# Essays on Financial Models

Abstract: This thesis consists of five essays exploring the validity of some extensively used financial models with a focus on the Swedish equity and derivative markets. The essays are of both an empirical and a theoretical nature. In the first paper, The Search for Chaos and Nonlinearities in Swedish Stock Index Returns, an investigation of the presence of nonlinearities in general and chaos in particular on the Swedish stock market is performed. Some properties of stock returns are hard to grasp with linear models. Nonlinearities must be introduced and can be of both a stochastic and a deterministic nature. In the former case, the movements are generated by external shocks, although these shocks may exhibit complicated interdependences. In the latter case, the movements are self-generating due to the nonlinear dynamics of the system, but still behave in a way seemingly indistinguishable from pure randomness. This is called chaotic motion, or chaos. Since the stock market crash of 1987 in particular, much effort has been made, trying to uncover the existence of different types of nonlinearities in financial and economic time series. Using the BDS test, we examine whether the rejection of the null hypothesis of IID stock returns arises from nonlinear or linear dependences in the conditional mean process, chaos, nonstationarities, or autoregressive conditional heteroscedasticity (ARCH). The results indicate that ARCH-effects are responsible for the rejections of IID. The second paper, The Compass Rose Pattern of the Stock Market: How Does it Affect Parameter Estimates, Forecasts, and Statistical Tests?, deals with the discrete nature of stock returns, imposed by the fact that stock prices move in discrete steps, or ticks. Recently, a geometrical pattern in a scatter plot of stock returns versus lagged stock returns, has been found. We believe that the effects of discreteness need a closer examination and, in this paper, we do Monte Carlo simulations on artificial stock prices with different degrees of rounding. We find AR-GARCH parameter estimates to be affected by the discreteness imposed by rounding. On basis of the compass rose and the discreteness we investigate different possibilities of improving predictions of stock returns, theoretically and empirically. The distributions of some correlation integral statistics, that is, the BDS test and Savit and Green's dependability index, are also influenced by the compass rose pattern. However, throughout the paper, we must impose heavy rounding of the stock prices to find significant effects on our estimates, forecasts, and statistical tests. Discreteness in stock returns is also the issue in the third paper, GARCH Estimation and Discrete Stock Prices. The results from the previous paper indicate the break-down of statistical models and tests based on state-continuity as the tick size to price ratio increases. Still, modeling such low-price stocks might be desirable in many situations. The continuous-state GARCH model is often used in modeling financial asset returns, but is misspecified if applied to returns calculated from discrete price series. I propose a modification of the above model for handling such cases, by modeling the dependent variable as an unobserved stochastic variable. The focus is on the GARCH framework, but the same ideas could also be used for other stochastic processes. Using Swedish stock price data and a stochastic optimization algorithm, that is, simulated annealing, I compare the parameter estimates and asymptotic standard errors from the approximative and the extended model. I find small deviations between the two models for longer time series, but larger differences for shorter series, mainly in the conditional variance parameters. None of the models provide continuous residuals. By constructing generalized residuals, I show how valid residual diagnostic and specification tests can be performed. The fourth paper, A Neural Network Versus Black-Scholes: A Comparison of Pricing and Hedging Performances, studies option pricing. The Black-Scholes formula is a well-known model for pricing and hedging derivative securities. It relies, however, on several highly questionable assumptions. This paper examines whether a neural network (MLP) can be used to find a call option pricing formula better corresponding to market prices and the properties of the underlying asset than the Black-Scholes formula. The neural network model is applied to out-of-sample pricing and delta-hedging of daily Swedish stock index call options from the period 1997-1999. The relevance of a hedge-analysis stressed in this paper. The Black-Scholes model with historical and implicit volatility estimates is used as benchmarks. Comparisons reveal that the neural network models outperform the benchmarks both in pricing and hedging performances. The moving block bootstrap procedure is used to test the statistical significance of the results. Although the neural networks are superior, the results are often insignificant at the 5% level. In the fifth paper, Comparison of Mean-Variance and Exact Utility Maximization in Stock Portfolio Selection, portfolio optimization is considered. The mean-variance approximation to expected utility maximization has been subject to much controversy ever since introduced by Markowitz. Given different correlated assets, how shall an investor create a portfolio maximizing his expected utility? The validity of the mean-variance approximation has been verified, but only in the limited case of choosing among 10-20 securities. This paper examines how well the approximation works in a larger allocation problem. The effects of limited short selling of the risky assets, as well as including synthetic options, that is, assets with high levels of skewness and kurtosis, in the security set is also explored. The results show that the mean-variance approximative portfolios have less skewness than the exact solution portfolios, but welfare losses, measured as the reduction in the certainty equivalent, are still small.

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