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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 the natural gas storage

facility, and the valuation of a toning contract on a electrical power plant.

And we're going to show you how some of the concepts of option pricing, can be

used to evaluate situations where operations play 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 in 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.

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The goal is to evaluate the value of a 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. And what I want to know is, what is the

value of such a cavern? So let It denote the gas stored in the

cavern on day, t. I would also refer sometimes to It, to be

the inventory. How much gas do you have?

Let zt greater than equal to 0, denote the gas pumped out of the cavern.

So whenever I am pumping gas out of the cavern, meaning I'm selling it to the

market, I'm going to assume that zt is greater than or equal to 0.

When I'm pumping gas into the cavern, I'm going to assume that zt is less than

equal to 0. So, fzi denote the loss of gas, if z

units are pumped out when the gas volume, or the inventory in the cavern is I.

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This is the discount, and this is the expectation with respect to the

risk-neutral method. Again, since I'm interested in pricing, I

need to ensure that the expectation is with respect to risk-neutral measure.

Not the real world probability, but then risk-neutral probability.

What are my constraints? I need to make sure that the inventory

level, always lies between 0 and C. These are operating constraints, these

are, this is where real, 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. And this says that the inventory has to

be positive, and cannot be more than the capacity C.

Down here tells you the dynamics of inventory.

Since I assume that zt is positive when I pull out, so the inventory and time t

plus 1, is going to be simply it minus zt.

The complicating factor in this particular problem is that zt, the

decisions of pumping out, ha, can be a function of past prices, and the

inventory level It, could be a arbitrary function of past prices inventory levels.

And, therefore, it's not something that I can compute at times 0, 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 gold mine 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 our option, and so on.

Except that here, we have to also keep into consideration what happens to 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 consist

of? It consists of value functions.

As in the case of option pricing, we have a value function that 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. And Vtip simply says, what is the value

of this lease starting at time t, when your inventory is i, and the gas price,

the current gas price is p. It's a maximum of p of z minus f z i.

So this is the current revenue. And this is the value from the future.

This is the value of 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 0 and C. We will put that in here, by making this

equal to minus infinity, if I minus Z is not in 0 to capital C.

And that way, we can ensure that we never take decisions, where we are violating

the constraints. How can one solve this dynamic program?

We can use a binomial lattice for the price speed.

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 innumerate all

possible inventory levels. Inventories are continuous variables,

which not clear how to do that. You can use approximate dynamic

programming, where the value function is approximated by factors.

You can sometimes use stimulation-based optimization, where you simulate the

prices from time 0 to time capital T. And 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 situation's that have

to do with operations. Here's another example of a real option.

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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 mode, 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 owner's.

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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 because, 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 will have to pay, because the plant was run.

Presumably, if you run the plant at a high capacity mode, they have to do,

maintenance more often, and therefore, they're interested in actions.

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.

That 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 the 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 are, can be used by the company while it's operating

the gas power plant. So, in order to compute the value, we'll

postulate a value function. Let's say v t s d t t g t, so three

states now, is the optimal profit over the time, little t to capital T, with the

current state being s t, current state of the plant being s t, either on or off.

Current price of electricity being Pt, and the current price of gas being Gt.

So what I want to do now, is to set up an 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 cost 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 dow,n 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 the 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 is 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

pay. So this is, 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, that's the option that you

are 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 SPD and GD, you take a maximum

overall possible action, so action takes value 0 or 1.

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The current cost of operating, so this is the current profit of operating,

sometimes it's negative ,meaning that you just have to pay, and this is a future

profit. This counted and again, risk-neutral,

because we're interested in pricing. Now the stake 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, then we'll have to correlate these two

binomial lattices. Each point in this lattice now will

actually represent two different points. S equals to 0 and s equal to 1, meaning

plant is down or plant is up. This is similar to the story that we

built in for the defordable 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. Born alive, or born defaulted.

Similarly, over here, we'll have a binary lattice, and every node will get split up

into two states. Plan up, plan down, we can do the

recursion, and compute out what the values of 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.