# Past Nomura Seminar

25 October 2013
16:00
Abstract
We analyse the impact of market makers' risk aversion on the equilibrium in a speculative market consisting of a risk neutral informed trader and noise traders. The unwillingness of market makers to bear risk causes the informed trader to absorb large shocks in their inventories. The informed trader's optimal strategy is to drive the market price to its fundamental value while disguising her trades as the ones of an uninformed strategic trader. This results in a mean reverting demand, price reversal, and systematic changes in the market depth. We also find that an increase in risk aversion leads to lower market depth, less efficient prices, stronger price reversal and slower convergence to fundamental value. The endogenous value of private information, however, is non-monotonic in risk aversion. We will mainly concentrate on the case when the private signal of the informed is static. If time permits, the implications of a dynamic signal will be discussed as well. Based on a joint work with Albina Danilova.
• Nomura Seminar
18 October 2013
16:00
Phelim Boyle
Abstract
The performance of the shares of a closed end bond fund is based on the returns of an underlying portfolio of bonds. The returns on closed end bond funds are typically higher than those of comparable open ended bond funds and this result is attributed to the use of leverage by closed end bond funds. This talk develops a simple model to assess the impact of leverage on the expected return and riskiness of a closed end bond fund. We illustrate the model with some examples
• Nomura Seminar
14 June 2013
16:00
Abstract
Advanced models such as Lévy models require advanced numerical methods for developing efficient pricing algorithms. Here we focus on PIDE based methods. There is a large arsenal of numerical methods for solving parabolic equations that arise in this context. Especially Galerkin and Galerkin inspired methods have an impressive potential. In order to apply these methods, what is required is a formulation of the equation in the weak sense. We therefore classify Lévy processes according to the solution spaces of the associated parabolic PIDEs. We define the Sobolev index of a Lévy process by a certain growth condition on the symbol. It follows that for Lévy processes with a certain Sobolev index b the corresponding evolution problem has a unique weak solution in the Sobolev-Slobodeckii space with index b/2. We show that this classification applies to a wide range of processes. Examples are the Brownian motion with or without drift, generalised hyperbolic (GH), CGMY and (semi) stable Lévy processes. A comparison of the Sobolev index with the Blumenthal-Getoor index sheds light on the structural implication of the classification. More precisely, we discuss the Sobolev index as an indicator of the smoothness of the distribution and of the variation of the paths of the process. An application to financial models requires in particular to admit pure jump processes as well as unbounded domains of the equation. In order to deal at the same time with the typical payoffs which can arise, the weak formulation of the equation has to be based on exponentially weighted Sobolev-Slobodeckii spaces. We provide a number of examples of models that are covered by this general framework. Examples of options for which such an analysis is required are calls, puts, digital and power options as well as basket options. The talk is based on joint work with Ernst Eberlein.
• Nomura Seminar
7 June 2013
16:00
Abstract
The martingale optimal transportation problem is motivated by model-independent bounds for the pricing and hedging exotic options in financial mathematics. In the simplest one-period model, the dual formulation of the robust superhedging cost differs from the standard optimal transport problem by the presence of a martingale constraint on the set of coupling measures. The one-dimensional Brenier theorem has a natural extension. However, in the present martingale version, the optimal coupling measure is concentrated on a pair of graphs which can be obtained in explicit form. These explicit extremal probability measures are also characterized as the unique left and right monotone martingale transference plans, and induce an optimal solution of the kantorovitch dual, which coincides with our original robust hedging problem. By iterating the above construction over n steps, we define a Markov process whose distribution is optimal for the n-periods martingale transport problem corresponding to a convenient class of cost functions. Similarly, the optimal solution of the corresponding robust hedging problem is deduced in explicit form. Finally, by sending the time step to zero, this leads to a continuous-time version of the one-dimensional Brenier theorem in the present martingale context, thus providing a new remarkable example of Peacock, i.e. Processus Croissant pour l'Ordre Convexe. Here again, the corresponding robust hedging strategy is obtained in explicit form.
• Nomura Seminar
31 May 2013
16:00
Ioannis Karatzas
Abstract
In an equity market with stable capital distribution, a capitalization-weighted index of small stocks tends to outperform a capitalization-weighted index of large stocks.} This is a somewhat careful statement of the so-called "size effect", which has been documented empirically and for which several explanations have been advanced over the years. We review the analysis of this phenomenon by Fernholz (2001) who showed that, in the presence of (a suitably defined) stability for the capital structure, this phenomenon can be attributed entirely to portfolio rebalancing effects, and will occur regardless of whether or not small stocks are riskier than their larger brethren. Collision local times play a critical role in this analysis, as they capture the turnover at the various ranks on the capitalization ladder. We shall provide a rather complete study of this phenomenon in the context of a simple model with stable capital distribution, the so-called Atlas model" studied in Banner et al.(2005). This is a Joint work with Adrian Banner, Robert Fernholz, Vasileios Papathanakos and Phillip Whitman.
• Nomura Seminar
24 May 2013
16:00
Harry Zheng
Abstract
In this paper we prove a weak necessary and sufficient maximum principle for Markov regime switching stochastic optimal control problems. Instead of insisting on the maximum condition of the Hamiltonian, we show that 0 belongs to the sum of Clarke's generalized gradient of the Hamiltonian and Clarke's normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle.
• Nomura Seminar
17 May 2013
16:00
Abstract
We discuss the superhedging problem under model uncertainty based on existence and duality results for minimal supersolutions of backward stochastic differential equations. The talk is based on joint works with Samuel Drapeau, Gregor Heyne and Reinhard Schmidt.
• Nomura Seminar
10 May 2013
16:00
David Hobson
Abstract
Suppose we are given a double continuum (in time and strike) of discounted option prices, or equivalently a set of measures which is increasing in convex order. Given sufficient regularity, Dupire showed how to construct a time-inhomogeneous martingale diffusion which is consistent with those prices. But are there other martingales with the same 1-marginals? (In the case of Gaussian marginals this is the fake Brownian motion problem.) In this talk we show that the answer to the question above is yes. Amongst the class of martingales with a given set of marginals we construct the process with smallest possible expected total variation.
• Nomura Seminar
29 April 2013
12:30
Abstract
The Ross Recovery Theorem gives sufficient conditions under which the market’s beliefs can be recovered from risk-neutral probabilities. His approach places mild restrictions on the form of the preferences of the representative investor. We present an alternative approach which has no restrictions beyond preferring more to less, Instead, we restrict the form and risk-neutral dynamics of John Long’s numeraire portfolio. We also replace Ross’ finite state Markov chain with a diffusion with bounded state space. Finally, we present some preliminary results for diffusions on unbounded state space. In particular, our version of Ross recovery allows market beliefs to be recovered from risk neutral probabilities in the classical Cox Ingersoll Ross model for the short interest rate.
• Nomura Seminar