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Friday, 28 November 2014 (Electrical Engineering Building, Room 407B)
| 8.00 - 8.45 Registration (Electrical Engineering Building) |
Morning session (Chair: Jan Obłoj)
8.50 - 9.00 Josef Teichmann: Opening remarks
9.00 - 9.45 Claude Martini: Investigating the extremal martingale measures with pre-specified marginals [Slides] [Hide]|
The extremal points in the set of all measures with pre-specified marginals, without the martingale constraint, have been extensively studied by many authors in the past (e.g. Denny, Douglas, Letac, Klopotowski to cite only a few). In this talk, we will focus on the characterization provided by Denny in the countable case: a key property is that the support of the probability Q has no 'cycle', otherwise a perturbation of Q can be constructed so that Q cannot be extremal.
In the context of the two-marginal martingale problem studied by Beiglböck-Juillet, with special cases provided by Henry-Labordère and Touzi, Hobson and Klimmeck, Hobson and Neuberger, and Campi, Laachir, M., we give some properties of the extremal points, and provide a candidate cycle-like property in the countable case. Joint work with Luciano Campi.
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9.45 - 10 COFFEE BREAK
10.00 - 10.45 Mathias Beiglböck: Brenier-type martingale transport plans [Hide]|
Motivated by problems in model-independent finance, different Brenier-type martingale transport plans have been discovered by Hobson, Neuberger, Klimmek, Julliet, Henry-Labordere, Touzi, and Stebegg. We will discuss a new, unifying approach that allows to strengthen the original results and establishes a connection with the Skorokhod embedding problem.
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11.30 - 1.00 LUNCH BREAK
Afternoon session (Chair: Nick Bingham)
1.00 - 1.45 David Hobson: Option pricing, fake Brownian motion and minimising the 1-variation [Slides] [Hide]|
Dupire argues that given a double continuum of call option prices (with sufficient regularity properties) it is possible to construct a martingale diffusion process whose marginal distributions are consistent with those calls. But can we construct other processes with the same one-marginals?
We give such a construction, and argue that the resulting process has mimimal total variation amongst all processes with the given marginals.
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Within a Markovian complete financial market, we consider the problem of hedging a Bermudan option with a given probability. Using stochastic target and duality arguments, we derive a backward numerical scheme for the Fenchel transform of the pricing function. This algorithm is similar to the usual American backward induction, except that it requires two additional Fenchel transformations at each exercise date. We provide numerical illustrations.
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We pursue robust approach to modelling in mathematical nance and investigate the pricing and hedging duality when prices of European options with n maturities are given. Motivated by the notion of prediction set in Mykland [18], we include in our setup modelling beliefs given by a set of paths deemed feasible, i.e. super-replication of a contingent claim is required only for paths falling in the given set. This allows us to interpolate between model-independent and model-specific settings. The former, with n = 1, was considered by Dolinsky and Soner [12] who showed that there is no pricing or hedging duality gap: superhedging price of a path-dependent European option is equal to the value of its martingale optimal transport problem. We obtain a similar duality result in the setup of multiple maturities. When non-trivial beliefs are included the resulting dual and primal problems need to be suitably modified.
The proof proceeds through a discretisation of the problem.
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2.30 - 3.45 COFFEE BREAK
3.00 - 3.45 Beatrice Acciaio: Arbitrage of the first kind and filtration enlargements in semimartingale financial models [Slides] [Hide]|
Given a financial market where no arbitrage profits are possible, and an agent with additional information with respect to it, I investigate whether the extra information can generate arbitrage profits. I will first justify why the right concept of arbitrage to consider here is the so-called Arbitrage of the First Kind (or, equivalently, Unbounded Profit with Bounded Risk). Then I will illustrate a simple and general condition ensuring that no arbitrage is available to the informed agent either. The preservation of No-Arbitrage under additional information is shown for a general semimartingale model both when this information is disclosed progressively in time and when it is fully added at the initial time (which correspond to the initial and to the progressive enlargement of filtration, respectively). In addition, I will provide a characterization of such a stability in a robust context, that is, for all possible semimartingale models.
This talk is based on a joint work with C. Fontana and K. Kardaras.
