1667 chance of moving to node F where the oi)ptice is 31.37. The second year cash flow is then 2 x 31.37 -—2 x 17 —6 = 22.74. The value ofsubsequent cash flows at node F is 21.42. The total value of the project if we move to node F istherefore 21.42 + 22.74 = 44.16. Similarly, the total value of the project if we move to nodes G andH are 10.35 and —13.93, respectively. The value of the project at node C is therefore
[0.1667 x 44.16 + 0.6666 x 10.35 + 0.1667 x (—13.93)Je!*! = 10.80
Figure 28.3 shows that the value of the project at the initial node, A, is 14.46. When the initialinvestment is taken into account the value of the project is therefore —0.54. This is in agreementwith our earlier calculations.
Suppose now that the company has the option to abandon the project at any time. We supposethat there is no salvage value and no further payments are required once the project has beenabandoned. Abandonment is an American put option with a strike price of zero and is valued inFigure 28.4. The put option should not be exercised at nodes E, F, and G because the value of theproject is positive at these nodes. It should be exercised at nodes H and I. The value of the putoption is 5.31 and 13.49 at nodes H and J, respectively. Rolling back through the tree, we see thatReal Options 673
the value of the abandonment put option at node D, if it is not exercised, is(0.1217 x 13.49 + 0.6566 & 5.31 +.0.2217 x O)eO'™! = 4.64
The value of exercising the put option at node D is 9.65. This is greater than 4.64, and so the putshould be exercised at node D. The value of the put option at node C is
[0.1667 x 0 + 0.6666 x 0 + 0.1667 x (5.31)Je?!*! = 0.80and the value at node A is(0.1667 x 0 +.0.6666 x 0.80 +0.1667 x 9.65)e°!*! = 1.94
The abandonment option is therefore worth $1.94 million. It increases the value of the project from—$0.54 million to +$1.40 million. A project that was previously unattractive now has a positivevalue to shareholders.
Suppose next that the company has no abandonment option. Instead it has the option at anytime to increase the scale of the project by 20%. The cost of doing this is $2 million. Oil productionincreases from 2.0 to 2.4 million barrels. Variable costs remain $17 per barrel and fixed costsincrease by 20% from $6.0 million to $7.2 million. This is an American call option to buy 20% ofthe base project in Figure 28.3 for $2 million. The option is valued in Figure 28.5. At node E, theoption should be exercised. The payoff is 0.2 x 42.24 -—2 = 6.45. At node F, it should also beexercised for a payoff of 0.2 x 21.42 — 2 = 2.28. At nodes G, H, and I, the option should not be
 
 
Node: A B Cc D E F G H I
Pu 0.1667 0.1217 0.1667 =—0.2217,—S-«0.8867,— 0.1217 0.1667 =—:0.2217 =~: 0.0867Pm 0.6666 0.6566 0.6666 = 0.6566 = 0.0266 )=— (0.6566 = (0.6666 = 0.6566 =: 0.0266Pd 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Figure 28.4 Valuation of option to abandon the project. Here, py, pm, aNd pg are the probabilities of“up”, “middle”, and “down” movements from a node674 CHAPTER 28
 
 
 
Node: A B Cc D E F G H I
Pu 0.1667 0.1217 0.1667) 0.2217, 0.8867) 0.1217 0.1667 0.2217 0.0867Pm 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266Pa 0.1667 = 0.2217, 0.1667) 0.1217 0.0867) 0.2217 0.1667 = 0.1217 0.8867
Figure 28.5 Valuation of option to expand the project. Here, p,, Pm, and py are the probabilities of“up”, “middle”, and “down” movements from a node
exercised. At node B, exercising is worth more than waiting and the option is worth0.2 x 38.32 — 2 = 5.66. At node C, if the option is not exercised, it is worth
(0.1667 x 2.28 + 0.6666 x 0.00 + 0.1667 x 0.00)e7°!*! = 0,34
If the option is exercised it is worth 0.2 x 10.80 —2 = 0.16. The option should therefore not beexercised at node C. At node A, the option is worth
(0.1667 x 5.66 + 0.6666 x 0.34 + 0.1667 x 0.00)e-°!*! = 1.06
If the option is not exercised, it is worth 0.2 x 14.46 — 2 = 0.89. Early exercise is therefore notoptimal at node A. In this case, the option increases the value of the project from —0.54 to +0.52.Again we find that a project that previously had a negative value now has a positive value.
‘The expansion option in Figure 28.5 is relatively easy to value because, once the option has beenexercised, all subsequent cash inflows and outflows increase by 20%. In the case where fixed costsremain the same or increase by less than 20%, it is necessary to keep track of more information atthe nodes of Figure 28.3. Specifically, we need to record:
1. The present value of subsequent fixed costs
2. The present value of subsequent revenues net of variable costs
The payoff from exercising the option can then be calculated.When a project has two or more options, they are in general not independent. The value ofhaving both option A and option B is typically not the sum of the values of the two options. ToReal Options 675
 
