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Valorização de opções em mercados de capitais ilíquidos
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This dissertation discusses option pricing within the framework of incomplete markets. Incompleteness in this work is caused by illiquidity of the underlying asset and is presented in Chapter 1. Here, illiquidity is related to the impossibility of
transacting the underlying asset and not directly related with with the cost of doing it. Traditional option pricing theory assumes that hedging portfolios may be adjusted at every point in time. However, this assumption is not valid when the underlying asset is illiquid and agents are not able to transact the underlying asset whenever they want. The aim of this work is to study the impact of this trading restriction on the pricing of options written on such assets.
In Chapter 2, bounds are obtained for the bid and ask prices of an option. These bounds are positively related with illiquidity of the underlying stock. In that Chapter, the analysis is extended to consider both the effect of longer time -to -maturity and
the effect of a higher level of illiquidity in the stock market. Then, the model is developed for the case of continuous time. One concludes that the bid and ask bounds for the option prices follow a modified Black and Scholes (1973) formula. The partial differential equation for the upper bound of the price of a call option is simply the Black and Scholes partial differential equation where the variance of the underlying asset has been increased, whereas in the case of the lower bound, the Black and Scholes formula works as if the variance of the underlying asset has been decreased.
The goal of Chapter 3 is to characterize equilibrium prices for options in incomplete markets. The methodology of Chapter 2 only allows the determination of bounds to the bid and ask option prices outside which a market -maker has a positive
profit with probability one. In other words, an arbitrage opportunity exists if the market -maker sells options above the upper bound or buys options below the lower bound. Clearly, this approach does not determine equilibrium prices. Considering
the possibility of transacting within the superreplicating bounds, the market -maker can no longer replicate exactly the payoff of the option and there are no arbitrage opportunities. In particular, there is some risk involved and the market -maker's attitude towards risk must be considered. Equilibrium in a financial market must verify that all agents maximize their utilities and markets clear. Hence, the determination of equilibrium prices involves both market -makers' and traders' decisions. Moreover, optimal decisions of each market -maker are restricted by the strategic behavior of other intermediaries. Therefore, one must also consider the degree of competition between market -makers. In the context of a simple two -period model, one characterizes the equilibrium bid and ask prices in both a monopolist and a competitive market.
Chapter 4 suggests an alternative explanation for an option pricing violation that Bakshi, et al. (2000) observe using the S&P 500 index options. These authors address the empirical question of whether call prices and the underlying stock always move in the same direction. In their study, Bakshi et al. conclude that the bid -ask mid -point call prices move in the opposite direction with the underlying asset between 7.2% and 16.3% of the time. These observed option -price movements are contrary to option pricing theory and as the authors say, "these occurrences cannot be treated as outliers since one cannot imagine throwing away as much as 17% of the observations". The authors suggest that the frequent occurrence of this pricing violation represents strong evidence against the predictions of one dimensional diffusion option pricing models.
They study the impact of including an additional state variable whose stochastic process is not perfectly correlated with the underlying asset. However, they conclude that the stochastic -volatility model results are not completely consistent with the
option data. In Chapter 4, illiquidity in the underlying asset is proposed as an alternative explanation for this type of option pricing violation.
Chapter 5 of the dissertation discusses the effect of the underlying asset's illiquidity in the computation of implied volatilities and presents a measure of the underlying asset's illiquidity. According to the Black and Scholes (1973) model, implied volatilities should be constant with moneyness and time to maturity. However, what is observed is that out -of -the -money and in -the -money options tend to have a higher implied volatility than at -the -money options. This is called the smile volatility effect. In general, option pricing models are adjusted to incorporate stochastic volatilities. In this work, a simpler explanation related with the impossibility of adjusting the hedging portfolio in every point in time is presented. Finally, a measure of an asset's illiquidity is suggested. This measure is computed using the bid -ask option prices written on
that asset and related to the number of periods between consecutive transactions.
Finally, Chapter 6 presents the main conclusions and drawbacks of this work, and guidelines for future research.