In this talk, we consider a set of the minimal copulas with respect to the concordance order.

The proposed set of minimal copulas includes d-countermonotonic copulas defined in Lee and Ahn (2014), which is known to be useful in many financial optimization problems.

(The talk is based on joint work with Jaeyoun Ahn and Woojoo Lee)
 

In this talk, Data Visualization of Multi Dimensional data is discussed with a key focus on Financial Risk. Today's leading investors are struggling to properly define and measure risk within their own investment portfolios and correctly communicate this to the outside world. Visualizations are essential to understanding the complex entanglement that one finds within the portfolios of large Pension Funds and Life Insurance Companies. Through different examples, key risk measures will be compared from a historical and forward looking aspect. Visualization of volatility curves will be a central part to this talk and a connection will be made to risk measures such as VaR, CVaR & Drawdown Risk.
 

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 LETFimplied 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 for comparing implied volatilities across leverage ratios, and test it with empirical price data.
 

In this talk, we discuss the issues of the applying the empirical mode decomposition to financial data analysis. Examples and empirical results will be given for illustration.
 

This paper attempts to provide a decision-theoretic foundation for the measurement of economic tail risk, which is not only closely related to utility theory but also relevant to statistical model uncertainty. The main result is that the only tail risk measure that satisfies a set of economic axioms for the Choquet expected utility and the statistical property of elicitability (i.e. there exists an objective function such that minimizing the expected objective function yields the risk measure) is median shortfall, which is the median of tail loss distribution. Elicitability is important for backtesting. We also extend the result to address model uncertainty by incorporating multiple scenarios. As an application, we argue that median shortfall is a better alternative than expected shortfall for setting capital requirements in Basel Accords.
 

This talk develops approximation methods for path-dependent functionals, which have been used in many applications involving path-dependent objective functions. In contrast to the traditional approach, this work provides a non-traditional convergence method in Monte Carlo analysis based on actual computations under the Skorohod topology. Some examples such as the approximation of discretely monitoring barrier option are considered.
 

In the previous paper "Mean Field Games and Systemic Risk", we proposed a simple homogeneous model of inter-bank borrowing and lending. We now consider heterogeneous grouping cases where parameters are identical within their own groups but different between groups. Due to this heterogeneity, a central bank has to keep deposits or provide extra cash flow instead of acting as a clearing house and systemic risk happens in the more complicated manner than the homogeneous case.
 

Derivative modeling starts with a knowledge about the statistics of certain market prices. New products are created based on the assumption on the probability distribution about the market price movement. The accounting practice and the replication trades are based on this probability distribution.

In practice, this distribution assumption is not fixed through the tenor of the product. Several facts, including the impact of new trading activities of the new product on the underlying market, and statistical uncertainties, force the practitioners to constantly search for a perfect model.

This talk provides a framework to describe the trader's behavior and their utility of minimizing the total hedging slippage. Choice of types of distribution, parameters in the distribution, choice of the rule of forming daily re-hedge portfolios, can be optimized to the above utility. Through the examples from interest rate and equity derivative products, we demonstrate that it is possible to stay with simple analytic model while still benefit from highly structured statistical knowledge, and it is practically "optimal" to employ models that are product dependent, defying a unification theory.
 

We consider a discrete-time model in which death benefits can depend on a stock price index, the logarithm of which is modeled as a random walk. Examples of such benefit payments include put and call options, barrier options, and lookback options. Because the distribution of the curtate-future-lifetime can be approximated by a linear combination of geometric distributions, it suffices to consider curtate-future-lifetimes with a geometric distribution. In the binomial tree model, closed-form expressions for the expectations of the discounted benefit payment are obtained for a series of options. They are based on results concerning geometric stopping of a random walk, in particular also on a version of the Wiener-Hopf factorization.

(This is a joint paper with Hans U. Gerber and Elias S.W. Shiu)
 

We consider the risk-constrained portfolio selection problems arising from an ordinary investor or an insurer who can invest her surplus into financial market. For an insurer, the optimal investment and reinsurance problem is studied. The goal is to maximize the expected utility of terminal wealth. In particular, a Markovian regime-switching environment is considered. The dynamic risk constraint is described by the maximal value-at-risk over different economic states. We will investigate the impacts of the risk constraint and switching regimes on the optimal strategies.

