Past Nomura Seminar

30 April 2015
16:00
Dr Harry Zheng
Abstract

In this talk we discuss a utility-risk portfolio selection problem. By considering the first order condition for the objective function, we derive a primitive static problem, called Nonlinear Moment Problem, subject to a set of constraints involving nonlinear functions of “mean-field terms”, to completely characterize the optimal terminal wealth. Under a mild assumption on utility, we establish the existence of the optimal solutions for both utility-downside-risk and utility-strictly-convex-risk problems, their positive answers have long been missing in the literature. In particular, the existence result in utility-downside-risk problem is in contrast with that of mean-downside-risk problem considered in Jin-Yan-Zhou (2005) in which they prove the non-existence of optimal solution instead and we can show the same non-existence result via the corresponding Nonlinear Moment Problem. This is joint work with K.C. Wong (University of Hong Kong) and S.C.P. Yam (Chinese University of Hong Kong).

30 March 2015
16:00
Dr Harry Zheng
Abstract

In this talk we discuss a utility-risk portfolio selection problem. By considering the first order condition for the objective function, we derive a primitive static problem, called Nonlinear Moment Problem, subject to a set of constraints involving nonlinear functions of “mean-field terms”, to completely characterize the optimal terminal wealth. Under a mild assumption on utility, we establish the existence of the optimal solutions for both utility-downside-risk and utility-strictly-convex-risk problems, their positive answers have long been missing in the literature. In particular, the existence result in utility-downside-risk problem is in contrast with that of mean-downside-risk problem considered in Jin-Yan-Zhou (2005) in which they prove the non-existence of optimal solution instead and we can show the same non-existence result via the corresponding Nonlinear Moment Problem. This is joint work with K.C. Wong (University of Hong Kong) and S.C.P. Yam (Chinese University of Hong Kong).

12 March 2015
16:00
Tim Siu-Tang Leung
Abstract

The growth of the exchange-traded fund (ETF) industry has given rise to the trading of options written on ETFs and their leveraged counterparts (LETFs). Motivated by a number of empirical market observations, we study the relationship between the ETF and LETF implied volatility surfaces under general stochastic volatility models. Analytic approximations for prices and implied volatilities are derived for LETF ​options, along with rigorous error bounds. In these price and IV expressions, we identify their non-trivial dependence on the leverage ratio. Moreover, we introduce a "moneyness scaling" procedure to enhance the comparison of implied volatilities across leverage ratios, and test it with empirical price data.

5 March 2015
16:00
Abstract
Systemic risk refers to the risk that the financial system is susceptible to failures due to the characteristics of the system itself. The tremendous cost of this type of risk requires the design and implementation of tools for the efficient macroprudential regulation of financial institutions. We propose a novel approach to measuring systemic risk.

Key to our construction is a rigorous derivation of systemic risk measures from the structure of the underlying system and the objectives of a financial regulator. The suggested systemic risk measures express systemic risk in terms of capital endowments of the financial firms. Their definition requires two ingredients: first, a random field that assigns to the capital allocations of the entities in the system a relevant stochastic outcome. The second ingredient is an acceptability criterion, i.e. a set of random variables that identifies those outcomes that are acceptable from the point of view of a regulatory authority. Systemic risk is measured by the set of allocations of additional capital that lead to acceptable outcomes. The resulting systemic risk measures are set-valued and can be studied using methods from set-valued convex analysis. At the same time, they can easily be applied to the regulation of financial institutions in practice.
 
We explain the conceptual framework and the definition of systemic risk measures, provide an algorithm for their computation, and illustrate their application in numerical case studies. We apply our methodology to systemic risk aggregation as described in Chen, Iyengar & Moallemi (2013) and to network models as suggested in the seminal paper of Eisenberg & Noe (2001), see also Cifuentes, Shin & Ferrucci (2005), Rogers & Veraart (2013), and Awiszus & Weber (2015). This is joint work with Zachary G. Feinstein and Birgit Rudloff
24 February 2015
12:30
Tobias Preis
Abstract

In this talk, I will outline some recent highlights of our research, addressing two questions. Firstly, can big data resources provide insights into crises in financial markets? By analysing Google query volumes for search terms related to finance and views of Wikipedia articles, we find patterns which may be interpreted as early warning signs of stock market moves. Secondly, can we provide insight into international differences in economic wellbeing by comparing patterns of interaction with the Internet? To answer this question, we introduce a future-orientation index to quantify the degree to which Internet users seek more information about years in the future than years in the past. We analyse Google logs and find a striking correlation between the country's GDP and the predisposition of its inhabitants to look forward. Our results illustrate the potential that combining extensive behavioural data sets offers for a better understanding of large scale human economic behaviour.

19 February 2015
16:00
Abstract

We consider the dynamic casino gambling model initially proposed by Barberis (2012) and study the optimal stopping strategy of a pre-committing gambler with cumulative prospect theory (CPT) preferences. We illustrate how the strategies computed in Barberis (2012) can be strictly improved by reviewing the entire betting history or by tossing random coins, and explain that such improvement is possible because CPT preferences are not quasi-convex. Finally, we develop a systematic and analytical approach to finding the optimal strategy of the gambler. This is a joint work with Prof. Xue Dong He (Columbia University), Prof. Jan Obloj, and Prof. Xun Yu Zhou.

