2. Consider a risk-neutral firm that has a project whose value evolves over time according to the following geometric Brownian motion: dV(t) = 0.02V (t)dt + 0.3V(t)dZ(t), where dZ(t) is the increment of a standard Wiener process, and the value of the project at t = 0 is $10 million. The instantaneous risk-free rate of interest is 5%. The firm can undertake the project at any time t by paying $10 million at that time to receive V (t). (a) As an approximation, the firm employs a discrete-time analysis to deter mine its optimal investment decision as well as the value of the investment option at t = 0. Specifically, the firm builds a four-step binomial tree to approximate the geometric Brownian motion of V(t), where each time step is of one year. In this framework, the firm assumes that the project only lasts for four years. What is the threshold value of the project (i.e., investment trigger) at which the firm undertakes the project? What is the value of the investment option at t = 0? (b) Now the firm uses a continuous-time analysis, taking into account that the investmentoption is perpetual. Let V ∗ be the threshold value of the project (i.e., the investment trigger) and F(V (t)) be the value of the investment option at time t. If V(t) ≥ V∗ (i.e., the stopping region), the project is immediately undertaken. If V (t) < V∗ (i.e., the continuation region), construct a portfolio that consists of the investment option and the project so as to establish a no-arbitrage relationship that governs F(V (t)). What is the resulting ordinary differential equation? Apply conditions to the boundaries of the continuation region, (0,V ∗), and solve the differential equation for the numerical values of F(V (0)) and V∗, where V (0) = $10 million. (c) Contrast the results in parts (a) and (b). How good is the approximation?