illustrate this, suppose that the company we have been considering has both abandonment andexpansion options. The project cannot be expanded if it has already be abandoned. Moreover thevalue of the put option to abandon depends‘on whether the project has been expanded.’
These interactions between the options in our example can be handled by defining four states ateach node:
1. Not already abandoned; not already expanded2. Not already abandoned; already expanded
3. Already abandoned; not already expanded
4, Already abandoned; already expanded
When we roll back through the tree, we calculate the combined value of the options at each nodefor all four alternatives. This approach to valuing path-dependent options is discussed in moredetail in Section 20.5.
When there are several stochastic variables, the value of the base project is usually determinedby Monte Carlo simulation. The valuation of the project’s embedded options is then more difficultbecause a Monte Carlo simulation works from the beginning to the end of a project. When wereach a certain point, we do not have information on the present value of the project’s future cashflows. However, the techniques mentioned in Section 20.9 for valuing American options usingMonte Carlo simulation can sometimes be used.
As an illustration of this point, Schwartz and Moon (2000) explain how their Amazon.comanalysis outlined in Section 28.4 could be extended to take account of the option to abandon(i.e., option to declare bankruptcy) when the value of future cash flows is negative.* At each timestep, a polynomial relationship between the value of not abandoning and variables such as thecurrent revenue, revenue growth rate, volatilities, cash balances, and loss carry forwards isassumed. Each simulation trial provides an observation for obtaining a least-squares estimate ofthe relationship at each time. This is the Longstaff and Schwartz approach of Section 20.9.
SUMMARY
 
 
In this chapter we have investigated how the ideas developed earlier in the book can be applied tothe valuation of real assets and options on real assets. We have shown how the risk-neutralvaluation principle can be used to value an asset dependent on a number of variables followingdiffusion processes. The drift rate of each variable is adjusted to reflect its market price of risk. Thevalue of the asset is then the present value of its expected cash flows discounted at the risk-free rate.
Risk-neutral valuation provides an internally consistent approach to capital investment appraisal.It also makes it possible to value the options that are embedded in many of the projects that areencountered in practice. We have illustrated the approach by applying it to the valuation ofAmazon.com at the end of 1999 and the valuation of an oil project.
7 As it happens, the two options do not interact in Figures 28.4 and 28.5. However, the interactions between the
options would become an issue if a larger tree with smaller time steps were built.
Roomy juste in Sartian 92 4 § phar hank + + hh hal. eatte teal + bros~ Lhe analysis in section 25.4 assumed that bankruptcy occurs When ihe casn Dalance [alls Below Zero, But tls Is Not
necessarily optimal for Amazon.com.
° FA. Longstaff and E. S. Schwartz, “Valuing American Options by Simulation: A Simple Least-SquaresApproach,” Review of Financial Studies, |4, no. | (Spring 2001), 113-47.676
CHAPTER 28
SUGGESTIONS FOR FURTHER READING
 
 
Amran, M., and N. Kulatilaka, Real Options, Harvard Business School Press, Boston, MA, 1999.
Copeland, T., T. Koller, and J. Murrin, Valuation: Measuring and Managing the Value of Companies,3rd edn., Wiley, New York, 2000.
Copeland, T., and V. Antikarov, Real Options: A Practitioners Guide, Texere, New York, 2001.
Schwartz, E. S., and M. Moon, “Rational Pricing of Internet Companies,” Financial Analysts Journal,May/June 2000, 62-75.
Trigeorgis, L., Real Options: Managerial Flexibility and Strategy in Resource Allocation, MIT Press,Cambridge, MA, 1996.
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
28.3.
28.4.28.5.28.6.
28.7.
 