(Joint work with J.Z. Liu, K. Siu and W.K. Ching)
 

To solve the high-dimensional Markowitz optimization problem, a new approach combining sparse regression and estimation of maximum expected return for a given risk level based on random matrix theory is proposed. We prove that under some sparsity assumptions on the underlying optimal portfolio, our estimated portfolio, the Response-estimated Sparse Regression Portfolio (ReSReP), asymptotically reaches the maximum expected return and meanwhile satisfies the risk constraint. To the best of our knowledge, this is the first time that these two goals are simultaneously achieved in the high-dimensional setting. The superior properties of ReSReP are demonstrated via simulation and extensive empirical studies.

(Based on joint work with Mengmeng Ao and Yingying Li)
 

In practice short term price movements inferred from order book data, which are sometimes referred as short term momentum indicators or short term momentum signals, play an emerging crucial role in the decision of child order placement. However, the modeling of momentum indicators has not been explicitly studied in the existing literature. In this work, we propose to model explicitly the short term momentum indicator and formulate the child order placement problem as an optimal multiple stopping problem. We provide theoretical study on the problem over infinite time horizon, and numerical approximation for that over finite time horizon. 

(Joint work with S.C.P. Yam)

We consider an N-player interacting strategic game in the presence of a (endogenous) dominating player, who gives direct influence on individual agents, through its impact on their control in the sense of Stackelberg game, and then on the whole community. Each individual agent is subject to a delay effect on collecting information, specifically at a delay time, from the dominating player. The size of his delay is completely known by the agent; while to others, including the dominating player, his delay plays as an hidden random variable coming from a common fixed distribution. By invoking a non-canonical fixed point property, we show that, for a general class of finite N-player games, each of them converges to the mean field counterpart which may possess an optimal solution that can serve as an epsilon-Nash equilibrium for the corresponding finite $N$-player game. Secondly, we provide, with explicit solutions, a comprehensive study on the corresponding linear quadratic mean field games of small agents with delay from a dominating player. Due to the non-Brownian nature of the filtration, for the representative agent, being information flow obtained from both the dominating player and the whole community via the mean field term, we propose to utilize Backward Stochastic Dynamics (instead of the common approach through BSDEs) for the construction of adjoint process for the resolution of his optimal control. A simple sufficient condition for the unique existence of mean field equilibrium is provided by tackling a class of non-symmetric Riccati equations. Finally, via a study of a class of forward backward stochastic functional differential equations, the optimal control of the dominating player is granted given the unique existence of the mentioned mean field equilibrium for small players.
 

This paper investigates the high-dimensional portfolio selection problem in which the number of risky assets (p) is greater than the number of observation times (n). We propose a combined estimation-optimization (CEO) approach that applies the $\ell_1$-constrained minimization to directly estimate the optimal control of the mean-variance portfolio (MVP) problems under single-period and multiple-period settings with historical data. We prove that the use of the traditional plug-in empirical mean and variance-covariance matrix makes the MVP strategy tends to a random stock picking (monkey picking) strategy, which offers positive Sharpe ratio with probability of 50% for p>>n and a large n. In addition, the multi-period solution has no significant improvement compared with the single-period strategy for p>>n and a large n. However, the CEO approach tends to the correct optimal solution for a large n for all aforementioned model settings. The distinctive advantages of the CEO approach over its competitive methods are the simple implementation scheme, the guarantee of existence of solution even when the empirical variance-covariance matrix is singular, the application beyond Gaussian distribution of stock returns, the application beyond single-period model and the genuine selection of stocks into the portfolio. In other words, the last advantage means that the scheme automatically filters out unfavourable stocks based on historical data so that the portfolio size N is much less than the number of available stocks p in the market. This also facilitates the 1/N portfolio strategy by considering appropriate stocks recommended by data. Numerical and empirical studies compare the performances between the CEO approach and other existing competitor schemes.