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Department of Mathematics

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Below you will find a list of the previous joint seminars organised by the Risk and Stochastics Group and the Department of Mathematics between the years of 2010 and 2013. Seminars are listed in reverse chronological order (most recent first).

9 December 2013 - Igor Evstigneev (Manchester)

**Modelling Dynamics and Equilibrium of Asset Markets: A Behavioral Approach**

Conventional models of dynamic equilibrium in asset markets are based on the principles of General Equilibrium theory (Walras, Arrow, Debreu, Radner and others). This theory in its classical form assumes that market participants are fully rational and their goals can be described in terms of the maximization of utilities subject to budget constraints. The objective of this work is to develop an alternative modelling approach admitting that market participants may have a whole variety of other patterns of behavior determined by their individual psychology, not necessarily reducible to fully rational utility maximization. The models developed do not rely upon restrictive hypotheses (perfect foresight) and avoid using unobservable agents’ characteristics such as individual utilities and beliefs, which makes them amenable to quantitative practical applications. The results obtained are concerned with fundamental questions and problems in Financial Economics such as equilibrium asset pricing and portfolio selection. The modelling frameworks combine stochastic dynamic games and evolutionary game theory. The methods employed are based on the stochastic stability analysis of nonlinear random dynamical systems.

Joint work with Rabah Amir (University of Iowa), Thorsten Hens (University of Zurich) and Klaus R. Schenk-Hoppé (University of Leeds).

**Optimal Liquidity Provision in Limit Order Markets**

A small investor provides liquidity at the best bid and ask prices of a limit order market. For small spreads and frequent orders of other market participants, we explicitly determine the investor's optimal policy and welfare. In doing so, we allow for general dynamics of the mid price, the spread, and the order flow, as well as arbitrary preferences of the liquidity provider under consideration.

Joint work with Christoph Kühn.

25 November 2013 **- Giorgia Callegaro** (Universita Degli Studi di Padova)

**An application to credit risk of a hybrid Monte Carlo–optimal quantization method
** In this paper, we use a hybrid Monte Carlo-Optimal quantization method to approximate the conditional survival probabilities of a firm, given a structural model for its credit default, under partial information.

We consider the case when the firm's value is a non-observable stochastic process ${(V_t)}_{t \ge 0}$ and investors in the market have access to a process ${(S_t)}_{t \ge 0}$, whose value at each time $t$ is related to $(V_s, 0 \le s \le t)$.

We are interested in the computation of the conditional survival probabilities of the firm given the ``investor's information''.

As an application, we analyze the shape of the credit spread curve for zero coupon bonds in two examples. Calibration to available market data is also analysed.

18 November 2013 - **Martin Larsson** (EPFL)

**Polynomial preserving diffusions and models of the term structure**

Polynomial preserving processes are multivariate Markov processes that extend the important class of affine processes. They are defined by the property that the semigroup leaves the space of polynomials of degree at most $n$ invariant, for each $n$, which lends significant tractability to models based on these processes. In this talk I will discuss existence and uniqueness of polynomial preserving diffusions, a task which is made nontrivial due to degenerate and non-Lipschitz diffusion coefficients, as well as a complicated geometric structure of the state space. I will then discuss how polynomial preserving processes can be used to build term structure models that accommodate three features that are otherwise difficult to combine: nonnegative short rates, tractable swaption pricing, and unspanned factors affecting volatility and risk premia.

11 November 2013 -**Claudio Fontana** (INRIA Paris)

**On honest times and arbitrage possibilities
** In the context of a general continuous financial market model, we study whether the additional information associated with an honest time T gives rise to arbitrage profits. By relying on the theory of progressive enlargement of filtrations, we explicitly show that arbitrage profits can never be realised strictly before T, while classical arbitrage opportunities can be realised exactly at T as well as after T. Moreover, arbitrages of the first kind can only be obtained by starting to trade as soon as T occurs. We carefully study the behavior of local martingale deflators and consider no-arbitrage-type conditions weaker than no free lunch with vanishing risk. Finally, we discuss extensions of the theory to the case of general semi martingale models.