. Explain the difference between the net present value approach and the risk-neutral valuation
approach for valuing a new capital investment opportunity. What are the advantages of the risk-neutral valuation approach for valuing real options?
. The market price of risk for copper is 0.5, the volatility of copper prices is 20% per annum, the
spot price is 80 cents per pound, and the six-month futures price is 75 cents per pound. What isthe expected percentage growth rate in copper prices over the next six months?
Consider a commodity with constant volatility, 0, and an expected growth rate that is a functionsolely of time. Show that, in the traditional risk-neutral world,
2In Sp ~ a(i F(T) — 5 T, owt)
where Sr is the value of the commodity at time 7 and F(t) is the futures price at time zero foracontract maturing at time t.
Prove equation (28.6).
Derive a relationship between the convenience yield of a commodity and its market price of risk.The correlation between a company’s gross revenue and the market index is 0.2. The excessreturn of the market over the risk-free rate is 6% and the volatility of the market index is 18%.What is the market price of risk for the company’s revenue?
A company can buy an option for the delivery of one million barrels of oil in three years at $25per barrel. The three-year futures price of oil is $24 per barrel. The risk-free interest rate is 5%
per annum with continuous compounding and the volatility of the futures price is 20% perannum. How much is the option worth?
. A driver entering into a car lease agreement can obtain the right to buy the car in four years for
$10,000. The current value of the car is $30,000. The value of the car, S, is expected to follow the
process .dS = wS dt + oS dz
where . = —0.25, o = 0.15, and dz is a Wiener process. The market price of risk for the car priceis estimated to be —0.1. What is the value of the option? Assume that the risk-free rate for allmaturities is 6%.Real Options 677
ASSIGNMENT QUESTIONS
 
 
 
28.9. Suppose that the spot price, 6-month futures price, and 12-month futures price for wheat are250, 260, and 270 cents per bushel, respectively. Suppose that the price of wheat follows theprocess in equation (28.9) with a = 0.05 and o = 0.15. Construct a two-time-step tree for theprice of wheat in a risk-neutral world.
A farmer has a project that involves an expenditure of $10,000 and a further expenditure of$90,000 in six months. It will increase wheat that is harvested and sold by 40,000 bushels inone year, What is the value of the project? Suppose that the farmer can abandon the project insix months and avoid paying the $90,000 cost at that time. What is the value of the abandonmentoption? Assume a risk-free rate of 5% with continuous compounding.
28.10. In the oil example considered in Section 28.6:
a. What is the value of the abandonment option if it costs $3 mitlion rather than zero?b. What is the value of the expansion option if it costs $5 million rather than $2 million? 
INSURANCE, WEATHER, ANDENERGY DERIVATIVES
In this chapter we examine some recent innovations in derivatives markets. We explain theproducts that have been developed to manage weather risk, energy price risk, and insurance risks.Some of the markets that we will talk about are in the early stages of their development. As theymature, we may well see significant changes in both the products that are offered and the ways theyare used.
29.1 REVIEW OF PRICING ISSUES
 
