The Geometry of Money and the Perils of Parameterization

Markets trade thousands of underlying, each one with tens or even hundreds of options, quoted throughout the day. Needless to say, the quotes are not generated manually. They are automated and derived from a functional form with a few parameters. If we know this parameterization, we know in advance how the prices tomorrow of many traded securities will belong to a low dimensional (number of parameters) manifold in a high dimensional (number of securities). If the vector of today prices does not belong to the convex hull of the manifold it creates arbitrage. We examine market practice (Black-Scholes, stochastic volatility models, interest rate interpolation by piecewise constant instantaneous forward rates converging implied volatility for extreme strikes in FX…) and show that many violate the no arbitrage condition

A kernel-free particle method: smile problem resolved

We present a novel Monte Carlo based LSV calibration algorithm that applies to all stochastic volatility models, including the non-Markovian rough volatility family. Our framework overcomes the limitations of the particle method proposed by Guyon and Henry-Labordère (2012) and theoretically guarantees a variance reduction without additional computational complexity. Specifically, we obtain a closed-form and exact calibration method that allows us to remove the dependency on both the kernel function and bandwidth parameter. This makes the algorithm more robust and less prone to errors or instabilities in a production environment. We test the efficiency of our algorithm on various hybrid (rough) local stochastic volatility models

Fast and Fair Lsv pricing : back to basics

In this presentation we go back to basics . P&L explanation . We show how to compute compute exotic Greeks explain the lab lv impact and give a fast pricing formula for all products highlighting three classic examples

Mixed Local-Vol model : as simple a Stochastic-Local Vol model as it gets

Stochastic volatility is all about generating a set of possible future scenarios for the volatility. The Mixed Local-Vol (MLV) approach is as simple a way of doing that as it gets. While being conceptually relatively simple (the stochastic volatility component is reduced to a small number of discrete states), MLV shares in fact many desirable features with its full-fledged Stochastic-Local-Volatility (SLV) sibling.

MLV seems to be the industry standard to price FX 1^st generation exotic options: it is fast and provides accurate prices and risks.

In this presentations, we explore the main theoretical and practical aspects of MLV, comparing it to well-known benchmarks as Local-Vol (Dupire), Vanna-Volga, Stochastic Vol (Heston) and Stochastic-Local-Vol models.

We price European-style options written on forward contracts in a commodity market, which we model with a state-dependent infinite-dimensional Heath-Jarrow-Morton (HJM) approach. We introduce a new class of volatility operators which map the square integrable noise into the Filipović space of forward curves, and we specify a deterministic parametrized version of it. For calibration purposes, we train a neural network to approximate the option price as a function of the model parameters. We then use it to calibrate the HJM parameters starting from (simulated) option market data. Finally we introduce a new loss function that takes into account bid and ask prices and offers a solution to calibration in illiquid markets. A key issue discovered is that the trained neural network might be non-injective, which could potentially lead to poor accuracy in calibrating the forward curve parameters, even when showing a high degree of accuracy in recovering the prices. This reveals that the original meaning of the parameters gets somehow lost in the approximation.

 Optimal Distributional Trading Gain: Generalizations of Merton’s Portfolio Problem with Implications to Bayesian Statistics

This presentation considers multiple market agents who have distinct distributional opinions about the state price density. We first determine the optimal trading positions of a utility maximizing market taker who trades Arrow-Debreu securities for prices set by the market maker.

We use calculus of variations to determine the solution of this problem for a general utility function. The choice of the logarithmic utility function leads to a solution in terms of a likelihood ratio of the densities corresponding to the market taker and the market maker and the resulting optimal utility is the Kullback-Leibler divergence. In particular, we obtain a trivial solution for Merton’s  portfolio problem in the traditional geometric Brownian motion model and and we show its immediate extension to the multivariate case. A further extension gives a solution for the market driven by a geometric Poisson process. In a market without the market maker, the distributional opinions of market takers reach an equilibrium in the form of the linear mixture of the distributions. We show that when the the result of the outcome is observed, the profit and loss from trading updates agents’ bankrolls in a Bayesian fashion, which provides one to one correspondence for the logarithmic utility maximazers’ profits and Bayesian statistics. We extend these results to the continuous time setup and show that the bankrolls of agents following Merton optimal portfolio strategy evolve as a posterior distribution.

 Multi SLV models in both Equity and FX contexts depend on certain deep-in-the-model parameters that are in practice often overlooked. In particular, the correlations between the stochastic variance processes play a significant role when pricing and risk managing popular products such as autocallables or FX correlation swaps. Ignoring these parameters may not only lead to significant mis-pricing of those products, but also mean giving up valuable degrees of freedom that could be used for calibration purposes

Machine Learning for Data Generation and Hedging in Finance

This talk presents applications of machine learning based methods to data generation and deep hedging in the context of trading and risk management. It is demonstrated how generative neural networks like variational autoencoders can be used as additional source for creating realistic market data like forward curves and volatility surfaces.

Local Volatility in Multi Dimensions

A tour de force of finance-related poor maths; from social media over financial institutions to public administration. Names will be named; is one of them yours?