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This paper evaluates the model risk of models used for forecasting systemic and market risk. Model risk, which is the potential for different models to provide inconsistent outcomes, is shown to be increasing with and caused by market uncertainty. During calm periods, the underlying risk forecast models produce similar risk readings; hence, model risk is typically negligible. However, the disagreement between the various candidate models increases significantly during market distress, further frustrating the reliability of risk readings. Finally, particular conclusions on the underlying reasons for the high model risk and the implications for practitioners and policy makers are discussed.
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To study finance problems in a model-free setting in continuous time we sometimes need to use 'stochastic calculus', for which a pathwise version was provided by Foellmer in the seminal paper 'Calcul d'Ito sans probabilités' (1980). We give an example concerning valuation of variance swaps. This application requires an Ito formula valid for functions more general than C2, leading us to study local time and the Tanaka-Meyer formula in the pathwise setting. We give some new results in this direction. This is joint work with Jan Obłoj and Pietro Siorpaes.
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Saturday, 29 November 2014 (Huxley Building, Room 139)
Morning session (Chair: Mikko Pakkanen)
9.00 - 9.45 Ronnie Sircar: Multiscale Perturbation Methods for Portfolio Choice Problems [Hide]|
Optimal investment in an environment of uncertain and changing market volatility is an issue where mathematics and statistics can and does play a guiding role. We review the Merton portfolio optimization problem, which has been a success story of stochastic control since 1969 in the case when volatility is assumed constant. We then study it in the presence of stochastic volatility using asymptotic approximations when the volatility process is characterized by its time-scales of fluctuation. This leads to a regular-singular perturbation problem for a nonlinear Hamilton-Jacobi-Bellman PDE. The asymptotics shares remarkable similarities with the linear option pricing problem, using the properties of the Merton risk-tolerance function, particularly that is satisfies a fast diffusion PDE. We give examples in the family of mixture of power utility functions, and also we use the asymptotic analysis to suggest a "practical" strategy which does not require tracking the fast-moving volatility factor. We also discuss extensions involving transaction costs, which entails perturbation analysis of an eigenvalue problem.
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9.50 - 10.15 COFFEE BREAK
9.45 - 10.30 Almut Veraart: Stochastic volatility in electricity markets [Hide]|
The talk will focus on the concept of stochastic volatility in electricity markets and will shed some light on the impact of renewable sources of electricity on the electricity price levels and their volatility. Suitable stochastic models will be discussed.
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We introduce a novel class of term structure models for variance swaps. The multivariate state process is characterized by a quadratic diffusion function. The variance swap curve is quadratic in the state variable and available in closed form, greatly facilitating empirical analysis. Various goodness-of-fit tests show that quadratic models fit variance swaps on the S&P 500 remarkably well and outperform affine models. We solve a dynamic optimal portfolio problem in variance swaps, index option, stock index and bond. An empirical analysis uncovers robust features of the optimal investment strategies.
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In this talk we consider the multivariate continuous-time GARCH(1,1) process driven by a Lévy process emphasising stationarity properties. The focus is on the volatility process which takes values in the positive semi-definite matrices.
In the univariate model existence and uniqueness of the stationary distribution as well as geometric ergodicity are well-understood, whereas for the multivariate model only an existence criterion is known as far as strict stationarity is concerned. We shall first review the multivariate COGARCH(1,1) model and its properties focussing on strict and weak stationarity. Thereafter, the main part of the talk is devoted to establishing sufficient conditions for geometric ergodicity and thereby for uniqueness of the stationary distribution and exponential strong mixing.
We follow a classical Markov/Feller process approach based on a Foster-Lyapunov drift condition on the generator. Apart from finding an appropriate test function for the drift criterion, the main challenge is to prove an appropriate irreducibility condition due to the degenerate structure of the jumps of the volatility process, which are all rank one matrices. We present a sufficient condition for irreducibility in the case of the driving Lévy process being compound Poisson.
This talk is based on ongoing joint work with Johanna Vestweber (Ulm University).
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12.30 - 2.00 LUNCH BREAK
Afternoon session (Chair: Alex Mijatović)
2.00 - 2.45 Christian Bayer: Asymptotics beats Monte Carlo: the case of correlated local volatility baskets [Slides] [Hide]|
We consider a basket of options with both positive and negative weights in the case where each asset has a smile, i.e., evolves according to its own local volatility and the driving Brownian motions are correlated. In the case of posi- tive weights, the model has been considered in a previous work by Avellaneda, Boyer-Olson, Busca, and Friz. We derive highly accurate analytic formulas for the prices and the implied volatilities of such baskets.