 
In Chapters 10 and 12 we explained the risk-neutral valuation result. This states that if we price aderivative on the assumption that investors are risk neutral then we get the right price—not just ina risk-neutral world, but in all other worlds as well. The correct approach to pricing a derivative isto calculate the expected payoff in a risk-neutral world and discount this expected payoff at therisk-free interest rate.
An alternative pricing approach, often used in the insurance industry, is sometimes referred to asthe actuarial approach. It involves using historical data to calculate the expected payoff anddiscounting this expected payoff at the risk-free rate to obtain the price. Historical data give anestimate of the expected payoff in the real world. It follows that the actuarial approach is correctonly when the expected payoff from the derivative is the same in both the real world and the risk-neutral world.
We showed in Section 10.7 that, when we move from the real world to the risk-neutral world, thevolatilities of variables remain the same but their expected growth rates are liable to change. Forexample, the expected growth rate of a stock market index decreases by perhaps 4% or 5% whenwe move from the real world to the risk-neutral world. Our discussion in Section 28.3 shows thatthe expected growth rate of a variable can reasonably be assumed to be the same in both the realworld and the risk-neutral world if the variable has zero systematic risk, so that percentage changesin the variable have zero correlation with stock market returns. We can deduce from this that theactuarial approach to valuing a derivative gives the right answer if all underlying variables havezero systematic risk.
A common feature of all the derivatives we will consider in this chapter is that the actuarialapproach can be used. The underlying variables can reasonably be assumed to have zerosystematic risk,
678Insurance, Weather, and Energy Derivatives 679
 
29.2. WEATHER DERIVATIVES
 
Many companies are in the position where their performance is liable to be adversely affected bythe weather.' It makes sense for these companies to consider hedging their weather risk in muchthe same way as they hedge foreign exchange or interest rate risks.
The first over-the-counter weather derivatives were introduced in 1997. To understand how theywork, we explain two variables:
HDD: Heating degree daysCDD: Cooling degree days
A day’s HDD is defined as
HDD = max(0, 65 — A)and a day’s CDD is defined as
CDD = max(0, A — 65)
where A is the average of the highest and lowest temperature during the day at a specified weatherstation, measured in degrees Fahrenheit. For example, if the maximum temperature during a day(midnight to midnight) is 68° Fahrenheit and the minimum temperature is 44° Fahrenheit, thenA = 56. The daily HDD is then 9 and the daily CDD is 0.
A typical over-the-counter product is a forward or option contract providing a payoff dependenton the cumulative HDD or CDD during a period. For example, an investment dealer could inJanuary 2002 sell a client a call option on the cumulative HDD during February 2003 at theChicago O'Hare Airport weather station with a strike price of 700 and a payment rate of $10,000per degree day. If the actual cumulative HDD is 820, the payoff is $1.2 million. Often contractsinclude a payment cap. If the payment cap in our example is $1.5 million, the contract is theequivalent of a bull spread. The client has a long call option on cumulative HDD with a strikeprice of 700 and a short call option with a strike price of 850.
A day’s HDD is a measure of the volume of energy required for heating during the day. A day’sCDD is a measure of the volume of energy required for cooling during the day. At the time ofwriting, most weather derivative contracts are entered into by energy producers and energyconsumers. But retailers, supermarket chains, food and drink manufacturers, health servicecompanies, agriculture companies, and companies in the leisure industry are also potential usersof weather derivatives. The Weather Risk Management Association (www.wrma.org) has beenformed to serve the interests of the weather risk management industry.
Au Sepueniuer LI77 Ue M1lltaZuU IVICLLALILNO LAULLALLES ucEeaU LIGA, WCALIIGL TULLELODS ailEuropean options on weather futures. The contracts are on the cumulative HDD and CDD fora month observed at a weather station.” The contracts are settled in cash just after the end of themonth once the HDD and CDD are known. One futures contract is on $100 times the cumulativeHDD or CDD. The HDD and CDD are calculated by a company, Earth Satellite Corporation,using automated data-collection equipment.
The temperature at a certain location can reasonably be assumed to have zero systematic risk. Itfollows from Section 29.1 that weather derivatives can be priced using the actuarial approachConsider, for example, the call option on the February 2003 HDD at Chicago O’Hare airport
! The U.S. Department of Energy has estimated that one-seventh of the U.S. economy is subject to weather risk.
? The CME has introduced contracts for 10 different weather stations (Atlanta, Chicago, Cincinnati, Dallas, DesMoines, Las Vegas, New York, Philadelphia, Portland, and Tucson).680 CHAPTER 29
mentioned earlier. We could collect 50 years of data and estimate a probability distribution for theIIDD. This in turn could be used to provide a probability distribution for the option payoff. Ourestimate of the value of the option would be the mean of this distribution discounted at the risk-free rate. We might want to adjust the probability distribution for temperature trends. Forexample, a linear regression might show that the cumulative February HDD is decreasing at arate of 10 per year on average. The output from the regression can then be used to estimate a trend-adjusted probability distribution for the HDD in February 2003.
29.3 ENERGY DERIVATIVES
 
