# Abstract

18.10.2019 Jon Armstrong (Kings College London)

Title: Isomorphisms of Markets

Abstract: Two markets should be considered isomorphic if they are financially indistinguishable. We will define a notion of isomorphism for financial markets in both discrete and continuous time. We then seek to identify the distinct isomorphism classes, that is to classify markets. We prove a number of classification theorems and show how the automorphisms of a market give rise to mutual fund theorems.

08.11.2019 Renyuan Xu(Oxford University)

Title: A Case Study on Pareto Optimality for Collaborative Stochastic Games

Abstract: Pareto Optimality (PO) is an important concept in game theory to measure global efficiency when players collaborate. In this talk, we start with the PO for a class of continuous-time stochastic games when the number of players is finite. The derivation of PO strategies is based on the formulation and analysis of an auxiliary N-dimensional central controller’s stochastic control problem, including its regularity property of the value function and the existence of the solution to the associated Skorokhod problem. This PO strategy is then compared with the set of (non-unique) NEs strategies under the notion of Price of Anarchy (PoA). The upper bond of PoA is derived explicitly in terms of model parameters. Finally, we characterize analytically the precise difference between the PO and the associated McKean-Vlasov control problem with an infinite number of players, in terms of the covariance structure between the optimally controlled dynamics of players and characteristics of the non-action region for the game. This is based on joint work with Xin Guo (UC Berkeley).

15.11.2019 Eric Renault (University of Warwick)

Title: Identification-Robust Inference for Risk Prices in Structural Stochastic Volatility Models

Abstract: In structural stochastic volatility asset pricing models, changes in volatility affect risk premia through two channels: (1) the investor’s willingness to bear high volatility in order to get high expected returns as measured by the market return risk price, and (2) the investor’s direct aversion to changes in future volatility as measured by the volatility risk price. Disentangling these channels is difficult and poses a subtle identification problem that invalidates standard inference. We adopt the discrete-time exponentially affine model of Han, Khrapov, and Renault (2019), which links the identification of the volatility risk price to the leverage effect. In particular, we develop a minimum distance criterion that links the market return risk price, the volatility risk price, and the leverage effect to well-behaved reduced-form parameters that govern the return and volatility’s joint distribution. The link functions are almost flat if the leverage effect is close to zero, making estimating the volatility risk price difficult. We translate the conditional quasi-likelihood ratio test that Andrews and Mikusheva (2016) develop in a nonlinear GMM framework to a minimum distance framework. The resulting conditional quasi-likelihood ratio test is uniformly valid. We invert this test to derive robust confidence sets that provide correct coverage for the risk prices regardless of the leverage effect’s magnitude.

22.11.2019 Andreas Kyprianou (Universtity of Bath)

Title: Entrance and exit at infinty for stable jump diffusions

Abstracts: In his seminal work from the 1950s, William Feller classified all one-dimensional diffusions on $-\infty\leq a<b\leq \infty$ in terms of their ability to access the boundary (Feller's test for explosions) and to enter the interior from the boundary. Feller's technique is restricted to diffusion processes as the corresponding differential generators allow explicit computations and the use of Hille-Yosida theory. In the present article we study exit and entrance from infinity for the most natural generalization, that is, jump diffusions of the form $dZ_t=\sigma(Z_{t-})\,dX_t,$ driven by stable L\'evy processes for $\alpha\in (0,2)$. Many results have been proved for jump diffusions, employing a variety of techniques developed after Feller's work but exit and entrance from infinite boundaries has long remained open. We show that the presence of jumps implies features not seen in the diffusive setting without drift. Finite time explosion is possible for $\alpha\in (0,1)$, whereas entrance from different kinds of infinity is possible for $\alpha\in [1,2)$. We derive necessary and sufficient conditions on $\sigma$ so that (i) non-exploding solutions exist and (ii) the corresponding transition semigroup extends to an entrance point at `infinity'. Our proofs are based on very recent developments for path transformations of stable processes via the Lamperti-Kiu representation and new Wiener-Hopf factorisations for L\'evy processes that lie therein. The arguments draw together original and intricate applications of results using the Riesz-Bogdan--\.Zak transformation, entrance laws for self-similar Markov processes, perpetual integrals of L\'evy processes and fluctuation theory, which have not been used before in the SDE setting, thereby allowing us to employ classical theory such as Hunt-Nagasawa duality and Getoor's characterisation of transience and recurrence.