The computational time required to implement these formulas is under two sec- onds even in the case of a basket on 100 assets. The combination of accuracy and speed makes these formulas potentially attractive both for calibration and for pricing. In comparison, simulation-based techniques are prohibitively slow in achieving a comparable degree of accuracy.
Joint work with Peter Laurence.
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The Heston stochastic volatility model was introduced over 20 years ago [1] and is arguably the most widely used stochastic volatility model in the industry.This is in large part due to analytical tractability and the existence of semi-closed formulae for European option pricing. Furthermore, asymptotics of the (spot) implied volatility smile have been thoroughly studied in Heston, giving insight into the behaviour of model-generated spot smiles. However, there are virtually no analytical results on the dynamics of model implied volatility smiles, a key model-risk metric for assessing the suitability of a model for exotic option pricing.
In this talk we will first derive small and large-maturity asymptotics for the Heston forward implied volatility smile ([2],[3],[4]) using the theory of sharp large deviations (and refinements). We will then use these results to gain insight into some core dynamical properties of the model. We will provide a number of cases of degenerate large deviations behaviour and we will show that it is exactly the analysis of these pathological cases that gives the most insight into the dynamical features of the model.
[1] S.Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options.The Review of Financial Studies, 6 (2): 327-342, 1993. [2] A. Jacquier and P. Roome. Asymptotics of forward implied volatility. Submitted, http://arxiv.org/abs/1212.0779. 2013. [3] A. Jacquier and P. Roome. Large-maturity regimes of the Heston forward smile. Submitted, http://arxiv.org/abs/1410.7206, 2014. [4] A. Jacquier and P. Roome. The small-maturity Heston forward smile. SIAM J. Finan. Math., 4 (1): 831-856, 2013. |
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In robust utility maximization in financial markets one considers a family of reference probability measures (the 'uncertainty set') and usually seeks the robust optimal portfolio and the worst measure in such set. With the case of uncertainty sets arising through moment constraints as main motivation, we provide a novel functional analytic approach based on so-called modular spaces that allow us to recover some of the existing duality results in the literature for the dominated case as well as provide new ones, even without assuming compactness of the uncertainty set as customarily done. Joint work with Joaquín Fontbona.
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We present a class of HJM models, which share numerical tractability with factor models, but allow for consistent re-calibration by today's yield curve. By consistency, we mean that one and the same model is used for simulation, calibration, and estimation of the yield curve.
From a mathematical point of view, a rich enough set of increment processes is described, whose concatenation converges to a limit process.
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4.00 - 4.30 COFFEE BREAK
4.30 - 5.15 Mathieu Rosenbaum: Volatility is rough, Part 1: Empirical facts and microstructural foundations [Slides] [Hide]|
Estimating volatility from recent high frequency data, we revisit the question of the smoothness of the volatility process. Our main result is that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. This leads us to adopt the fractional stochastic volatility (FSV) model of Comte and Renault. We call our model Rough FSV (RFSV) to underline that, in contrast to FSV, H less than 1/2. We demonstrate that our RFSV model is remarkably consistent with financial time series data; one application is that it enables us to obtain improved forecasts of realized volatility. Furthermore, we find that although volatility is not long memory in the RFSV model, classical statistical procedures aiming at detecting volatility persistence tend to conclude the presence of long memory in data generated from it. This sheds light on why long memory of volatility has been widely accepted as a stylized fact. Finally, we provide a quantitative market microstructure-based foundation for our findings, relating the roughness of volatility to high frequency trading and order splitting.
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In 'Volatility is Rough: Part 1', by Gatheral, Jaisson and Rosenbaum we showed that log-volatility behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable time scale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We now show how the RFSV model can be used to price claims on both the underlying and integrated volatility. Moreover, we show how, in the case of SPX, VIX options may be used to calibrate the model parameters. The resulting fit to SPX options is markedly better than that of conventional Markovian stochastic volatility models, and is achieved with fewer parameters.
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