 
Energy companies are among the most active and sophisticated users of derivatives. Many energyproducts trade in both the over-the-counter market and on exchanges. In this section we willexamine the trading in crude oil, natural gas, and electricity derivatives.
Crude Oil
Crude oil is one of the most important commodities in the world with global demand amountingto about 65 million barrels (8.9 million tonnes) daily. Ten-year fixed-price supply contracts havebeen commonplace in the over-the-counter market for many years. These are swaps where oil at afixed price is exchanged for oil at a floating price.
In the 1970s the price of oil was highly volatile. The 1973 war in the Middle East led to a triplingof oil prices. The fall of the Shah of Iran in the late 1970s again increased prices. These events ledoil producers and users to a realization that they needed more sophisticated tools for managing oilprice risk. In the 1980s both the over-the-counter market and the exchange-traded marketdeveloped products to meet this need.
In the over-the-counter market, virtually any derivative that is available on common stocks orstock indices is now available with oil as the underlying asset. Swaps, forward contracts, andoptions are popular. Contracts sometimes require settlement in cash and sometimes requiresettlement by physical delivery (i.e., by delivery of the oil).
Exchange-traded contracts are also popular. New York Mercantile Exchange (NYMEX) andthe International Petroleum Exchange (IPE) trade a number of oil futures and futures optionscontracts. Some of the futures contracts are settled in cash; others are settled by physical delivery.For example, the Brent crude oil futures traded on the IPE has cash settlement based on the Brentindex price; the light sweet crude oil futures traded on NYMEX requires physical delivery. In bothcases the amount of oil underlying one contract is 1,000 barrels.) NYMEX also trades popularcontracts an two refined products: heating oil and gasoline. In both cases one contract is for the
an en ing nd @ one ntract 18
delivery of 42,000 gallons.
Natural Gas
The natural gas industry throughout the world has been going through a period of deregulation andthe elimination of state monopolies. The supplier of natural gas is now not necessarily the samecompany as the producer of the gas. Suppliers are faced with the problem of meeting daily demand.
A typical over-the-counter contract is for the delivery of a specified amount of natural gas at aroughly uniform rate over a one-month period. Forward contracts, options, and swaps areavailable in the over-the-counter market. The scller of gas is usually responsible for moving theInsurance, Weather, and Energy Derivatives 681
gas through pipelines to the specified location. NYMEX trades a contract for the delivery of10,000 million British thermal units of natural gas. The contract, if not closed out, requiresphysical delivery to be made during the delivery month at a roughly uniform rate to a particularhub in Louisiana. The IPE trades a similar contract in London.
Electricity
Electricity is an unusual commodity because it cannot easily be stored.? The maximum supply ofelectricity in a region at any moment is determined by the maximum capacity of all the electricity-producing plants in the region. In the United States there are 140 regions known as control areas.Demand and supply are first matched within a control area, and any excess power is sold to othercontrol areas. It is this excess power that constitutes the wholesale market for electricity. The abilityof one control area to sell power to another control area depends on the transmission capacity ofthe lines between the two areas. Transmission from one area to another involves a transmissioncost, charged by the owner of the line, and there are generally some transmission or energy losses.
A major use of electricity is for air-conditioning systems. As a result the demand for electricity,and therefore its price, is much greater in the summer months than in the winter months. Thenonstorability of electricity causes occasional very large movements in the spot price. Heat waveshave been known to increase the spot price by as much as 1000% for short periods of time.
Like natural gas, electricity has been going through a period of deregulation and the eliminationof state monopolies. This has been accompanied by the development of an electricity derivativesmarket. NYMEX now trades a futures contract on the price of electricity, and there is an activeover-the-counter market in forward contracts, options, and swaps. A typical contract (exchangetraded or over the counter) allows one side to receive a specified number of megawatt hours for aspecified price at a specified location during a particular month. In a 5 x 8 contract, power isreceived for five days a week (Monday to Friday) during the off-peak period (11 p.m. to 7 a.m.) forthe specified month. In a 5 x 16 contract, power is received five days a week during the on-peakperiod (7 a.m. to 11 p.m.) for the specified month. In a 7 x 24 contract, it is received around theclock every day during the month. Option contracts have either daily exercise or monthly exercise.In the case of daily exercise, the option holder can choose on each day of the month (by givingone day’s notice) to receive the specified amount of power at the specified strike price. When thereis monthly exercise a single decision on whether to receive power for the whole month at thespecified strike price is made at the beginning of the month.
An interesting contract in electricity and natural gas markets is what is known as a swing option ortake-and-pay option. In this a minimum and maximum for the amount of power that must bepurchased at a certain price by the option holder is specified for each day during a month and for themonth in total. The option holder can change (or swing) the rate at which the power is purchasedduring the month, but usually there is a limit on the total number of changes that can be made.
Modeling Energy Prices
As discussed in Section 28.5 a realistic model for a energy and other commodity prices shouldincorporate both mean reversion and volatility. A reasonable model is equation (28.9):
dinS =[0(t) —aln S|dt +odz (29.1)
3 Electricity producers with spare capacity often use it to pump water to the top of their hydroelectric plants so thatit can be used to produce electricity at a later time. This is the closest they can get to storing this commodity.682 CHAPTER 29
 