7 November 2013 - **Kostas Kardaras** (LSE)

**Equilibrium in risk-sharing games
** A market is considered with several acting financial agents, whose aim is to increase their utility by efficiently sharing their random endowments. Given the endogenously derived optimal sharing rules, we consider the situation where agents do not reveal their true endowments, but instead report as endowments the random quantities that maximise their utility when the sharing rules are applied. Under exponential utilities (coinciding with entropic risk measures), an analysis of Nash equilibrium is carried out, where it is shown in particular that the optimal contract of each agent possesses endogenous bounds only depending on the agents' risk tolerance, and not on their random endowment. Existence and uniqueness of Nash equilibrium for the 2-player game is obtained. Furthermore, it is discussed how such an equilibrium benefits extremely high risk tolerance agents and results in risk-sharing inefficiency.

Joint work with M. Antropelos.

24 October 2013 - **Budhi Arta Surya** (SBM ITB)

**Optimal Capital Structure with Scale Effects under Spectrally Negative Levy Models
** The optimal capital structure model with endogenous bankruptcy was first studied by Leland (1994) and Leland and Toft (1996), and was later extended to the spectrally negative Levy model by Hilberink and Rogers (2002) and Kyprianou and Surya (2007). This paper incorporates the scale effects by allowing the values of bankruptcy costs and tax benefits dependent on the firm's asset value. These effects have been empirically shown, among others, in Warner (1976), Ang et al. (1982), and Graham and Smith (1999). By using the fluctuation identities for the spectrally negative Levy process, we obtain a candidate bankruptcy level as well as a sufficient condition for optimality. The optimality holds in particular when, monotonically in the asset value, the value of tax benefits is increasing, the loss amount at bankruptcy is increasing, and its proportion relative to the asset value is decreasing. The solution admits a semi-explicit form, and this allows for instant computation of the optimal bankruptcy levels, equity/debt/firm values and optimal leverage ratios. A series of numerical studies are given to analyze the impacts of scale effects on the default strategy and the optimal capital structure.

Joint work with Kazutoshi Yamazaki

Link: http://arxiv.org/abs/1109.0897

30 May 2013 - **Mikhail Urusov** (University of Duisburg-Essen)

**On The Boundary Behaviour of Diffusions and the Martingale Property of the Associated Local Martingales
** Abstract

15 May 2013 - **Michael Schröder** (Vrije Universiteit Amsterdam)

**Mechanisms for no-arbitrage term-structure modelling, with applications to interest-rates and realized-variance
** Suppose that the sentiment is changing in some financial market, or that conditions have changed recently. Examples include volatility levels which are expected to change, or interest-rates expected to be adjusted.

How do we quantify the effects of these changes on derivatives positions?

We will discuss mechanisms for the construction of `no-arbitrage' term structures which enable this; these retain tractability in valuing derivatives and comply with stylized facts like mean-reversion and positivity of rates. This will be illustrated in a paradigm valuation of typical fixed-income derivatives.

20 March 2013 - **Mikhail Zhitlukhin** (University of Manchester and Steklov Mathematical Institute Moscow)

**General Bayesian changepoint detection problems for Brownian** **motion**

We consider changepoint detection problems for Brownian motion in a general Bayesian setting. A changepoint (another name - "disorder") is an unknown moment of time when the drift of an observable Brownian motion changes. The goal is to find a stopping time which is close as possible to the moment of disorder. We solve the problem for a wide class of penalty functions and prior distributions of the moment of disorder. Using the results obtained we also study optimal stopping problems for a Brownian motion and a geometric Brownian motion with a changepoint. The latter result can be used to model asset price processes with changing trend. In the last part of the talk we apply the method to real financial data and show it gives comparatively good results.

