On Pricing American Put Option on a Fixed Term: A Monte Carlo Approach

This paper focuses primarily on pricing an American put option with a fixed term where the price process is geometric mean-reverting. The change of measure is assumed to be incorporated. Monte Carlo simulation was used to calculate the price of the option and the results obtained were analyzed. The option price was found to be $94.42 and the optimal stopping time was approximately one year after the option was sold which means that exercising early is the best for an American put option on a fixed term. Also, the seller of the put option should have sold $0.01 assets and bought $ 95.51 bonds to get the same payoff as the buyer at the end of one year for it to be a zero-sum game. In the simulation study, the parameters were varied to see the influence it had on the option price and the stopping time and it showed that it either increases or decreases the value of the option price and the optimal stopping time or it remained unchanged.

Hamiltonian Berge cycles in random hypergraphs

In this note we study the emergence of Hamiltonian Berge cycles in random r-uniform hypergraphs. For $r\geq 3$ we prove an optimal stopping time result that if edges are sequentially added to an initially empty r-graph, then as soon as the minimum degree is at least 2, the hypergraph with high probability has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erdős–Rényi random r-graph, and we also show that the 2-out random r-graph with high probability has such a cycle. We obtain similar results for weak Berge cycles as well, thus resolving a conjecture of Poole.

Optimal stopping time, consumption, labour, and portfolio decision for a pension scheme

In this work we solve in a closed form the problem of an agent who wants to optimise the inter-temporal recursive utility of both his consumption and leisure by choosing: (1) the optimal inter-temporal consumption, (2) the optimal inter-temporal labour supply, (3) the optimal share of wealth to invest in a risky asset, and (4) the optimal retirement age. The wage of the agent is assumed to be stochastic and correlated with the risky asset on the financial market. The problem is split into two sub-problems: the optimal consumption, labour, and portfolio problem is solved first, and then the optimal stopping time is approached. We compute the solution through both the so-called martingale approach and the solution of the Hamilton–Jacobi–Bellman partial differential equation. In the numerical simulations we compare two cases, with and without the opportunity, for the agent, to work after retirement, at a lower wage rate.