Selected Publications and Preprints:

Joint work with Jing Li, "Minimizing conditional Value-at-Risk under constraint on expected value", preprint, 2009. Here is the PDF file. Abstract: This paper gives a complete solution to the problem of the type:

inf E[(x - X)^+] subject to E[X] = z; ~E[X] = x_r; x_d <= X <= x_u a.s.;

where the constants satisfy -infinity < x_d < x_r < x_u <= infinity, x in R, z in R. The expectations E[] and ~E[] are taken under two equivalent probability measures P and ~ P under the assumption that the Radon-Nikodym derivative has a continuous distribution. The result is then used to find the optimal dynamic portfolio which minimizes the Conditional Value-at-Risk in a complete market model. We show the efficient frontier for a mean-CVaR portfolio selection problem in the Black-Scholes' model.

"Infinite horizon optimal search problem with hiring and firing options", preprint, 2009. Here is the PDF file. Abstract: As in the classic `Secretary Problem': the candidates arrive sequentially. In this paper, they are represented with i.i.d. Ito diffusion processes. Two interwoven sequences of optimal stopping times are decided which signify the hiring and fring of each candidate. The goal is to choose the stopping times to maximize expected sum of benefit and costs and the time horizon is infinite. The optimality conditions in terms of Martingale Characterization, Least Superharmonic Majorant, and Variational Inequalities are given. The calculation for the simple Brownian case with linear cost/benefit functions is demonstrated.

Joint work with Jing Li, "Risk minimizing portfolio optimization and hedging with conditional Value-at-Risk", Review of Futures Markets, 16, 471-506, 2008. Here is the PDF file. Abstract: We look at the problem of how to find a dynamic optimal portfolio so that the Conditional Value-at-Risk (CVaR) is minimized under the condition where the returns are bounded. CVaR is a coherent risk measure based on the popular VaR. In a complete market setting, we derive the exact optimal conditions. Then we provide applications in two classic complete market models: the Binomial model and the Black-Scholes model. In these cases, the procedures to find the optimal strategies are given with exact formulas. Numerical results show, as expected, dynamic portfolio provide much lower CVaR risk than static portfolios.

Joint work with Lloyd Blenman, "Joint ventures, risk sharing and optimal contract design", preprint, 2009. Here is the PDF file. A brief description: We study risk sharing in a joint venture. It is basically a Principal-Principal problem (vs. Principal-Agent problem). We look for the conditions under which both venture partners will agree on a risk sharing rule where their expected utilities are simultaneously maximized so that the result is robust to renegotiation. This is a much stronger optimality condition than the usual Pareto Optimal. The key is that we allow the design of the risk sharing contract to be characterized by an additional parameter. Then it is often the case that there exists a contract so that the strong optimality condition is satisfied. Therefore, the optimal contract design allows the existence of a robust renegotiation proof result of risk sharing, which very surprisingly, turns out to an invariant of the contract-specific characteristics, and it is not a consequence of the simple CARA utility functions we used. We also impose a super-martingale criterion so that there is no incentive for the venture partners to delay the initiation of the risk sharing contract.

Joint work with Lloyd Blenman, "Joint ventures and risk sharing", Journal of Business and Entrepreneurship, 21, 96-107, 2009. Here is the PDF file.

Joint work with Libor Pospisil and Jan Vecer, "The cost of negative returns", preprint, 2007. Here is the PDF file. Abstract: We study the impact of negative returns on the health of a given financial portfolio. It is often the case that a series of significant negative returns trigger a credit event such as a downgrade in rating, or even a default of the portfolio owner. We focus our attention on a Weighted Average of Ordered Returns, which is a statistic that allows us to weight returns according to their relative adverse impact. We use an option pricing approach to derive the theoretical price and properties of a forward, a swap contract, a call and a put option written on the Weighted Average of Ordered Returns under different assumptions about the distribution of returns. The models of returns considered in this paper are defined by the following underlying price processes: geometric Brownian motion, Merton model with Poisson jumps, and GARCH model. We present a convergence result which states that the price of a forward on the Weighted Average of Ordered Returns converges to the theoretical law invariant coherent risk measure. Finally, we show that the forward price process itself satisfies the axioms of a dynamic coherent risk measures.

