July 17, 2024

Financial Mathematics Seminars - November 22, 2000

4:30 - 5:30 p.m.

On the Nature of Options

Dr. Peter Carr, Bank of America Securities

We consider the role of options when markets in its underlying asset are frictionless and when the stochastic process for this underlying is unknown. By adding a static option position to a particular dynamic hedging strategy in its underlying, we show that the option allows the investor to trade at the option's strike price, even if the market jumps over this price level. Thus, an option provides liquidity at its strike even when the market doesn't. We then present two methods for extending this local liquidity to every price between the pre and post jump level. The first method involves holding a continuum of options of all strikes. The second method holds one option, but adjusts the dynamic hedging strategy. We discuss the advantages and disadvantages of each approach and consider the benefits of combining them.

6:00 - 7:00 p.m.

Estimation of Continuous Time Processes via the Empirical Characteristic Function

Professor George Jiang, Schulich School of Business, York University

This paper examines a particular class of continuous-time stochastic processes commonly known as affine diffusions (AD) and affine jump-diffusions (AJD) in which the drift, the diffusion and the jump coefficients are all affine functions of the state variables. By deriving the joint characteristic function associated with a vector of observed state variables for such models, we are able to examine the statistical properties of these diffusions and jump-diffusions as well as develop an efficient estimation technique based on empirical characteristic functions (ECF) and a GMM estimation procedure based on exact moment conditions. The estimators developed in this paper are in stark contrast to those available in the literature in the sense that our methods require neither discretization nor simulation. We demonstrate that our methods are, in particular, useful for the AD and AJD models with latent variables, i.e. the case where some of the state variables are unobserved. We illustrate our approach with a detailed examination of the continuous-time square-root stochastic volatility (SV) model, along with an empirical application using S&P 500 index returns.