<p>The thesis will have two main parts. First, let us start with an example. In finance, the standard version of the Black-Scholes formula is a beautiful closed form solution used to price European options. This famous formula is ingenious, but has a flaw that relegates it to something that should be admired, and perhaps not be used in the real world. It relies on the assumption that prices of shares evolve according to geometric Brownian motion. This means that we are willing to accept that extreme shocks to prices are almost impossible. Is this a realistic assumption? Of course not. The stock market crashes of 1929, 1987 are great examples to show that extreme events do happen. More recently, the 1997 Asian crisis and 2000 crash of the NASDAQ show that in addition, such events are not so rare. These jumps occur even more frequently and are larger in magnitude for share prices of individual companies. This problem is by no means new, and a plethora of models and pricing techniques have been developed. The standard Black-Scholes formula is just one example, but this is simply illustration of the matter at hand. The process that we use to model a financial time series is of paramount importance, whether we do it for forecasting purposes or for pricing financial derivatives. If we choose to use a model that does not capture the key empirical aspects of the data, then any subsequent inference may be very unfavourably biased. It is because of this problem that we should investigate the more standard modeling that assumes continuity and normal or log-normal distribution of financial time series. We will begin from the very basics and we will see that this is a wonderful piece of theory, deserving of the reputation it has in being simple, groundbreaking and extremely useful. This work should bring us to a position where we can evaluate a second goal. Stochastic processes with jumps and "heavy-tails" have existed for some time, but have begun to filter through to the financial industry only recently. This lag is due to the perceived added conceptual difficulty in the introduction of such models, although we will see that this should not be the case. There is plenty of real evidence that nancial time series exhibit discontinuous behaviour and that these series are far from normally or log-normally distributed. Rather than looking at standard models as correct, and jump or stochastic volatility models as complicated, we should look upon standard models as educational but not sufficient for the real world. Stochastic volatility or jump models should instead be viewed as natural. The theme of the thesis is the importance of choosing a correct model for the underlying process. Although we may speak of the implications of some models to hedging, we will not actually look at specific hedging techniques. The particular aspect of pricing is also not considered in full scope although we will see the Black-Scholes pricing formula. We will consider that the main problem is to specify the model correctly where the method of pricing is a subsequent technicality. In examples we may take pricing tools like Monte-Carlo simulation as a given. We will not strive for full generality or formality, but rather take a physical approach and aim for clarity and understanding. Let us now move on to the beginning, with the introduction of our primary source of randomness.</p>
Why Black-Scholes Equations Are Effective Beyond Their Usual Assumptions: Symmetry-Based Explanation
