# A Utility Approach: Strategy Analysis and Optimization

University essay from Lunds universitet/Matematisk statistik

Abstract: Utility theory and Monte Carlo simulations are used to calculate optimal allocation for long term as well as, risk averse investors with a portfolio consisting of one risky asset and one risk-free bank account. The problems solved in this thesis are divided into two types, static and dynamic. A strategy is given for a static problem, such as Buy and Hold (B\&H) or Constant Weights (CW) and optimal weights are calculated and analyzed. Optimal allocations for different static strategies can be compared with Bootstrap. Analysis is done with four different equity models. The four models are constructed so that the two first moments are almost equal in order for them to be comparable. The main conclusion is that when stochastic jumps are introduced (the Merton model), it does not affect the optimal weight in the risky asset to decrease as much as when stochastic volatility is introduced (the Heston model). A volatility control (VC) model was analyzed and compare with CW and it was found that VC wasn't a better option for a long term investor. Dynamic programming is used to solve the Bellman-equation for optimal dynamic strategies. Validation is used by comparing the results with problems where theoretical result are known. It was found when solving a consumption and investment problem, that the consumption result are much more robust compared to the optimal weights. The error when using $10^5$ simulations for the validation problems is only $\sim \pm 10^{-9}$ for consumption but $\sim \pm 0.01$ for optimal weight. Underlying processes are introduced e.g., stochastic volatility, and an algorithm that uses a proxy function on the maximum argument instead of the objective function is presented. The main reason for this is that the algorithm is much more flexible compared to the using regression on the objective function i.e., the same algorithm can be used for any underlying asset without changing the algorithm much. The downside is that many trajectories are needed to get the same confidence interval. There is left to prove theoretically that the algorithm converges to the right solution in a general case but several validation tests hint at convergence.