Semi-Parametric Estimation of American Option Prices

Abstract

An American option provides the right to perform a specified financial transaction (sell, buy, exchange) on or before the contract maturity. Many different contracts traded on centralized and OTC markets are of this kind. In particular, a plain vanilla American option is a contract between two parties concerning the possibility of selling or buying a reference asset (underlying) at a specified price (strike price). Setting the contract price and choosing the best moment for its exercise are two of the most studied problems in finance during the last 40 years. In financial markets, the behavior of the underlying is not predictable. Thus, a description of the probability law governing its stochastic evolution is necessary for the determination of the contract price and the optimal exercise decision.The majority of the existing literature focuses on mathematical and numerical procedures for computing the option price and determining the optimal exercise policy for a given law of motion of the underlying. For these purposes, only a model for the dynamics of the underlying under the risk-neutral distribution is required. When this approach is put into practice, typically a parametric model for such distribution is adopted and the parameters are calibrated on a cross-section of available option prices.On the contrary, in this PhD thesis, that summarizes the research conducted to obtain the degree of Doctor of Philosophy in Economics at the University of Lugano, an econometric framework for the empirical pricing of American options is developed. In this framework, a statistical model for the dynamics of the underlying is specified by the researcher and estimated on available data. Data include both time series of relevant state variables and cross-sections of observed option prices. The estimated model is then used to estimate the price of contracts that are not currently actively traded on the market.The econometric approach proposed in this thesis features three major characteristics. First, it is based on a coherent specification of both historical and risk-neutral dynamics. Second, the statistical model for the dynamics of the underlying is more general than most of the models previously considered in the literature. Third, the model parameters can be consistently estimated even when the amount of option data is limited.In the first three chapters of the thesis, the problem, the proposed solution and an empirical application of the novel method are presented. Chapter 1 introduces the price of an American option as the expected value of the contract at the most remunerative time for exercising it. Some different pricing techniques based on this representation and the way they are used to handle with real data are briefly reviewed. Chapter 2 presents the novel empirical methodology developed in the PhD research.Chapter 3 describes an application of this methodology for the analysis of IBM shares and plain vanilla American options written on them. In the last two chapters of the thesis, the regularity assumptions

Empirical Study:

“…The American put option price for any time-to-maturity and strike price can be expressed as the product of the underlying share price and the American put option-to-share price ratio p. To do so, let us use a result in Gagliardini and Ronchetti [2010] (GR). They show that in the framework of the present paper American-style options with payoff at exercise that is linearly homogeneous w.r.t.…”

“…At every day in July and August 2008, the estimation of the SDF is done in three different ways: using only a time series of share and risk-free asset prices, using only a cross-section of share and risk-free asset prices and option mid-quotes, or using both time series and cross-section. More precisely, the three used estimation techniques are a Generalized Method of Moments estimation (GMM, see Hansen [1982] and Hansen and Singleton [1982]), a cross-sectional calibration and an Extended Method of Moments estimation (XMM, see Gagliardini, Gouriéroux and Renault [2011] and Gagliardini and Ronchetti [2010]). Each methodology finds the values for the SDF parameters that best satisfy a set of empirical counterparts of the no-arbitrage restrictions on the base of a particular criterion.…”

Option pricing with maximum entropy densities: The inclusion of higher‐order moments

“…Nonparametric approaches, however, may suffer from the curse of dimensionality. To overcome the curse of dimensionality, researchers introduce semiparametric estimators for option valuation (Fengler & Hin, 2015; Gagliardini & Ronchetti, 2013).…”