10 May 2012 - **Brenda Lopez Cabrera** (Humboldt-Universität zu Berlin)

**State Price Densities implied from Weather Derivatives
** A State Price Density (SPD) is the density function of a risk neutral equivalent martingale measure for option pricing, and is indispensible for exotic option pricing and portfolio risk management. Many approaches have been proposed in the last two decades to calibrate a SPD using financial options from the bond and equity markets. Among these, non and semi parametric methods were preferred because they can avoid model mis-specification of the underlying and thus give insight into complex portfolio propelling. However, these methods usually require a large data set to achieve desired convergence properties. Despite recent innovations in financial and insurance markets, many markets remain incomplete, and there exists an illiquidity issue. One faces the problem in estimation by e.g. kernel techniques that there are not enough observations locally available. For this situation, we employ a Bayesian quadrature method because it allows us to incorporate prior assumptions on the model parameters and hence avoids problems with data sparsity. It is able to compute the SPD of both call and put options simultaneously, and is particularly robust when the market faces the illiquidity issue. As illustration, we calibrate the SPD for weather derivatives, a classical example of incomplete markets with financial contracts payoffs linked to nontradable assets, namely, weather indices.

3 May 2012** -** **Larbi Alili** (University of Warwick)

**On some involutive inversions of one dimensional diffusions
** For full details, please click here.

15 March 2012 - **Josef Teichmann** (ETH, Zürich)

**Finite dimensional realizations for the CNKK-volatility surface model
** We show that parametrizations of volatility surfaces (and even more involved multivariate objects) by time-dependent Lévy processes (as proposed by Carmona-Nadtochiy-Kallsen-Krühner) lead to quite tractable term structure problems. In this context we can then ask whether the corresponding term structure equations allow for (regular) finite dimensional realization, which necessarily leads to models driven by

an affine factor process. This is another confirmation that affine processes play a particular role in mathematical finance. The analysis is based on a careful geometric analysis of the term structure equations by methods from foliation theory.

8 March 2012 - **Johan Tysk** (Uppsala University)

For full details, please click here.

23 February 2012 - **Vicky Henderson** (Oxford)

**Executive Stock Options: Portfolio Effects
** Executives compensated with stock options generally receive grants periodically and so on any given date, may have a portfolio of options of differing strikes and maturities on their company's stock. Non-transferability and trading restrictions in the company stock result in the executive facing unhedgeable risk. We employ exponential utility indifference pricing to analyse the optimal exercise thresholds for each option, option values and cost of the options to shareholders. Portfolio interaction effects mean that each of these differ, depending on the composition of the remainder of the portfolio. We demonstrate that the exercise threshold for a particular option can be discontinuous at the time that the option's position in the exercise order changes.

The cost to shareholders of an option portfolio is lowered relative to its cost computed on a per-option basis. The model can explain a number of empirical observations - which options are attractive to exercise first, how exercise changes following a new grant, and early exercise.

Joint work with Jia Sun and Elizabeth Whalley (WBS).

16 February 2012 - **Mike Tehranchi** (Cambridge)

**Put-call symmetry and self-duality
** We discuss generalisations of the notions of put-call symmetry and

self-duality. These notions have found applications in the pricing and

hedging of certain path-dependent contingent claims. Our results include a

classification of the possible forms of self-duality in one-dimension: in

addition to the arithmetic and geometric duality already appearing the

literature, there exists exactly one other type among continuous models. We

also give a description of the possible forms of put-call symmetry for

common models: in dimension greater than two, interesting new symmetries

appear.

9 February 2012 - **Daniel Hernández** (CIMAT, Mexico)

**Dynamic risk measures for exponential Levy market models
** The study of robust utility maximization problems for Levy processes is closely related with risk measures. In this talk we shall present recent results on the form of the penalization function associated with risk measures defined in a proper set of absolutely continuous measures, for a Levy market model.

9 February 2012 - **Markus Riedle** (King's College)

For full details, please click here.

2 February 2012 **- Curdin Ott** (University of Bath)

For full details, please click here.

19 January 2012 - **Damiano Brigo** (King's College London)

**Arbitrage-free valuation of counterparty credit risk
** Although explicit pricing of counterparty credit risk goes way back to 1994 in the financial modelling literature, only after the eight credit events that happened in one month of 2008 the research environment has become increasingly active in modelling credit valuation adjustments (CVA). Basel III is also imposing heavy capital requirements on CVA after noticing that about 2/3 of the losses during the crisis are due to CVA mark to market volatility rather than to actual defaults.

