Gambling theory and stock option models

This thesis investigates problems both in gambling theory and in stock option models. In gambling theory, we study the difference between the Vardi casino and the Dubins-Savage casino. In the simple Dubins-Savage casino there is only one table in which a sub-fair gamble is available fixed odds ratio, r and the problem is to change a fortune of size f to a fortune of size 1 with maximum probability before going broke. Vardi proposed the casino where there is available a table for each odds ratio r. Since the Dubins-Savage casino can be duplicated in the Vardi casino, it is clear that the Vardi casino will provide a bigger probability to achieve the goal than the Dubins-Savage casino. A main result of the thesis is to show that the advantage of the Vardi casino is surprisingly small. This implies the surprising conclusion that it does not really help the gambler to have a variety of gambles available, and raises the question of why casinos in the real world have such a variety of gambles. In particular, the optimal probabilities of the Vardi casino and the Dubins-Savage casino with odds ratio r = 1 (red-and-black) agree to three decimal places. We further conjecture that the largest difference between the Vardi and the Dubins-Savage optimal probabilities occurs at f = 1/3. The thesis also studies the two classic stochastic models involved in finance and economics, the additive Bachelier model and the multiplicative Black-Scholes model. Both models have advantages and shortcomings. Chen et al. [6] introduced a general class of models with decreasing-return- to-scale indexed by a parameter interpolating between the additive (theta = 0) and the multiplicative (theta = 1) cases. We study the American and the Russian option under the decreasing-return-to-scale models and give the optimal policy of each option for these new models. The two parts of the thesis are related through the fact that gambling is involved in each case, this despite the fact that investors often prefer to believe there is no gambling involved in their activity. Of course gamblers often believe this as well. Furthermore, among the stocks with the same negative drift, in order to maximize the probability to achieve a particular amount of fortune to survive for the gamblers problem of stocks (see [29] [30]), they need to buy those stocks with big volatilities (odds ratios).