Volume 2003http://edoc.hu-berlin.de/18452/3602021-09-23T03:34:18Z2021-09-23T03:34:18ZEfficient point methods for probabilistic optimization problemsDentcheva, DarinkaLai, BogumilaRuszczynski, Andrzejhttp://edoc.hu-berlin.de/18452/89592020-03-07T04:14:17Z2003-10-20T00:00:00ZEfficient point methods for probabilistic optimization problems
Dentcheva, Darinka; Lai, Bogumila; Ruszczynski, Andrzej
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
We consider nonlinear stochastic programming problems with probabilistic constraints. The concept of a p-efficient point of a probability distribution is used to derive equivalent problem formulations, and necessary and sufficient optimality conditions. We analyze the dual functional and its subdifferential. Two numerical methods are developed based on approximations of the p-efficient frontier. The algorithms yield an optimal solution for problems involving r-concave probability distributions. For arbitrary distributions, the algorithms provide upper and lower bounds for the optimal value and nearly optimal solutions. The operation of the methods is illustrated on a cash matching problem with a probabilistic liquidity constraint.
2003-10-20T00:00:00ZPortfolio optimization with stochastic dominance constraintsDentcheva, DarinkaRuszczynski, Andrzejhttp://edoc.hu-berlin.de/18452/89582020-03-07T04:14:17Z2003-09-30T00:00:00ZPortfolio optimization with stochastic dominance constraints
Dentcheva, Darinka; Ruszczynski, Andrzej
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return. We develop optimality and duality theory for these models. We construct equivalent optimization models with utility functions. Numerical illustration is provided.
2003-09-30T00:00:00ZArbitrage pricing simplifiedKallio, MarkkuZiemba, William T.http://edoc.hu-berlin.de/18452/89572020-03-07T04:14:17Z2003-09-30T00:00:00ZArbitrage pricing simplified
Kallio, Markku; Ziemba, William T.
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
The paper derives fundamental arbitrage pricing results in finite dimensions in a simple unified framework using Tucker's theorem of the alternative. Frictionless results plus those with dividends, periodic interest payments, transaction costs, different interest rates for lending and borrowing, shorting costs and constrained short selling are presented. While the results are mostly known and appear in various places, our contribution is to present them in a coherent and comprehensive fashion with very simple proofs. The analysis yields a simple procedure to prove new results and some are presented for cases with frictions.
2003-09-30T00:00:00ZOn Leland's option hedging strategy with transaction costsZhao, YongganZiemba, William T.http://edoc.hu-berlin.de/18452/89562020-03-07T04:14:17Z2003-09-30T00:00:00ZOn Leland's option hedging strategy with transaction costs
Zhao, Yonggan; Ziemba, William T.
Higle, Julie L.; Römisch, Werner; Sen, Surrajeet
Nonzero transaction costs invalidate the Black-Scholes (1973) arbitrage argument based on continuous trading. Leland (1985) developed a hedging strategy which modifies the Black-Scholes hedging strategy with a volatility adjusted by the length of the rebalance interval and the rate of the proportional transaction cost. Leland claimed that the exact hedge could be achieved in the limit as the length of rebalance intervals approaches zero. Unfortunately, the main theorem (Leland 1985, P1290) is in error. Simulation results also confirm opposite findings to those in Leland (1985). Since standard delta hedging fails to exactly replicate the option in the presence of transaction costs, we study a pricing and hedging model which is similar to the delta hedging strategy with an endogenous parameter, namely the volatility, for the calculation of delta over time. With transaction costs, the optimally adjusted volatility is substantially different from the stock's volatility under the criterion of minimizing the mean absolute replication error weighted by the probabilities that the option is in or out of the money. This model partially explains the phenomenon that the implied volatilities with equity options are skewed. Data on S&P500 index cash options from January to June 2002 are used to illustrate the model. Option prices from our model are highly consistent with the Black-Scholes option prices when transaction costs are zero.
2003-09-30T00:00:00Z