In this talk we introduce the mathematics of CVA and explain why it is a difficult hybrid derivatives valuation problem. Subtleties on payoff and modelling mathematics including wrong way risk, closeout conventions, first to default risk, collateral modelling, re-hypothecation and gap risk are investigated with quantitative case studies from a few asset classes. General conclusions on the mathematical difficulties involved in CVA pricing and risk management are presented.

12 January 2012 - **Johannes Ruf** ( Oxford)

**On the Hedging of Options on Exploding Exchange Rates
** Recently strict local martingales have been used to model exchange rates. In such models, put-call parity does not hold if one assumes minimal superreplicating costs as contingent claim prices. I will illustrate how put-call parity can be restored by changing the definition of a contingent claim price. More precisely, I will discuss a change of numeraire technique when the underlying is only a local martingale. Then, the new measure is not necessarily equivalent to the old measure. If one now defines the price of a contingent claim as the minimal superreplicating costs under both measures, then put-call parity holds. I will discuss properties of this new pricing operator. To illustrate this techniques, I will discuss the class of "Quadratic Normal Volatility" models, which have drawn much attention in the financial industry due to their analytic tractability and flexibility.

This talk is based on joint work with Peter Carr and Travis Fisher.

8 December 2011**- Alex Miljatovic** (University of Warwick)

For full details, please click here .

24 November 2011**- Kees van Schaik** (University of Manchester)

For full details, please click here.

17 November 2011**- Mete Soner** (ETH)

For full details, please click here.

10 November 2011**- Kevin Warner** (Tower Research Capital)

For full details, please click here.

3 November 2011 **- Jordan Stoyanov** (Newcastle)

**Moment Analysis of Distributions: Classical and Recent Results
** The main discussion will be on characterization/properties of distributions in terms of the moments. This turns to be important for stochastic models in many areas, including in finance and risk modelling. Some distributions are uniquely determined by the moments (M-determinate), others are non-unique (M-indeterminate). Along classical criteria, some recent developments will be presented and used to analyze the moment determinacy of distributions of random variables or stochastic processes. All statements and criteria will be well illustrated by examples involving popular distributions (N, LogN, SN, Exp, Po, IG, etc.) Several facts will be reported, sme of them are not so well-known, they are surprising and even shocking. It will be shown that the moment determinacy of the distributions is essential in inference problems. Some challenging open questions will be outlined.

27 October 2011**- Huyên Pham** (University Paris Diderot)

**Optimal High Frequency Trading with Limit and Market Orders
** We propose a framework for studying optimal market making policies in a limit order book (LOB). The bid-ask spread of the LOB is modelled by a Markov chain with finite values, multiple of the tick size, and subordinated by the Poisson process of the tick-time clock. We consider a small agent who continuously submits limit buy/sell orders at best bid/ask quotes, and may also set limit orders at best bid (resp. ask) plus (resp. minus) a tick for getting the execution order priority, which is a crucial issue in high frequency trading. By trading with limit orders, the agent faces an execution risk since her orders are executed only when they meet counterpart market orders, which are modelled by Cox processes with intensities depending on the spread and on her limit prices. By holding non-zero positions on the risky asset, the agent is also subject to the inventory risk related to price volatility. Then the agent can also choose to trade with market orders, and therefore get immediate execution, but at a least favourable price because she has to cross the bid-ask spread.

The objective of the market maker is to maximize her expected utility from revenue over a short term horizon by a tradeoff between limit and market orders, while controlling her inventory position. This is formulated as a mixed regime switching regular/impulse control problem that we characterize in terms of quasi-variational system by dynamic programming methods. In the case of a mean-variance criterion with martingale reference price or when the asset price follows a Levy process and with exponential utility criterion, the dynamic programming system can be reduced to a system of simple equations involving only the inventory and spread variables.

Calibration procedures are derived for estimating the transition matrix and intensity parameters for the spread and for Cox processes modelling the execution of limit orders. Several computational tests are performed both on simulated and real data, and illustrate the impact and profit when considering execution priority in limit orders and market orders.

20 October 2011**- Ragnar Norberg** (LSE and University of Lyon)

For full details, please click here.

