In this talk we discuss a utilityrisk 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 “meanfield terms”, to completely characterize the optimal terminal wealth. Under a mild assumption on utility, we establish the existence of the optimal solutions for both utilitydownsiderisk and utilitystrictlyconvexrisk problems, their positive answers have long been missing in the literature. In particular, the existence result in utilitydownsiderisk problem is in contrast with that of meandownsiderisk problem considered in JinYanZhou (2005) in which they prove the nonexistence of optimal solution instead and we can show the same nonexistence 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).
Past Nomura Seminar
In this talk we discuss a utilityrisk 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 “meanfield terms”, to completely characterize the optimal terminal wealth. Under a mild assumption on utility, we establish the existence of the optimal solutions for both utilitydownsiderisk and utilitystrictlyconvexrisk problems, their positive answers have long been missing in the literature. In particular, the existence result in utilitydownsiderisk problem is in contrast with that of meandownsiderisk problem considered in JinYanZhou (2005) in which they prove the nonexistence of optimal solution instead and we can show the same nonexistence 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).
The growth of the exchangetraded 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 nontrivial 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.
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 setvalued and can be studied using methods from setvalued convex analysis. At the same time, they can easily be applied to the regulation of financial institutions in practice.
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 futureorientation 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. 
We consider the dynamic casino gambling model initially proposed by Barberis (2012) and study the optimal stopping strategy of a precommitting 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 quasiconvex. 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.
New simple methods of simulating multivariate diffusion bridges, approximately and exactly, are presented. Diffusion bridge simulation plays a fundamental role in simulationbased likelihood inference for stochastic differential equations. By a novel application of classical coupling methods, the new approach generalizes the onedimensional bridgesimulation 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, 645675.
Bladt, M., Finch, S. and Sørensen, M. (2014): Simulation of multivariate diffusion bridges. arXiv:1405.7728, pp. 130.
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 Ftmeasurable function defined as an ”esssup” 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 Qconditional expectation is defined up to a Qnull set and second the sup of a noncountable 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] BionNadal J., Martingale problem approach to path dependent diffusion processes with jumps, in preparation. [2] BionNadal 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. 209244 (1975). [4] Stroock D. and Varadhan S., Diffusion processes with continuous coefficients, I and II, Communications on Pure and Applied Mathematics, 22, pp 345400 (1969).

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 (highfrequency 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 epsilonoptimal 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 ]