statistically estimated in a model-free manner from direct observations of a trajectory but also from discrete observations of an antiderivative––a situation that corresponds to estimating the roughness of volatility from observations of the realized variance. As a consequence, we obtain strong consistency theorems in the context of several rough volatility models. The talk is based on joint work with Xiyue Han and Zhenyuan Zhang. |
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10/31/2024
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Johannes Wiesel (CMU) Title: Empirical martingale projections via the adapted Wasserstein distance
Abstract: Given a collection of multidimensional pairs {(Xi,Yi):1≤i≤n}, we study the problem of projecting the associated suitably smoothed empirical measure onto the space of martingale couplings (i.e. distributions satisfying [Y|X]=X) using the adapted Wasserstein distance. We call the resulting distance the smoothed empirical martingale projection distance (SE-MPD), for which we obtain an explicit characterization. We also show that the space of martingale couplings remains invariant under the smoothing operation. We study the asymptotic limit of the SE-MPD, which converges at a parametric rate as the sample size increases if the pairs are either i.i.d. or satisfy appropriate mixing assumptions. Additional finite-sample results are also investigated. Using these results, we introduce a novel consistent martingale coupling hypothesis test, which we apply to test the existence of arbitrage opportunities in recently introduced neural network-based generative models for asset pricing calibration.
This talk is based on joint work with Jose Blanchet, Jonghwa Park, Erica Zhang and Zhenyuan Zhang.
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11/8/2024 |
Walter Schachermayer (Vienna) Joint with probability seminar, details at http://www.math.columbia.edu/d |
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11/14/2024 |
NO SEMINAR |
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11/21/2024 |
Anran Hu (Columbia University)
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11/28/2024 |
NO SEMINAR (Thanksgiving) |
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12/5/2024 |
NO SEMINAR |
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