13 October 2011 **- Almut Veraart** (Imperial College London)

**Ambit Stochastics with Applications to Energy Markets
** This talk gives a brief introduction into the new area of ambit stochastics, which constitutes a general probabilistic framework for tempo-spatial modelling. Certain classes of random fields and stochastic processes within the framework of ambit stochastics will be presented and their applicability to modelling spot, forward and futures prices from energy markets will be discussed.

This is joint work with Ole. E. Barndorff-Nielsen (Aarhus University) and Fred Espen Benth (University of Oslo)

6 October 2011**- Roman Muraviev** (ETH)

For full details, please click here.

9 June 2011 - **Guilia Di Nunno** (University of Oslo)

**Dynamic no-good-deal bounds and no-good-deal pricing measures
** We study price systems consistent with no-good-deal pricing measures for given bounds on the Sharpe ratio and we introduce the definition of dynamic no-good-deal bounds and pricing measure. The development of the theory requires a sandwich preserving extension theorem for linear operators, which we present in some generality. We then show how this result can be applied to obtain static and dynamic no-good-deal pricing measures. If time permits, we can also provide other examples of reasonably restricted classes of equivalent martingale measures that can be obtained.

This presentation is based on a paper with Dr. Jocelyn Bion-Nadal (CNRS-Ecole Polytechnique, France).

19 May 2011 - **Bernt Oksendal ** (University of Oslo)

**Optimal pricing strategies and Stackelberg equilibria in time- delayed stochastic differential games
** In the classical newsvendor problem there are two agents: (i) The manufacturer, who today (i.e. at time t-\delta) decides the unit price to sell the manufactured goods for to the retailer, with delivery tomorrow (at time t); (ii) The retailer, who then today (at time t-\delta) decides the quantity to order from the manufacturer and the price to sell each item for to the public the next day. What is the optimal price set by the manufacturer and the optimal quantity to order and the optimal retailer price? The problem is that neither of these agents know what the demand will be the next day, only its probabilistic distribution. This is a problem that occurs in many situations, for example in the pricing of electricity in a liberated electricity market. We generalize this classical newsvendor problem to continuous time and a jump diffusion setting, and formulate it as a problem to find the Stackelberg equilibrium of a stochastic differential game with delayed information flow. We find a maximum principle for this type of control problem, and use it to solve the optimal pricing problem in some specific cases.

presentation is based on recent joint work with Leif Sandal and Jan Ubøe, both at NHH, Bergen, Norway.

24 March 2011 - **Peter Bank** (Technische Universität Berlin and Quantitative Products Laboratory)

**Market indifference prices
** We discuss the pricing and wealth dynamics in a market where a large trader's orders are filled at indifference prices. As we will see, this indifference principle is mathematically best described by a nonlinear SDE for the market makers' utility process. We will derive this SDE and discuss its solvability in terms of Malliavin derivatives and Sobolev embedding results for stochastic integrals.

10 March 2011- **Lioudmila Vostrikova** (Université d'Angers)

**F-divergence minimal martingale measures and optimal portfolios for exponential Levy models with a change-point
** We study exponential Levy models with change-point which is a random variable, independent from initial Levy processes. On canonical space with initially enlarged ltration we describe all equivalent martingale measures for change-point model and we give the conditions for the existence of f-minimal equivalent martingale measure. Using the connection between utility maximisation and f-divergence minimisation, we obtain a general formula for optimal strategy inchange-point case for initially enlarged ltration and also for progressively en-larged ltration when the utility is exponential. We illustrate our results consid-ering the Black-Scholes model with change-point.

3 March 2011- **Carlos G. Pacheco González** (CINVESTAV, Mexico City)

**The Kac semi-group and applications to stochastic control
** In this talk we present the Kac semi-group within the context of Markov processes, and we show applications in stochastic control problems with non-constant discounted criteria. In particular we set a Hamilton-Jacobi-Bellman equation for a problem where the discounted process is the Cox-Ingersol-Ross model.