12 February 2015
16:00
Idris Kharroubi
Abstract
We propose a new probabilistic numerical scheme for fully nonlinear equations of Hamilton-Jacobi-Bellman (HJB) type associated to stochastic control problems, which is based on the a recent Feynman-Kac representation by means of control randomization and backward stochastic differential equation with nonpositive jumps. We study a discrete time approximation for the minimal solution to this class of BSDE when the time step goes to zero, which provides both an approximation for the value function and for an optimal control in feedback form. We obtained a convergence rate without any ellipticity condition on the controlled diffusion coefficient.
5 February 2015
16:00
Michael Sørensen
Abstract

New simple methods of simulating multivariate diffusion bridges, approximately and exactly, are presented. Diffusion bridge simulation plays a fundamental role in simulation-based likelihood inference for stochastic differential equations. By a novel application of classical coupling methods, the new approach generalizes the one-dimensional bridge-simulation method proposed by Bladt and Sørensen (2014) to the multivariate setting. A method of simulating approximate, but often very accurate, diffusion bridges is proposed. These approximate bridges are used as proposal for easily implementable MCMC algorithms that produce exact diffusion bridges. The new method is more generally applicable than previous methods because it does not require the existence of a Lamperti transformation, which rarely exists for multivariate diffusions. Another advantage is that the new method works well for diffusion bridges in long intervals because the computational complexity of the method is linear in the length of the interval. The usefulness of the new method is illustrated by an application to Bayesian estimation for the multivariate hyperbolic diffusion model.

 

The lecture is based on joint work presented in Bladt, Finch and Sørensen (2014).References:

Bladt, M. and Sørensen, M. (2014): Simple simulation of diffusion bridges with application to likelihood inference for diffusions. Bernoulli, 20, 645-675.

Bladt, M., Finch, S. and Sørensen, M. (2014): Simulation of multivariate diffusion bridges. arXiv:1405.7728, pp. 1-30.

29 January 2015
16:00
Jocelyne Bion-Nadal
Abstract

Dynamic risk measuring has been developed in recent years in the setting of a filtered probability space (Ω,(Ft)0≤t, P). In this setting the risk at time t is given by a Ft-measurable function defined as an ”ess-sup” of conditional expectations. The property of time consistency has been characterized in this setting. Model uncertainty means that instead of a reference probability easure one considers a whole set of probability measures which is furthermore non dominated. For example one needs to deal with this framework to make a robust evaluation of risks for derivative products when one assumes that the underlying model is a diffusion process with uncertain volatility. In this case every possible law for the underlying model is a probability measure solution to the associated martingale problem and the set of possible laws is non dominated.

In the framework of model uncertainty we face two kinds of problems. First the Q-conditional expectation is defined up to a Q-null set and second the sup of a non-countable family of measurable maps is not measurable. To encompass these problems we develop a new approach [1, 2] based on the “Martingale Problem”.

The martingale problem associated with a diffusion process with continuous coefficients has been introduced and studied by Stroock and Varadhan [4]. It has been extended by Stroock to the case of diffusion processes with Levy generators [3]. We study [1] the martingale problem associated with jump diffusions whose coefficients are path dependent. Under certain conditions on the path dependent coefficients, we prove existence and uniqueness of a probability measure solution to the path dependent martingale problem. Making use of the uniqueness of the solution we prove some ”Feller property”. This allows us to construct a time consistent robust evaluation of risks in the framework of model uncertainty [2].

References

[1] Bion-Nadal J., Martingale problem approach to path dependent diffusion processes with jumps, in preparation.

[2] Bion-Nadal J., Robust evaluation of risks from Martingale problem, in preparation.

[3] Strook D., Diffusion processes asociated with Levy generators, Z. Wahrscheinlichkeitstheorie verw. Gebiete 32, pp. 209-244 (1975).

[4] Stroock D. and Varadhan S., Diffusion processes with continuous coefficients, I and II, Communications on Pure and Applied Mathematics, 22, pp 345-400 (1969).

 

22 January 2015
16:00
Sebastian Jaimungal
Abstract

This paper introduces a mean field game framework for optimal execution with continuous trading. We generalize the classical optimal liquidation problem to a setting where, in addition to the major agent who is liquidating a large portion of shares, there are a number of minor agents (high-frequency traders (HFTs)) who detect and trade along with the liquidator. Cross interaction between the minor and major agents occur through the impact that each trader has on the drift of the fundamental price. As in the classical approach, here, each agent is exposed to both temporary and permanent price impact and they attempt to balance their impact against price uncertainty. In all, this gives rise to a stochastic dynamic game with mean field couplings in the fundamental price. We obtain a set of decentralized strategies using a mean field stochastic control approach and explicitly solve for an epsilon-optimal control up to the solution of a deterministic fixed point problem. As well, we present some numerical results which illustrate how the liquidating agents trading strategy is altered in the presence of the HFTs, and how the HFTs trade to profit from the liquidating agents trading.

[ This is joint work with Mojtaba Nourin, Department of Statistical Sciences, U. Toronto ]

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