Minicourses
Jim Gatheral: Rough volatility
The scaling properties of historical volatility time series, which now appear to be universal, together with the scaling properties of implied volatility smiles, motivate a new class of stochastic volatility models where paths of volatility are rougher than those of Brownian motion. Rough volatility connects the microstructure of financial markets with the large-scale behavior of volatility as encoded in the implied volatility surface. Rough volatility models are typically very parsimonious yet are in remarkable agreement with both econometric data and option prices. Practical applications include efficient forecasting of future integrated variance and option pricing.
This mini-course will include the following topics:
Background material of the lectures
Paolo Guasoni: Long term investments
Individuals planning for retirement and institutions investing for present and future generations face significantly different risks than short or medium term investors with only a few years' horizons. This minicourse will explore the challenges of investing and spending over a long horizon in the presence of (i) fluctuating investment opportunities; (ii) reliance on the steady spending levels, i.e., loss aversion; (iii) aging and health spending; and (iv) permanent vs. temporary wealth shocks. Fluctuations in investment opportunities involve both variations in interest rates and in risk premia of asset classes. Such variations imply that a long run investor does not make investment decisions merely on the prospect of the immediate return and risk of an asset, but also in relation to the assets' own covariations with shocks in investment opportunities. We discuss the mathematical techniques that make such models tractable and their resulting intuition and implications for asset pricing. Reliance on steady spending is typical of university endowments, sovereign funds, and other foundations that fund long-term commitments that cannot be easily reduced before their original schedule. Family offices and trust funds may also face similar challenges. For such investors, it is critical to design and finance a spending schedule that is significantly less volatile than wealth, even if this means that spending must be significantly lower in earlier times. We see how to embed such a preferences in a dynamic optimization problem, and how to derive rational and self-financed spending policies and their investment counterparts. Health expenses rise significantly in retirement and are different from other forms of consumption - they do not generate utility per se, but rather help reducing mortality, thereby increasing lifetimes. We solve a model of investment, consumption, and healthcare spending in the presence of aging, with mortality increasing according to the Gompertz' law, and healthcare partially reducing mortality growth. While portfolio composition is insensitive to aging, the composition of expenditures markedly shifts toward health expenses in time of retirement.
Some asset classes entail exposure to shocks that gradually recede over time, in contrast to the permanent risks that are assumed in most standard models. We discuss how portfolio choice intuition changes for such asset classes, and the preference restrictions that lead to a the different treatment of non-permanent shocks. We conclude with a summary of results and an overview of open questions and applications. (slides)
Special invited lectures
Jean-Philippe Bouchaud:
Market impact: a review
Price impact refers to the correlation between an incoming order (to buy or to sell) and the subsequent price change. That a buy (sell) trade should push the price up (down) is intuitively obvious and is easily demonstrated empirically. Such a mechanism must, in fact, be present for private information to be incorporated into market prices. But it is also a sore reality for trading firms for which price impact induces large (but often overlooked) extra costs. Monitoring and controlling impact has therefore become one of the most active domains of research in quantitative finance since the mid-nineties. A large amount of empirical results has accumulated over the years concerning the dependence of impact on traded quantities, the time evolution of impact, the impact of metaorders, cross-impact, etc. In this lecture, I will present some of the most salient empirical findings, and a variety of theoretical ideas that have been proposed to rationalise them. Some remaining puzzles and open problems will be discussed as well. (slides) References [1] M. Benzaquen & J.-P. Bouchaud (2018), Market impact with multi-timescale liquidity, Quantitative Finance, 18:11, 1781-1790. (pdf version) [2] J. Donier, J. Bonart, I. Mastromatteo and J.-P. Bouchaud (2015), A fully consistent, minimal model for non-linear market impact, Quantitative Finance, 15(7), 1109-1121. (pdf version)
Stéphane Crépey:
When capital is a funding source: the XVA anticipated BSDEs
XVAs refer to various financial derivative pricing adjustments accounting for counterparty risk (CVA) and its funding (FVA) and capital (KVA) implications for a bank. We show that the XVA equations are well posed, including in the realistic case where capital (including capital at risk) is deemed fungible as a source of funding for variation margin. This intertwining of capital at risk and FVA, added to the fact that the KVA is part of capital at risk, lead to a system of anticipated McKean BSDEs (ABSDEs) for the FVA and the KVA, with coefficients entailing a conditional risk measure of the one-year-ahead increment of the martingale part of the FVA. We first consider the resulting XVA ABSDEs in the case of a hypothetical bank without debt. In the practical case of a defaultable bank, the XVA ABSDEs, which are stopped before the default of the bank, are solved by reduction to a reference stochastic basis, corresponding to the pricing model used by the base (as opposed to XVA) traders of the bank. This reduction methodology, which is of independent interest, generalizes several previous default intensity pricing formulas in the credit risk literature. The FVA reduction provided by the use of capital as a funding source is found very significant numerically, as high as one half or more on a real banking (uncollateralized) portfolio. (slides) References [1] Crépey, S., R. Élie, W. Sabbagh, and S. Song (2018). When capital is a funding source: The XVA Anticipated BSDEs, working paper. [2] Albanese, C., S. Caenazzo, and S. Crépey (2017). Credit, funding, margin, and capital valuation adjustments for bilateral portfolios. Probability, Uncertainty and Quantitative Risk 2(7).