24 February 2011- **Sergey Nadtochiy** (Oxford University)

**An approximation scheme for the optimal investment strategy in incomplete market**

Characterizing and constructing the solutions to stochastic optimization problems of optimal portfolio choice is a long standing problem. In this talk, I will discuss a new method based on a splitting scheme for the associated Hamilton-Jacobi-Bellman equation in a two-factor stochastic volatility model for the stock price. The scheme converges to a solution of the corresponding PDE, and yields an explicit uniform approximation of the optimal investment strategy. This solution approach offers, among others, insightful observations on how market incompleteness is processed and how it affects the 'infinitesimal' investment preferences. This is joint work with Thaleia Zariphopoulou.

3 February 2011 - **Chris Rogers**(Cambridge)

**Diverse beliefs and market selection**

This talk presents the basic framework for equilibrium pricing where agent heterogeneity is characterized by diverse beliefs. This turns out to be a tractable and sensible modelling framework in which to study various phenomena, which we will illustrate with several examples, drawn in the main from the literature on market selection. The Market Selection Hypothesis loosely speaking proposes that agents with `inferior' beliefs will eventually be `eliminated' from the market, but these terms need to be defined. Once they are, we are able to prove some results about when agents are indeed eliminated from the market; these results only partly confirm the intuition of the Market Selection Hypothesis. We have some surprising examples which show that some very unexpected phenomena may occur.

8 November 2010 - **Luitgard Veraart** (LSE)

**The relaxed investor with partial information
** We consider an investor in a financial market consisting of a riskless bond and several risky assets. The price processes of the risky assets are geometric Brownian motions where either the drifts are modelled as random variables assuming a constant volatility matrix or the volatility matrix is considered random and drifts are assumed to be constant. The investor is only able to observe the asset prices but not all the model parameters and hence information is only partial. A Bayesian approach is used with known prior distributions for the random model parameters.

We assume that the investor can only trade at discrete time points which are multiples of h>0 and investigate the loss in expected utility of terminal wealth which is due to the fact that the investor cannot trade and observe continuously.

It turns out that in general a discretization gap appears, i.e for $h \to 0$ the expected utility of the h-investor does not converge to the expected utility of the continuous investor. This is in contrast to results under full information in(Rogers, L.C.G. 2001. The relaxed investor and parameter uncertainty. Finance and Stochastics, 5 (2), 131-154).

We also present simple asymptotically optimal portfolio strategies for the discrete-time problem. Our results are illustrated by some numerical examples.

This is joint work with Nicole Bäuerle and Sebastian Urban.

29 April 2010 - **Yuliya Mishura** (Kyiv University)

**Financial applications of the models with long-range dependence
** In our work we move away from the semimartingale model of financial market and consider the models with so called long-range dependence. The arbitrage problems are discussed in the most general setting.

In particular, we consider financial market with risky asset governed by both the Wiener process and fractional Brownian motion with Hurst parameter $H>3/4$. Using Hitsuda and Cheridito representations for the mixed Brownian--fractional Brownian process, we present the solution of the problem of quantile hedging and clarify in

this case the dependence of maximal possible success probability on the available initial capital $\nu <H_0 $. More general problem of efficient hedging is also solved.

11 March 2010 - **Hans Rudolf Lerche** (University of Freiburg)

**Blackwell Prediction**** **

Let *x*_{1} , *x*_{2} *, . . .* be a (not necessarily random) infinite 0-1 sequence. We wish to sequentially predict the sequence. This means that, for each *n ≥* 1, we will guess the value of *x _{n}*

5 Feb 2010 - **Tomas Björk** (Stockholm School of Economics)

**Time inconsistent stochastic control**

We present a theory for stochastic control problems which, in various ways, are time inconsistent in the sense that they do not admit a Bellman optimality principle. We attach these problems by viewing them within a game theoretic framework, and we look for subgame perfect Nash equilibrium points.

For a general controlled Markov process and a fairly general objective functional we derive an extension of the standard Hamilton-Jacobi-Bellman equation, in the form of a system of non-linear equations, for the determination for the equilibrium strategy as well as the equilibrium value function. All known examples of time inconsistency in the literature are easily seen to be special cases of the present theory. We also prove that for every time inconsistent problem, there exists an associated time consistent problem such that the optimal control and the optimal value function for the consistent problem coincides with the equilibrium control and value function respectively for the time inconsistent problem. We also study some concrete examples.

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