11. Finance

Denada Prifti
28 July 2004

Option valuation is a major accomplishment of modern finance. It spurred the development and widespread use of familiar financial options, such as puts and calls in common assets, as well as exotic options. This thesis reflects both option valuation theory and practice. It aims to provide an accessible and interactive approach with which to explore the theory of option valuation, sensitivities and implied volatility, integrated with user-friendly software.

The accessibility to users of various backgrounds is facilitated by organizational design. After a brief summary within each section, the software application is presented. The main focus hereby is a comprehensive coverage of implementation with XploRe procedures, so called quantlets. The related theory is thereafter outlined. Such an approach offers users, who are confident with finance, to bypass the theory and focus directly on computational tools in software application. For didactic purposes, a quantlet is repeated in different subsections, when designed to carry out several tasks. For example, the quantlet 20527 european () is used either to analytically calculate the price of a European option using the Black-Scholes formula or to calculate its implied volatility, and is therefore presented both in subsections 11.2.3 and 11.5.1.

The basis of any option valuation model involves a description of the stochastic process followed by the underlying asset. Therefore this thesis begins with the analysis of asset price dynamics in section 11.1. Turbulence in financial markets, which cause increases in volatility, are captured by continuous-time

stochastic processes. Shocks, such as market crashes, which cause discontinuity on the observed underlying price, are incorporated within a jump-diffusion process.

Two well-known approaches to option valuation are presented and illustrated through computational examples: the continuous-time Black-Scholes model and the binomial tree model.

The Black-Scholes model has become the basic benchmark for pricing equity and commodity options. It is also used in modified form to price Eurodollar future options, foreign currency, Treasury bond options, caps and floors. Section 11.2 firstly describes the classical Black-Scholes model for pricing European options both on dividend and non-dividend underlying stocks. It then extends to the class of quadratic approximation methods for pricing American options within the Black-Scholes world.

Section 11.3 discusses the binomial tree model, which provides discrete approximation to the continuous process underlying the Black-Scholes model. It basically solves the same equation, using a numerical procedure, as opposed to an analytical approach. In doing so, it offers opportunities along the way to check for early exercise of American options. This is the advantage that the binomial model has compared to the classical Black-Scholes model. The binomial model is very important, because it shows how to get away from a reliance on closed form solutions and is a means of valuing options that relies on simple, fast and accurate numerical methods.

The benefit of option valuation models, is not necessarily to provide the "right" price. The best pricing method adopted until now is the market price - an efficient market will provide the best and the truest price for options. The true benefits of option valuation models are that they provide an accurate "snap shot" of the current market conditions, and more importantly, they break the option market price into each of the factors that comprises it. Thus each factor can be examined separately and its contribution to the determination of the option price assessed. Relying on option valuation models, it is possible to predict how the market price of the option should change, when for example the key factor - volatility -, or any other factor changes. The sensitivities of the option price to these factors are known as Greeks. Section 11.4 addresses these sensitivities and their importance to option traders.

An additional benefit of option valuation models is that they can be used to extract information from option markets. For example, extracting market expectations from options data is the purpose of calculating implied volatilities from option prices. Implied volatility is the risk perceived by the market today and is built into the time value of the option premium. It is therefore a crucial indicator that traders observe to assist them in assessing the value of an option. However, when the Black Scholes formula is inverted to imply volatilities from option market prices, the volatility estimates vary with both strike and expiration. The convex shape of the implied volatility with respect to differing strike values, is referred to as the smile effect. Section 11.5 explores implied volatilities, their computation and the smile effect. This section then continues with discussing implied tree theories extending the Black-Scholes theory, making it consistent with the smile. Implied trees are constructed in order for local volatility to vary from node to node, resulting in a flexible tree, where the market price of all standard options can be matched.