In the realm of Martin Hairer's regularity structures we aim to introduce topologies on spaces of modelled distributions, which enable on the one hand reconstruction and which allow on the other hand a rich class of modelled distribution valued semi-martingales. This is done to have tools from regularity structures and semi-martingale theory at hand. Examples from the theory of term structures in mathematical Finance are shown. Joint work with David Prömel, ETH Zürich.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

This talk will introduce a class of game-theoretic supermartingales, whose main advantage over their measure-theoretic counterparts is that they do not presuppose a given probability measure; instead, they can be used to define an outer measure motivated by economic considerations combined only with topological (but not statistical) assumptions. Under the continuity assumption, it is possible to show that a typical continuous price path "looks like Brownian motion" with a possibly deformed time axis. A weaker assumption of boundedness of jumps still implies the almost sure existence of pathwise stochastic integrals of functions with finite p-variation for some p with respect to cadlag price paths with bounded jumps.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

Vovk recently introduced a pathwise approach to continuous time mathematical finance which does not require any measure-theoretic foundation and allows us to describe properties of “typical price paths” or “game-theoretic martingales" by only relying on superhedging arguments. I will show how to construct a model free Itô integral in this setting. We will also see that every typical price paths a rough path in the sense of Lyons. Based on joint work with David Prömel.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We present a pathwise Tanaka formula for absolutely continuous functions with weak derivative of finite q-variation provided the local time is of finite p-variation with 1/p + 1/q >1. To justify the assumption on the local time, we follow Vovk's hedging based approach to model free financial mathematics. We prove that it is possible to make an arbitrarily large profit by investing in those one-dimensional paths which do not possess local times fulfilling the aforementioned assumptions. This talk is based on a joint work with Nicolas Perkowski.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

Since Hobson's seminal paper the connection between model-independent pricing and the skorokhod embedding problem has been a driving force in robust finance. We establish a general pricing-hedging duality for financial derivatives which are susceptible to the Skorokhod approach. Using Vovk's approach to mathematical finance we derive a model-independent super-replication theorem in continuous time, given information on finitely many marginals. Our result covers a broad range of exotic derivatives, including lookback options, discretely monitored Asian options, and options on realized variance.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We study a notion of pathwise integral with respect to paths of finite quadratic variation, defined as the limit of non-anticipative Riemann sums, as defined by Follmer (1979) and extended by Cont & Fournie (2010). We prove a pathwise isometry property for this integral, analogous to the well-known Ito isometry for stochastic integrals. This property is then used to represent the integral as a continuous map on an appropriately defined vector space of integrands. Finally, we obtain a pathwise 'signal plus noise' decomposition, which is the pathwise analog of the semimartingale decomposition, for a large class of irregular paths obtained through functional transformations of a reference path with non-vanishing quadratic variation. The relation with controlled rough paths is discussed.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

In his seminal paper "Calcul d'Ito sans probabilités", Hans Föllmer proposed a non-probabilistic version of the Itô formula, which was recently generalized by Rama Cont and David-Antoine Fournié in a functional framework. Using the notion of pathwise quadratic variation, we derive first a pathwise isometry formula for functionals of a given path. This formula allows to generalize the notion of vertical derivatives and allows to define a weak version of vertical derivatives for functionals which are not necessarily smooth in the classical sense. The whole approach involves only pathwise arguments and does not rely on any probability notions. Nevertheless, we show that when applying to a stochastic process, this notion of weak derivatives coincides with the weak derivatives proposed by Cont and Fournié in a probabilistic framework.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We develop a pathwise calculus for functionals of integer-valued measures. We show that smooth functionals in the sense of this pathwise calculus are dense in the space of square-integrable (compensated) integrals with respect to a large class of integer-valued random measures. Using these results, we extend the framework of Functional Itô Calculus to functionals of integer-valued random measures. We construct a 'stochastic derivative' operator with respect to such integer-valued random measures and obtain an explicit martingale representation formula for square-integrable martingales with respect to the filtration generated by such integer-valued random measures. Our results hold beyond the class of Poisson random measures and allow for random and time-dependent compensators. This is joint work with R. Cont.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

