In this module, we're going to show you examples of natural gas and electricity related options and show how Option Theory can be used to value operations with built in optionality. In this module, we're going to talk about two real options, the valuation of a natural gas storage facility and the valuation of a tolling contract on an electrical power plant. We're going to show you how some of the concepts of option pricing can be used to evaluate situations where operations lay a big important role. Caverns can be used to store gas and profit from temporal variation of price. So the idea is you buy natural gas when the price is low, store it into a cavern, and then pump it out and sell it into the market when the price is high. Typically, natural gas in the United States is used for heating purposes, and the demand and therefore, the price of natural gas goes up in winter whereas in summer it's cheaper. So you buy in summer and you sell in winter. The goal is to evaluate the value of the lease on a cavern with capacity C. So we have a cavern. Here's my cavern. I can pump gas into it or I can release gas out of it. What I want to know is what is the value of such a cavern? So let I_t denote the gas stored in the cavern on day t. I will also refer sometimes to I_t to be the inventory. How much gas do you have? Let z_t greater than or equal to 0 denote the gas pumped out of the cavern. So whenever I'm pumping gas out of the cavern, meaning I'm selling into the market, I'm going to assume that z_t is greater than or equal to zero. When I'm pumping gas into the cavern, I'm going to assume that z_t is less than or equal to zero. So f(z,I) denote the loss of gas if z units are pumped out when the gas volume or the inventory in the cavern is I. Most of the loss because the gas is actually used to drive the pumps that are necessary for pumping in and pumping out the gas. Some part of it is leakage. That P of t denote the price of gas at time t. Then the optimization problem that I'm interested in solving is the following. P of t is the price, z of t is the amount I pumped out,f(z,I) is the loss that happened at time t when I pumped out z_t in the inventory, or the amount that I stored was I_t. So this entire thing gives me the revenue from selling gas. This is the discount and this is the expectation with respect to the risk-neutral measure. Again, since I'm interested in pricing, I need to ensure that the expectation is with respect to the risk-neutral measure, not the real-world probability, but then risk-neutral probability. What are I constraints? I need to make sure that the inventory level always lies between zero and C. These are operating constraints. This is where the real option part of the problem comes in. This is not a financial instrument alone, this is a financial instrument coupled with something physical. This says that the inventory has to be positive and cannot be more than the capacity C. Down here, it tells you the dynamics of inventory. Since I assume that z_t is positive when I pull out, so the inventory at time t plus one is going to be simply I_t minus z_t. The complicating factor in this particular problem is that z_t, the decisions of pumping out, can be a function of past prices and the inventory level, I_t, could be an arbitrary function of past prices inventory levels. Therefore, it's not something that I can compute a times zero, as the prices evolve, I need to compute it dynamically as we go along. This is not a new concept, even with American option pricing and the goldmine equipment upgrade option that we saw in the last module, this story was there. We had to decide as we went along whether we want to upgrade, whether we want to exercise the option and so on. Except that here, we have to also keep into consideration what happens with the inventory, what happens to the cost of actually pumping in or pumping out the gas that I need. So we can set it up as a dynamic program. So what does this dynamic program consists of? It consists of value functions. As in the case of option pricing, we have a value function which tells me what is the value of being in a particular state? Except, in this particular problem, there are two states. This is the gas price, and this is the inventory. The decisions that you make depend on both the price and the inventory. V_t(I,P) simply says what is the value of this lease starting from time t when your inventory is I and the gas price the current gas price is P? Is the maximum of P of z minus f(z, I). So this is the current revenue, and this is the value from the future, and that is discounted back by e to the minus r to bring it back into present dollars. Remember, there was this constraint that we needed to make sure that the inventory lies between zero and C. We will put that in here by making this equal to minus infinity if I minus z is not in zero to capital C. That way we can ensure that we never take decisions where we're violating the constraints. How can one solve this dynamic program? We can use a binomial lattice for the price P. So if that was the only state, we can do the backward recursion that we've been using in the past. However, one has to enumerate all possible inventory levels. Inventory the continuous variables, so it's not clear how to do that. You can use approximate dynamic programming where the value function is approximated by factors. You can sometimes use simulation based optimization where you simulate the prices from time zero to time capital T, then use some kind of predictive control to figure out how the inventory is going to behave and how you're going to get the value function out of it. I'm not telling you the details of how these operations work, the point of this module is simply to expose you to problems where one is using option pricing in situations that had to do bit operations.