Roger Lord:
Optimal contours and controls in semi-analytical option pricing revisited
For models with analytically available characteristic functions, Fourier inversion is an important computational method for a fast and accurate calculation of plain vanilla option prices. To improve the numerical stability of the inversion, Lord and Kahl [2007] suggested a method to find an optimal contour of integration. Joshi and Yang [2011] built on Andersen and Andreasen's [2002] suggestion, and used the Black-Scholes formula as a control variate. This presentation gives an overview of work done in this area, and shows new work on the effectiveness of combining controls and contours. Joint work with Christian Kahl. References [1] Lord, R. and C. Kahl (2007). Optimal Fourier Inversion in semi-analytical option pricing, Journal of Computational Finance, 10(4), 1-30. (available at SSRN). [2] Joshi, M. and C. Yang (2011). Fourier Transforms, Option Pricing and Controls (available at SSRN). [3] Andersen, L.B.G. and J. Andreasen (2002). Volatile volatilities. Risk, 15(12):163-168. Short lectures
Misha van Beek:
Conditional scenario generation
Scenario generation is an integral part of financial economics, and often relies on Monte-Carlo simulation. However, many application require conditional paths rather than unconditional simulation. For example, in policy analysis a central bank is interested in future scenarios conditional on a certain policy implementation. In stress testing, banks and insurers want to know future paths conditional on an impending recession. In portfolio construction, portfolio managers look to construct portfolios conditional on analyst views of certain assets. In this talk I bring together macro-economic models and factor models of asset returns in a common framework, and show how to derive analytical distributions of future macro-economic variables and asset returns, conditional on any set of views about the future of the economy, factor returns or asset returns. I show that the framework is a multi-period, multi-factor, macro-informed generalization of the Black-Litterman model.
Anastasia Borovykh:
Pricing Bermudan options under local Lévy models with default
In this presentation we will discuss an efficient method for pricing Bermudan options under a so-called local Lévy model. We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Lévy-type martingale. This class of models allows for a local volatility, local default intensity and a locally dependent Lévy measure, therefore being a very flexible framework for modelling asset dynamics. We present a pricing method for Bermudan options based on an analytical approximation of the characteristic function combined with the COS method. Due to a special form of the obtained characteristic function the price can be computed using a fast Fourier transform-based algorithm resulting in a fast and accurate calculation. (slides) References [1] Matthew Lorig, Stefano Pagliarani, Andrea Pascucci, A family of density expansions for Lévy-type processes. [2] Anastasia Borovykh, Andrea Pascucci, Cornelis W. Oosterlee, Pricing Bermudan options under local Lévy models with default.
Shuaiqiang Liu:
Pricing options and computing implied volatilities using artificial neural networks
We proposed a data-driven approach, by means of an Artificial Neural Network (ANN), to value financial options and calculate implied volatilities with the aim of accelerating the corresponding numerical methods. In order to solve advanced financial asset models, different numerical methods have been developed, e.g. finite differences PDE techniques, Fourier methods and Monte Carlo simulation. However, fast and efficient computation is increasingly important, especially in the case of model calibration or financial risk management. It is well-known that ANNs are powerful universal function approximators without assuming any mathematical form. Recent advances in data science have shown that by deep learning techniques even highly nonlinear multi-dimensional functions can be accurately represented. Besides, training an ANN is expensive, but running the trained ANN is cheap. Based on the above facts, we develop an ANN solver under a unified data-driven frame to further speed up the computation of the solution to an asset model. More precisely, an "ANN solver" is typically decomposed into two separate phases, a training phase and a testing (or prediction) phase. During the training phase, the ANN "learns" the existed solver, which is usually time consuming, however, it will be done once and off-line. During the testing phase, the trained ANN delivers the derivative prices or other quantities highly efficiently and can be employed as an agent of the original solver in practice. We test this approach on different types of solvers, including the analytic solution for the Black-Scholes equation, the COS method for the Heston stochastic volatility model and the iterative root-finding method for the calculation of implied volatilities. The numerical results show that the ANN solver can reduce the computing time significantly. References [1] Shuaiqiang Liu, Cornelis W. Oosterlee, Sander M.Bohte, Pricing options and computing implied volatilities using neural networks.
Rogier Quaedvlieg:
Realized semicovariances: looking for signs of direction inside the covariance matrix
We propose a new decomposition of the realized covariance matrix into four "realized semicovariance" components based on the signs of the underlying high-frequency returns. We derive the asymptotic distribution for the different components under the assumption of a continuous multivariate semimartingale and standard infill asymptotic arguments. Based on high-frequency returns for a large cross-section of individual stocks, we document distinctly different features and dynamic dependencies in the different semicovariance components. We demonstrate that the accuracy of portfolio return variance forecasts may be significantly improved by exploiting these differences and "looking inside" the realized covariance matrices for signs of direction. Joint work with Tim Bollerslev and Andrew Patton. Poster presentations
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