Davis, Obloj and Raval (2013) developed a theory of robust pricing and hedging of weighted variance swaps given market prices of co-maturing put options. They make use of Föllmer’s quadratic variation for continuous paths, and of an analogous notion of local time. Here we develop a theory of pathwise local time, defined as a limit of suitable discrete quantities along a general sequence of partitions of the time interval. We provide equivalent conditions for the existence of pathwise local time. Our approach agrees with the usual (stochastic) local times for a.e. path of a continuous semimartingale. We establish pathwise versions of the Itô-Tanaka, change of variables and change of time formulae. Finally, we study in detail how the limiting objects, the quadratic variation and the local time, depend on the choice of partitions. In particular, we show that an arbitrary given non-decreasing process can be achieved a.s. by the pathwise quadratic variation of a standard Brownian motion for a suitable sequence of (random) partitions; however, such degenerate behavior is excluded when the partitions are constructed from stopping times.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We explore the robust replication of forward-start straddles given quoted (Call and Put options) market data. One approach to this problem classically follows semi-infinite linear programming arguments, and we propose a discretisation scheme to reduce its dimensionality and hence its complexity. Alternatively, one can consider the dual problem, consisting in finding optimal martingale measures under which the upper and the lower bounds are attained. Semi-analytical solutions to this dual problem were proposed by Hobson and Klimmek (2013) and by Hobson and Neuberger (2008). We recast this dual approach as a finite dimensional linear programme, and reconcile numerically, in the Black-Scholes and in the Heston model, the two approaches.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We study the Monge-Kantorovich transport between two probability measures, where the transport plans are subject to a probabilistic constraint. For instance, in the martingale optimal transport problem, the transports are laws of martingales. Interesting new couplings emerge as optimizers in such problems. Constrained transport arises in the context of robust hedging in mathematical finance via linear programming duality. We formulate a complete duality theory for general performance functions, including the existence of optimal hedges. This duality leads to an analytic monotonicity principle which describes the geometry of optimal transports. Joint work with Mathias Beiglböck, Florian Stebegg and Nizar Touzi.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We consider robust (pathwise) approach to pricing and hedging. Motivated by the notion of prediction set in Mykland (2003), we include in our setup modelling beliefs by allowing to specify a set of paths to be considered, e.g. super-replication of a contingent claim is required only for paths falling in the given set. The framework interpolates between model--independent and model--specific settings. We establish a general pricing--hedging duality. The setup is parsimonious and includes the case of no traded options as well as the so-called martingale optimal transport duality of Dolinsky and Soner (2013) which we extend to multiple dimensions and multiple maturities. In presence of non-trivial beliefs, the equality is obtained between limiting values of perturbed problems indicating that the duality holds only if the market is stable under small perturbations of the inputs. Our framework allows to quantify the impact of making assumptions or gaining information. We focus in particular on the latter and study if the pricing-hedging duality is preserved under additional information. Joint work with Zhaoxu Hou and Anna Aksamit.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We consider a continuous-time financial market that consists of securities available for dynamic trading, and securities only available for static trading. We work in a robust framework and discuss two different ways of including additional information. In the first case, the informed agent's information flow is modeled by a filtration which is finer that the one of the uninformed agent. This clearly leads to a richer family of trading strategies, and to a smaller set of pricing measures. In the second case, we assume that the additional information consists in being able to exclude some evolution of the asset price process. In particular, super-replication of a contingent claim is required only along paths falling in the smaller set of admissible paths, and the pricing measures to be considered are only those supported on this set. The talk is based on joint works with Martin Larsson, Alex Cox and Martin Huesmann.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We consider a market including a traded asset whose forward price St is unambiguously defined and on which put options are traded with maturity/strike pairs {(Tj,Kji), i = 1, . . . , ij, j = 1, . . . , n}. The prices of these options, and the underlying asset price, are known at the current time t = 0, and are assumed to satisfy the Davis-Hobson (2007) conditions for consistency with an arbitrage-free model. Given a path-dependent contingent claim with exercise value ö(ST1, . . . , STn) we look for the cheapest semi-static superhedging portfolio, consisting of static positions in the traded options together with dynamic trading in the underlying where rebalancing takes place only at the option exercise times Tj. This problem is naturally formulated as an infinite-dimensional linear program (LP) and (under stated conditions) we can apply interior point conditions to show that there is no duality gap, the dual problem being maximization of expectation over martingale measures. One advantage of this approach is that computations can be done by finite-dimensional LP algorithms, following a 2-stage discretization process where we firstly restrict the dynamic trading integrands to finite linear combinations of basis functions, and then discretize the state space; we present some examples. Finally, we comment on possible extensions of these results to models with transaction costs. This is joint work with Sergey Badikov and Antoine Jacquier.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

Path-dependent Kolmogorov equations are a class of infinite dimensional partial differential equations on the space of cadlag functions which extend Kolmogorov's backward equation to path-dependent functionals of stochastic processes. Solutions of such equations are non-anticipative functionals which extend the notion of harmonic function to a non-Markovian, path-dependent setting. We discuss existence, uniqueness and properties of weak and strong solutions of path-dependent Kolmogorov equations using the Functional Ito calculus. Time permitting, some applications to mathematical finance and non-Markovian stochastic control will be discussed.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

In the first part of the talk we revisit the basic theory of functional Ito calculus, using the regularization approach. This allows us to explore its relations with the corresponding Banach space stochastic calculus. In the second part of the talk, we introduce a viscosity type solution for path-depenendent partial differential equations, called strong-viscosity solution, with the peculiarity that it is a purely analytic object. We discuss its properties and we present an existence and uniqueness result for strong-viscosity solutions to semilinear parabolic path-dependent partial differential equations.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

In this talk, we consider a class of parabolic semilinear path-dependent PDEs that can be associated with a class of stochastic integral equations, which may depend on the entire sample paths of a time-inhomogeneous diffusion process. For instance, such integral equations can determine the log-Laplace functionals of historical superprocesses. By exploiting this relationship, we show uniqueness, existence and non-extendibility of mild solutions, and verify that every mild solution turns out to be a viscosity solution of the path-dependent PDE in question.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)