Tolling agreements on a gas-fired power plant, allow a company renting a gas power plant to operate it and then use typically natural gas to generate electricity and to earn money which is the difference between the cost of natural gas and the cost of electricity. What we want to evaluate in this particular problem is the optimal operating policy for a company that is renting a two regime gas power plant, over a time period 0 to t. So what does this two regime power plant mean? It means that the plant can be operated in a low capacity mode. In which case, the output is Q lower bar, the gas consumption is H lower bar. In the high-capacity mood, the output is going to be Q upper bar and the gas consumption is going to be H upper bar. The company does not own the power plant, and therefore it has to pay rent to the owners. If the power plant is shut down, it is not providing any electricity, then the rent they have to pay is K. If the plant is being operated in the low capacity mode, then they have to pay rent K lower bar and if they are operating it in the high capacity mode then they pay rent K upper bar. The reason the rent is different for different states is because, implicitly in the rent, you are also capturing the maintenance cost. If it shut down, then the owner of the plant is simply taking rent from you for allowing you the usage of the plant. If you run it on a low capacity mode, then the rent include both the cost of actually giving you the operations as well as the future cost of maintenance that they would have to pay because the plant was run. Presumably, if you run the plant or the high capacity mode they have to do maintenance more often, and therefore they try to get a higher rent from you. So what is the state of the plant? The state of the plant is either 0 or 1, either it's on or it's off. If the plant is on and you want to turn it off, you have to pay a ramp down cost of CD. If the plant is off and you want to turn it on, you have to pay a ramp up cost CU. The actions that are available in given state is to turn the plant on or turn the plant off. So as before, in order to compute out what the value of a particular option in this case renting, the plant as well as trying to decide when to use a low capacity mode and when to use the high capacity mode, when to shut down the plant when to bring it back, these are all options that can be used by the company while its operating the gas power plant. So in order to compute the value, we'll postulate a value function. Let's say V_t, s_t, T_t, G_t, so three states now, is the optimal profit over the time little t to capital T. But the current state being st, current state of the plant being s_t either on or off, current price of electricity being P_t and the current price of gas being G_t. So what I want to do now is to setup a recursion for this problem. In order to set up this recursion, I have to tell you what happens when you take actions. So let c(s,a) denote the costs of taking action a when the plant is in state s, s takes two values 0 or 1, a takes two values 0 or 1. So plant is up or down and action a basically says either we bring the plant up or we bring the plant down. Let u(s,a) denote the state of the plant when action a is taken in state s. So here are the expression: If the state is 0, meaning that the plant is down, and we take the action as down, we do not bring it up, then the cost of this action is simply to pay the rent K. If the state was 0, and we decide to bring it up, then you pay the ramp up cost plus the rent K. If the state was up and you put it down, then you pay the ramp down cost plus the rent K. Now, suppose this state of the plant was up and you continued operating it, you took the action of being up, then, you have the option of deciding to run the plant either at the low capacity mode or the high capacity mode. So this maximum is actually also evaluating an option. If you run it at the low capacity mode, you have Q lower bar as the production, H lower bar as the consumption of gas, and K lower bar as the rent that you want to play. So this is the profit in low capacity, and this is the profit in high capacity. You take the maximum of these two and decide that's the option that you're going to exercise. What about the next state? If the current state s is equal to 0 and the action a is equal to 0, the next state is 0. If the action is equal to 1, the next state is 1. If the action here is 0 the next state is 0, action is 1 the next state is 1, so the next state is basically whatever action that you do. The dynamic program that is going to be underlying this, is that in a particular state s p t and gt, you take a maximum over all possible actions, so action takes values 0 or 1, the current cost of operating, so this is the current profit of operating, sometimes its negative meaning that you just have to pay and this is a future profit discounted, and again risk-neutral, because we're interested in pricing. Now the state here consists of the price of electricity, the price of gas, and the state of the plant 0, 1. One can solve this dynamic program by constructing a binomial lattice for a gas price and electricity price separately. If you want to get a correlations and we'll have to correlate these two binomial lattices. Each point in this lattice now will actually represent two different points, s equal to 0 and s equal to 1, meaning plant is gone or plant is up. This is similar to the story that we built in for the defaultable bonds. We had a term structure for interest rates which was a binomial lattice and in every state we split it up into two states, bond alive or bond defaulted. Similarly, over here we'll have a binary lattice and every node will get split up into two states, plant up, plant down. We can do the recursion and compute out what the value of the option is going to be. Again, I'm not going to be showing you how to compute this in practice, it's a module where we are trying to introduce this idea that option theory and financial engineering is starting to make an impact in other applications where people have to also include the cost of operations.