29.11.2019 Miryana Grigorova (University of Leeds)

Title: A non-linear incomplete market model with default: Pricing of European and American options

Abstract:

We present an incomplete market model with default which consists of one risky asset with
dynamics driven by two "sources of risk", namely a Brownian motion and a compensated default
martingale. Additionally to this feature, the wealth processes follow non-linear dynamics with
a non-linear driver f, which allows to incorporate a number of imperfections in the market.
We thus face a non-linear incomplete market with default. We provide a dual formulation of the seller's superhedging price for a European option in terms of the supremum, over a suitable set of equivalent probability measures Q, of the non-linear f-evaluation/expectation under Q of the payoff. We also provide some related criteria for replicability of a given pay-off. By a form of symmetry, we derive corresponding results for the buyer. Our results rely on fi rst establishing a non-linear optional decomposition for processes which are (non-linear) f-strong supermartingales under Q, for all Q. This decomposition is the analogue in our
framework of the well-known optional decomposition from the linear case. We also show that the non-linear optional decomposition is equivalent to a non-linear predictable decomposition with constraints. This result allows us to show an in finitesimal characterization of the seller's (superhedging) price process as the minimal supersolution of a constrained BSDE with default. We will also discuss corresponding results for the seller's superhedging price of an American option.

24.01.2020 Ernst Eberlein (University of Freiburg, Germany)

Title: Multiple curve interest rate modelling
Abstract:
The global nancial crisis which started in 2007 changed the xed income markets in a fundamental way. Due to a new perception of risk, a number of interest rates, which until then had been roughly equivalent, drifted apart. The basic rates, which are relevant for the interbank market, became tenor-dependent after market participants became aware of credit, liquidity and funding risks in this market segment. These risks had been assumed to be negligible before. In the new reality classical modelling approaches which are based on arbitrage considerations assuming tenor-independence cannot reflect the market behaviour any more. More sophisticated approaches, so-called multiple curve models, are needed to take the increased diversity of risks into account.

We explain rst the empirical facts and develop then three dierent approaches to multiple curve modelling: a forward rate (HJM-type), a forward process and a swap rate model. In all three approaches time-inhomogeneous Levy processes are used as drivers. Negative interest rates can be taken into account in a natural way. We derive valuation formulas for standard interest rate nancial products such as caps, floors and swaptions. Calibration results are presented where we also consider data in the setting of a two price economy, thus exploiting explicitly bid and ask quotes.

21.02.2020 Daniela Escobar (London School of Economics)

Title: Robust Pricing for insurance contracts, dynamic problems and possible extensions
Abstract:

This talk focuses on model ambiguity for different pricing problems. First, we consider a risk measure, which is also a pricing principle in insurance: the Distortion Premium. We study its properties and robust solutions with respect to the initial model. We also study a more complex system, which is the pricing of a real option. In this case, decisions, states, and modeling of different exogenous variables are considered with the goal to maximise a profit. In the first case, we will propose to calculate the robust price using the Wasserstein distance. In the second case, we will propose a new distance, which is consistent with the dynamic principle. Possible extensions for each case will be discussed.

13.03.2020 Drik Becherer (Humboldt University Berlin, Germany)

Title: Optimal trade execution with transient relative price impact and directional views:
A variational approach to a 3-dimensional non-convex free boundary problem

Abstract:

This talk focuses on model ambiguity for different pricing problems. First, we consider a risk measure, which is

We solve the optimal execution problem to trade a large ﬁnancial asset position within finite time in an illiquid market, where price impact is transient and possibly non-linear (in log-prices), like in models by e.g. J.P.Bouchaud, F.Lillo, or Gatheral. The mechanical impact can be seen as a multiplicative variant of seminal models by Obizhaeva/Wang (2013), Predoiu/Shaikhet/Shreve (2011) or Schied et al (2010) - but for impact being in relative percentage terms as Bertsimas/Lo (1998, ch.3) to avoid the possibility of negative prices. We admit for non-zero drift in the fundamental price process, that is directional views about short-term price trends, cf. Almgren/Chriss (2000, ch.4). We prove an explicit description for the (regular) three-dimensional free boundary surface which separates the no-/action regions of the singular control problem by a family of characteristic curves. Whereas the free boundary description by ODEs is almost as simple as in Obizhaeva/Wang, our analysis is very different and more complex, for lack of an apparent convexity structure. Yet, a key argument to prove global optimality turns out to show at first a local optimality for a candidate free boundary under smooth perturbations by variational arguments.

For the application problem, the results shed light on interesting phenomena like, e.g.,
a) how to profit from directional views (signals) on price trends by optimal (non-trivial) round-trips;