Joint work with Kiseop Lee, "Parameter estimation from multinomial trees to jump diffusions with K means clustering", Risk, 21, 82-86, 2008. Here is the PDF file. Abstract: Ever since the pioneering work of Cox, Ross and Rubinstein [12], tree models have been popular among asset pricing methods. On the other hand, statistical estimation of parameters of tree models has not been studied as much. In this paper, we use K Means Clustering method to estimate the parameters of multinomial trees. By the weak convergence property of multinomial trees to continuous-time models, we show that this method can be in turn used to estimate parameters in continuous time models, illustrated by an example of jump-diffusion model.

Joint work with Masahiko Egami, "A continuous-time search model with job switch and jumps", to appear in Mathematical Methods of Operations Research, 2008. Here is the PDF file. Abstract: We study a new search problem in continuous time. In the traditional approach, the basic formulation is to maximize the expected (discounted) return obtained by taking a job, net of search cost incurred until the job is taken. Implicitly assumed in the traditional modeling is that the agent has no job at all during the search period or her decision on a new job is independent of the job situation she is currently engaged in. In contrast, we incorporate the fact that the agent has a job currently and starts searching a new job. Hence we can handle more realistic situation of the search problem. We provide optimal decision rules as to both quitting the current job and taking a new job as well as explicit solutions and proofs of optimality. Further, we extend to a situation where the agent’s current job satisfaction may be affected by sudden downward jumps (e.g. de-motivating events), where we also find an explicit solution; it is rather a rare case that one finds explicit solutions in control problems using a jump diffusion.

"Risk measure pricing and hedging in incomplete markets", Annals of Finance, 2, 51-71, 2006. Here is the PDF file. Abstract: This article attempts to extend the complete market option pricing theory to in-complete markets. Instead of eliminating the risk by a perfect hedging portfolio, partial hedging will be adopted and some residual risk at expiration will be tolerated. The risk measure (or risk indifference) prices charged for buying or selling an option are associated to the capital required for dynamic hedging so that the risk exposure will not increase. The associated optimal hedging portfolio is decided by minimizing a convex measure of risk. I will give the definition of risk-efficient options and confirm that options evaluated by risk measure pricing rules are indeed risk-efficient. Relationships to utility indifference pricing and pricing by valuation and stress measures will be discussed. Examples using the shortfall risk measure and average VaR will be shown.

Joint work with Jan Vecer, "Pricing Asian options in a semimartingale model", Quantitative Finance, 4, 170-175, 2004. Here is the PDF file. Abstract: In this article we study arithmetic Asian options when the underlying stock is driven by special semimartingale processes. We show that the inherently path dependent problem of pricing Asian options can be transformed into a problem without path dependency in the payoff function. We also show that the price satisfies a simpler integro-differential equation in the case the stock price is driven by a process with independent increments, Levy process being a special case.

Joint work with Jan Vecer, "Mean comparison theorem cannot be extended to Poisson case, Journal of Applied Probability, 41, 4, 1199-1202, 2004. Here is the PDF file. Abstract: In this paper, we show that the Mean Comparison Theorem which is valid for Brownian motion, cannot be extended to Poisson process. A counter example in the Poisson case, for which the Mean Comparison Theorem does not hold, is provided.

Joint work with Steven Shreve, "Minimizing shortfall risk using duality approach - an application to partial hedging in incomplete markets", Ph.D. thesis, 2004. Here is the PDF file. Abstract: Duality results are established for expected shortfall risk minimization in a semimartingale model. These in turn provide upper bounds for the minimal risk and a jump-diffusion example is given. Also being proved is a particular predictable criterion for the dual space (Kramkov-Schachermayer sense) in the jump-diffusion context.