# Past Nomura Seminar

26 April 2013
16:00
Mete Soner
Abstract
The original transport problem is to optimally move a pile of soil to an excavation. Mathematically, given two measures of equal mass, we look for an optimal bijection that takes one measure to the other one and also minimizes a given cost functional. Kantorovich relaxed this problem by considering a measure whose marginals agree with given two measures instead of a bijection. This generalization linearizes the problem. Hence, allows for an easy existence result and enables one to identify its convex dual. In robust hedging problems, we are also given two measures. Namely, the initial and the final distributions of a stock process. We then construct an optimal connection. In general, however, the cost functional depends on the whole path of this connection and not simply on the final value. Hence, one needs to consider processes instead of simply the maps S. The probability distribution of this process has prescribed marginals at final and initial times. Thus, it is in direct analogy with the Kantorovich measure. But, financial considerations restrict the process to be a martingale Interestingly, the dual also has a financial interpretation as a robust hedging (super-replication) problem. In this talk, we prove an analogue of Kantorovich duality: the minimal super-replication cost in the robust setting is given as the supremum of the expectations of the contingent claim over all martingale measures with a given marginal at the maturity. This is joint work with Yan Dolinsky of Hebrew University.
• Nomura Seminar
8 March 2013
16:00
Abstract
A celebrated financial application of convex duality theory gives an explicit relation between the following two quantities: (i) The optimal terminal wealth X*(T) := Xφ* (T) of the classical problem to maximise the expected U-utility of the terminal wealth Xφ(T) generated by admissible portfolios φ(t); 0 ≤ t ≤ T in a market with the risky asset price process modeled as a semimartingale; (ii) The optimal scenario dQ*/dP of the dual problem to minimise the expected V -value of dQ/dP over a family of equivalent local martingale measures Q. Here V is the convex dual function of the concave function U. In this talk we consider markets modeled by Itô-Lėvy processes, and we present in a first part a new proof of the above result in this setting, based on the maximum principle in stochastic control theory. An advantage with our approach is that it also gives an explicit relation between the optimal portfolio φ* and the optimal scenario Q*, in terms of backward stochastic differential equations. In a second part we present robust (model uncertainty) versions of the optimization problems in (i) and (ii), and we prove a relation between them. We illustrate the results with explicit examples. The presentation is based on recent joint work with Bernt ¬Oksendal, University of Oslo, Norway.
• Nomura Seminar
1 March 2013
16:00
Abstract
The banking industry lost a trillion dollars during the global financial crisis. Some of these losses, if not most of them, were attributable to complex derivatives or securities being incorrectly priced and hedged. We introduce a new methodology which provides a better way of trying to hedge and mark-to-market complex derivatives and other illiquid securities which recognise the fundamental incompleteness of markets and the presence of model uncertainty. Our methodology combines elements of the No Good Deals methodology of Cochrane and Saa-Requejo with the Robustness methodology of Hansen and Sargent. We give some numerical examples for a range of both simple and complex problems encompassing not only financial derivatives but also “real options”occurring in commodity-related businesses.
• Nomura Seminar
8 February 2013
16:00
Dirk Becherer
Abstract
We discuss sparse portfolio optimization in continuous time. Optimization objective is to maximize an expected utility as in the classical Merton problem but with regularizing sparsity constraints. Such constraints aim for asset allocations that contain only few assets or that deviate only in few coordinates from a reference benchmark allocation. With a focus on growth optimization, we show empirical results for various portfolio selection strategies with and without sparsity constraints, investigating different portfolios of stock indicies, several performance measures and adaptive methods to select the regularization parameter. Sparse optimal portfolios are less sensitive to estimation errors and performance is superior to portfolios without sparsity constraints in reality, where estimation risk and model uncertainty must not be ignored.
• Nomura Seminar
1 February 2013
16:00
Teemu Pennanen
Abstract
We study portfolio optimization and contingent claim valuation in markets where illiquidity may affect the transfer of wealth over time and between investment classes. In addition to classical frictionless markets and markets with transaction costs, our model covers nonlinear illiquidity effects that arise in limit order markets. We extend basic results on arbitrage bounds, attainable claims and optimal portfolios to illiquid markets and general swap contracts where both claims and premiums may have multiple payout dates. We establish the existence of optimal trading strategies and the lower semicontinuity of the optimal value of portfolio optimization under conditions that extend the no-arbitrage condition in the classical linear market model.
• Nomura Seminar
25 January 2013
16:00
Abstract
A multi-dimensional extension of the structural default model with firms' values driven by diffusion processes with Marshall-Olkin-inspired correlation structure is presented. Semi-analytical methods for solving the forward calibration problem and backward pricing problem in three dimensions are developed. The model is used to analyze bilateral counter- party risk for credit default swaps and evaluate the corresponding credit and debt value adjustments.
• Nomura Seminar
18 January 2013
16:00
Shige Peng
Abstract
The models of Brownian motion, Poisson processes, Levy processes and martingales are frequently used as basic formulations of prices in financial market. But probability and/or distribution uncertainties cause serious problems of robustness. Nonlinear expectations (G-Expectations) and the corresponding martingales are useful tools to solve them.
• Nomura Seminar
30 November 2012
16:00
Albert Ferreiro-Castilla
Abstract
In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was introduced for a large family of L\'evy processes that is based on the Wiener-Hopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology. We also provide here a theoretical analysis of the new Monte Carlo simulation technique in Kuznetsov et al. (2011) and of its multilevel variant. We find that the rate of convergence is uniformly with respect to the jump activity'' (e.g. characterised by the Blumenthal-Getoor index).
• Nomura Seminar