Differential machine learning combines automatic adjoint differentiation (AAD) with modern machine learning (ML) in the context of risk management of financial Derivatives. We introduce novel algorithms for training fast, accurate pricing and risk approximations, online, in real-time, with convergence guarantees. Our machinery is applicable to arbitrary Derivatives instruments or trading books, under arbitrary stochastic models of the underlying market variables. It effectively resolves computational bottlenecks of Derivatives risk reports and capital calculations.

Differential ML is a general extension of supervised learning, where ML models are trained on examples of not only inputs and labels but also differentials of labels wrt inputs. It is also applicable in many situations outside finance, where high-quality first-order derivatives wrt training inputs are available. Applications in Physics, for example, may leverage differentials known from first principles to learn function approximations more effectively.

In finance, AAD computes pathwise differentials with remarkable efficacy so differential ML algorithms provide extremely effective pricing and risk approximations. We can produce fast analytics in models too complex for closed-form solutions, extract the risk factors of complex transactions and trading books, and effectively compute risk management metrics like reports across a large number of scenarios, backtesting and simulation of hedge strategies, or regulations like XVA, CCR, FRTB or SIMM-MVA.

TensorFlow implementation is available on https://github.com/differential-machine-learning

Differential machine learning combines automatic adjoint differentiation (AAD) with modern machine learning (ML) in the context of risk management of financial Derivatives. We introduce novel algorithms for training fast, accurate pricing and risk approximations, online, in real-time, with convergence guarantees. Our machinery is applicable to arbitrary Derivatives instruments or trading books, under arbitrary stochastic models of the underlying market variables. It effectively resolves computational bottlenecks of Derivatives risk reports and capital calculations.

Differential ML is a general extension of supervised learning, where ML models are trained on examples of not only inputs and labels but also differentials of labels wrt inputs. It is also applicable in many situations outside finance, where high-quality first-order derivatives wrt training inputs are available. Applications in Physics, for example, may leverage differentials known from first principles to learn function approximations more effectively.

In finance, AAD computes pathwise differentials with remarkable efficacy so differential ML algorithms provide extremely effective pricing and risk approximations. We can produce fast analytics in models too complex for closed-form solutions, extract the risk factors of complex transactions and trading books, and effectively compute risk management metrics like reports across a large number of scenarios, backtesting and simulation of hedge strategies, or regulations like XVA, CCR, FRTB or SIMM-MVA.

TensorFlow implementation is available on https://github.com/differential-machine-learning

Semi-Static and Sparse Variance-Optimal Hedging“  – Among theoretical results, we show that a variance swap can be reasonably hedged by three static option positions and dynamic hedging in the underlying

Model-free bounds for multi-asset options using option-implied information

We consider derivatives written on multiple underlyings in a one-period financial market, and we are interested in the computation of model-free upper and lower bounds for their arbitrage-free prices. We work in a completely realistic setting, in that we only assume the knowledge of traded prices for other single- and multi-asset derivatives, and even allow for the presence of bid-ask spread in these prices. We provide a fundamental theorem of asset pricing for this market model, as well as a superhedging duality result, that allows to transform the abstract maximization problem over probability measures into a more tractable minimization problem over vectors, subject to certain constraints. Then, we recast this problem into a linear semi-infinite optimization problem, and provide two algorithms for its solution. These algorithms provide upper and lower bounds for the prices that are ε-optimal, as well as a characterization of the optimal pricing measures. Moreover, these algorithms are efficient and allow the computation of bounds in high-dimensional scenarios (e.g. when d=60). Numerical experiments using synthetic data showcase the efficiency of these algorithms, while they also allow to understand the reduction of model-risk by including additional information, in the form of known derivative prices.

The valuation of variable annuities

In this work we analyze the valuation of variable annuities. We utilize a market-consistent framework and the QP-rule for this. The variable annuiity may have several features, including surrender benefiit. The latter is the most challenging part and we discuss model risk and valuation issues. The valuation in a hybrid Levy model is illustrated which allows on the one side the perfect calibration to the financial market and on the other side the use of statistical information on insurance-related quantitiies. Also the hedging scheme is discussed.

The talk bases on joint work with Philippe Artzner, Laura Ballotta, Ernst Eberlein, Karl-Theodor Eisele, and Raghid Zeineddine.

Title: Correlation stress testing of stock and credit portfolios

Abstract: We develop a general approach for stress testing correlations in stock and credit portfolios. Using Bayesian variable selection methods, we build a sparse factor structure, linking individual names or stocks with country and industry factors. Based on methods from modelling correlations in interest rate modelling, especially in the context of market models, we calibrate a parametric correlation matrix, where correlations of stocks / names are represented as a function of the country and industry factors. Economically meaningful stress scenarios on the factors can then be translated into stressed correlations. The method also lends itself as a reverse tress testing framework: using e.g. the Mahalanobis distance on the joint risk factor distribution, allows to infer worst-case correlation scenarios. We give examples of stress tests including an application to analyse a USD 6.2 bn loss by JP Morgan in 2012, known as the “London Whale”. (This is joint work with Fabian Woebbeking)