A power company is considering renting a two-regime gas-fired power plant over the discrete) time [0, T]. In order to figure out whether or not this is worthwhile, the company must figure out the optimal operating policy for the plant. At any time t the company can operate the plant in "shut-down" mode, low-capacity mode or high-capacity mode. The relevant details for each of these three operating states are: Shut-down mode: plant rent K Low capacity mode: output Q, gas consumption H, plant rent K High capacity mode: output Q, gas consumption H, plant rent K If the plant is currently in operation then shutting the plant down will incur a ramp-down cost of Cd. Similarly, if the plan is currently shut down, then moving it into a low / high production mode will incur a ramp-up cost of Cy. We use st {0 = plant shut-down, 1 = plant operating} to denote the time t state of the plant. In each period there are two possible actions a {0 = operate in shut-down mode, 1 = operate in production mode}. We let V(s, P, G) denote the optimal profit of the plant over the horizon [t, T] when the current state is s, the current price of electricity is P, and the current price of gas is G. We also have the following additional notation: c(s, a): cost of taking action a when the plant is in state s us, a): new state of the plant when action a is taken in state s Introduction to Real Options 9 Specific values are: 0 1 0 1 c(0,0) K u(0,0) c(0,1) Cu +K u(0,1) c(1,0) Ca+K u(1.0) c(1,1) min {K+GH - PO, K +G,H - - PQ} -(1,1) Note that when the plant is in production mode one has the choice to run it at the low or high capacity mode. We can now write the standard dynamic programming iteration: V. (84, P, Gt) = max {-c(st,a) +e=rA!E [V:+1(u(su, a), Pr+1,Ge+1)]} (5) a {0,1} where E [:] denotes the usual risk-neutral expectation conditional on all known-information at time t, r is the continuously compounded risk-free rate and At is the length of a period. There are two sources of uncertainty here: the price dynamics of electricity and gas. It is straightforward to construct lattices for these prices and solve (5) numerically beginning at time T and working backwards. A power company is considering renting a two-regime gas-fired power plant over the discrete) time [0, T]. In order to figure out whether or not this is worthwhile, the company must figure out the optimal operating policy for the plant. At any time t the company can operate the plant in "shut-down" mode, low-capacity mode or high-capacity mode. The relevant details for each of these three operating states are: Shut-down mode: plant rent K Low capacity mode: output Q, gas consumption H, plant rent K High capacity mode: output Q, gas consumption H, plant rent K If the plant is currently in operation then shutting the plant down will incur a ramp-down cost of Cd. Similarly, if the plan is currently shut down, then moving it into a low / high production mode will incur a ramp-up cost of Cy. We use st {0 = plant shut-down, 1 = plant operating} to denote the time t state of the plant. In each period there are two possible actions a {0 = operate in shut-down mode, 1 = operate in production mode}. We let V(s, P, G) denote the optimal profit of the plant over the horizon [t, T] when the current state is s, the current price of electricity is P, and the current price of gas is G. We also have the following additional notation: c(s, a): cost of taking action a when the plant is in state s us, a): new state of the plant when action a is taken in state s Introduction to Real Options 9 Specific values are: 0 1 0 1 c(0,0) K u(0,0) c(0,1) Cu +K u(0,1) c(1,0) Ca+K u(1.0) c(1,1) min {K+GH - PO, K +G,H - - PQ} -(1,1) Note that when the plant is in production mode one has the choice to run it at the low or high capacity mode. We can now write the standard dynamic programming iteration: V. (84, P, Gt) = max {-c(st,a) +e=rA!E [V:+1(u(su, a), Pr+1,Ge+1)]} (5) a {0,1} where E [:] denotes the usual risk-neutral expectation conditional on all known-information at time t, r is the continuously compounded risk-free rate and At is the length of a period. There are two sources of uncertainty here: the price dynamics of electricity and gas. It is straightforward to construct lattices for these prices and solve (5) numerically beginning at time T and working backwards
