The existence of optimal singular controls for stochastic differential equations

We study a singular control problem where the state process is governed by an Ito stochastic differential equation allowing both classical and singular coutrols. By reformulating the state equation as a martingale problem on an appropriate canonical space, it is shown, under mild continuity conditions on the data, that an optimal control exists. The dynamic programming principle for the problem is established through the method of conditioning and concatenation. Moreover, it is shown that there exists a family of optimal controls such that the corresponding states form a Markov process. When the data is Lipschitz continuous, the value function is shown to be uniformly con tinuous and to be the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman variational inequality. We also provide a description of the continuation region, the region in which the optimal state process is continuous, and we show that there exists a family of optimal controls which keeps the state inside the region after a possible initial jump. The last part is independent of the rest of the thesis. Through stretching of time, the singular control problem is transformed into a new problem that involves only classical control. Such problems are relatively well understood. As a result, it is shown that there exists an optimal control where the classical control variable is in Markovian form and the increment of the singular control variable on any time interval is adapted to the state process on the same time interval.Science, Faculty ofMathematics, Department ofGraduat

Ravnanje s človeškimi viri in identifikacija zaposlenih z delovno organizacijo

Researchers such as Derman and Kani (1994), Dupire (1994), and Rubinstein (1994) have proposed a one-factor model for asset prices that is exactly consistent with all European option prices. In this model, which we refer to as the implied volatility function (IVF) model, the asset price volatility is a function of both time and the asset price. Practitioners often use the IVF model to price exotic options. This paper explores the validity of this. It does so by assuming a two-factor stochastic volatility model for the asset price and examining the way the IVF model prices compound options and barrier options. We find the model works well for compound options, but sometimes gives rise to large pricing errors for barrier options

Volatility surfaces: theory, rules of thumb, and empirical evidence

Implied volatilities are frequently used to quote the prices of options. The implied volatility of a European option on a particular asset as a function of strike price and time to maturity is known as the asset's volatility surface. Traders monitor movements in volatility surfaces closely. In this paper we develop a no-arbitrage condition for the evolution of a volatility surface. We examine a number of rules of thumb used by traders to manage the volatility surface and test whether they are consistent with the no-arbitrage condition and with data on the trading of options on the S&P 500 taken from the over-the-counter market. Finally we estimate the factors driving the volatility surface in a way that is consistent with the no-arbitrage condition.Implied volatility, Volatility surface, Dynamics, No-arbitrage, Empirical results,