where S is the energy price, and a and o are constant parameters. The 6(t) term capturesseasonality and trends. Section 28.5 shows how to construct a trinomial tree for this model fromfutures prices.
The correlation between changes in energy prices and the return on the market is low. It isprobably reasonable, therefore, to assume that energy prices behave in the same way in the realworld and the risk-neutral world. The parameters in equation (29.1) can therefore be estimatedfrom historical data.
The parameters are different for different sources of energy. For crude oil, the reversion rateparameter, a, in equation (29.1) is about 0.5 and the volatility parameter, a, is about 20%; fornatural gas, a is about 1.0 and o is about 40%; for electricity, a is typically between 10 and 20 whilea is 100 to 200%. The seasonality of electricity prices is also greater.
How an Energy Producer Can Hedge Risks
There are two components to the risks facing an energy producer: one is the price risk; the other is
the volume risk. Although prices do adjust to reflect volumes, there is a less-than-perfect relationship
between the two, and energy producers have to take both into account when developing a hedging
strategy. The price risk can be hedged using the energy derivative contracts discussed in this section.
The volume risks can be hedged using the weather derivatives discussed in the previous section.Define:
Y: Profit for a monthP: Average energy prices for the monthT: Relevant temperature variable (HDD or CDD) for the monthAn energy praducer can use historical data to obtain a best-fit linear regression relationship of the
formY=a+bP+cT +e
where ¢ is the error term. The energy producer can then hedge risks for the month by taking aposition of —4 in energy forwards or futures and a position of —c in weather forwards or futures.The relationship can also be used to analyze the effectiveness of alternative option strategies.
29.4 INSURANCE DERIVATIVES
 
When derivative contracts are used for hedging purposes, they have many of the same character-istics as insurance contracts. Both types of contracts are designed to provide protection againstadverse events. It is not surprising that many insurance companies have subsidiaries that tradederivatives and that many of the activities of insurance companies are becoming very similar tothose of investment banks.
Traditionally the insurance industry has hedged its exposure to catastrophic (CAT) risks such ashurricanes and earthquakes using a practice known as reinsurance. Reinsurance contracts can takea number of forms. Suppose that an insurance company has an exposure of $100 million toearthquakes in California and wants to limit this to $30 million. One alternative is to enter into
* For a fuller discussion of the spot price behavior of energy products, see D. Pilipovic, Energy Risk, McGraw-Hill,New York, 1997.Insurance, Weather, and Energy Derivatives 683
annual reinsurance contracts that cover on a pro rata basis 70% of its exposure. If Californiaearthquake claims in a particular year