Rendiconti per gli Studi Economici Quantitativi (original) (raw)

Essays on Portfolio Optimization, Simulation and Option Pricing

2014

This thesis consists of three papers which cover the efficient Monte Carlo simulation in option pricing, the application of realized volatility in trading strategies and geometrical analysis of a four asset mean variance portfolio optimization problem. The first paper studies different efficient simulation methods to price options with different characters such as moneyness and maturity times. The incomplete market environments are also been considered. The second paper uses realized volatility based on high frequency data to improve the volatility trading strategy. The performance is compared with that using the implied volatility. The last paper reexamines the Markowitz's portfolio optimization problem using a general case. It also extends the problem to four assets, it describes the exact mean variance efficient fronter in the weight space and studies the frontier in the mean variance space. The thesis may serve to help our understanding of how to apply numerical and analytical methods to solve financial problems.

Discrete and continuous time approximations of the optimal exercise boundary of American options

2002

The valuation of American-style options gives rise to an optimal stopping problem involving the computation of a time dependent exercise boundary over the whole life of the contract. An exact computational formula for this time dependent optimal boundary is not known. Nevertheless, some numerical approaches can be proposed to approximate the optimal boundary. In particular, in this contribution we study three different numerical techniques: an improved lattice method, a randomization approach based on the American option valuation procedure proposed by Carr and an analytic approximation procedure proposed by Bunch and Johnson. The three techniques studied are tested and compared through a wide empirical analysis.

HEDGING BY SEQUENTIAL REGRESSION: AN INTRODUCTION TO THE MATHEMATICS OF OPTION TRADING

1989

It is widely acknowledge that there has been a major breakthrough in the mathematical theory of option trading. This breakthrough, which is usually sum- marized by the Black-Scholes formula, has generated a lot of excitement and a certain mystique. On the mathematical side, it involves advanced probabilistic techniques from martingale theory and stochastic calculus which are accessible only to a

Numerical Valuation of European and American Options under Kou's Jump-diffusion Model

Numerical methods are developed for pricing European and American options under Kou's jump-diffusion model, which assumes the price of the underlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is log-double-exponentially distributed. The price of a European option is given by a partial integro-differential equation (PIDE), while American options lead to a linear complementarity problem (LCP) with the same operator. Spatial differential operators are discretized using finite differences on nonuniform grids, and time stepping is performed using the implicit Rannacher scheme. For the evaluation of the integral term easy-to-implement re-cursion formulas are derived which have optimal computational cost. When pricing European options the resulting dense linear systems are solved using a stationary iteration. For American options two ways to solve the LCPs are described: an operator slitting method and a penalty method. Numerical experiments confirm that the developed methods are very efficient, as fairly accurate option prices can be computed in a few milliseconds on a PC.