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It is well established that the standard Black-Scholes model does a very poor job in matching the prices of vanilla European options. The implied volatility varies by both time to maturity and by the moneyness of the option. One approach to this problem is to use the market option prices to back out a local volatility function that reproduces the market prices. Since option price observations are only available for a limited set of maturities and strike prices, most algorithms require a smoothing technique to implement this approach. In this paper we modify the implementation of Andersen and Brotherton-Ratcliffe to provide another way of dealing with this issue. Numerical examples indicate that our approach is reasonably successful in reproducing the input prices. Mathematics Subject Classification (2000): 91B28Journal of Economic Literature Classification: C61, C63

The dynamic portfolio selection problem with bankruptcy and nonlinear transaction costs is studied. The portfolio consists of a risk-free asset, and a risky asset whose price dynamics is governed by geometric Brownian motion. The investor pays transaction costs as a (piecewise linear) function of the traded volume of the risky asset. The objective is to find the stochastic controls (amounts invested in the risky and risk-free assets) that maximize the expected value of the discounted utility of terminal wealth. The problem is formulated as a non-singular stochastic optimal control problem in the sense that the necessary condition for optimality leads to explicit relations between the controls and the value function. The formulation follows along the lines of Merton (Merton 1969 Rev. Econ. Stat. 51 , 247–257; Merton 1971 J. Econ. Theory 3 , 373–413) and Bensoussan & Julien (Bensoussan & Julien 2000 Math. Finance 10 , 89–108) in the sense that the controls are the amounts of the risky asset bought and sold, and they are bounded. It differs from the works of Davis & Norman (Davis & Norman 1990 Math. Oper. Res. 15 , 676–713), who use, in the presence of proportional transaction costs, a singular-control formulation in which the controls are rates of buying and selling of the risky asset, and they are unbounded. Numerical results are presented for buy/no transaction and sell/no transaction interfaces, which characterize the optimal policies of a constant relative risk aversion investor. The no transaction region, in the presence of nonlinear transaction costs, is not a cone. The Merton line, on which no transaction takes place in the limiting case of zero transaction costs, need not lie inside the no transaction region for all values of wealth.

The multi-period portfolio-selection problem is formulated as a Markowitz meanvariance optimization problem. It is shown that the single-period Markowitz quadratic programming algorithm can be used to solve the multi-period assetallocation problem with suitable modifications in the covariance and linear constraint matrices. It is assumed that the number of shares invested in risky assets is deterministic and the amount of money invested in the risk-free asset is random at future trading dates when short sales on assets are allowed. The general covariance matrix in the multi-period setup contains intertemporal correlations between assets, in addition to correlations between assets at all trading dates. Analytical solutions for the optimal trading strategy, which is linear in the risk-aversion parameter, are obtained. The efficient frontier is a straight line in the expected return/standard deviation of the portfolio space. When the dynamics of the risky assets follow geometric Brownian motion, it is shown that the time-zero allocations to the risky assets coincide with those obtained by Merton in the continuous-time framework. When short sales are not allowed on assets, the values of the portfolio at future trading dates may not be conserved. In the modified Markowitz mean-variance formulation, the value of the portfolio at future trading dates is conserved in the expected sense, and the optimal trading strategy is selected so that the deviation is minimized in the least-square sense. The efficient frontier is a parabola in the expected return/total variance space when short-sales are allowed and when the portfolio consists of risky assets only. When short sales are not allowed, it is shown that finding the optimal trading strategy is equivalent to solving the single-period Markowitz quadratic programming problem, with suitable modifications in the covariance and linear constraint matrices. By solving the Karush-Kuhn-Tucker conditions, analytical solutions are obtained for a two-period two-assets case, and the Markowitz mean-variance principle is illustrated by solving a test problem numerically using the revised simplex method.