Real Options
Real Options: Types
Expand
Delay
Abandon
Using a Binomial Model
Sometimes using a closed-form solution like the Black-Scholes formula is impossible.
Many options (especially in energy) dont have the properties necessary for the Black-Scholes formula to apply.
In many cases, you can use a binomial model with Monte-Carlo simulations.
Thats where were heading next.
Binomial Model
To make the multi-period problem more manageable, allow the binomial tree to recombine (we call it recombining binomial lattice)
t=0
t=1
t=2
t=3
=>
t=0
t=1
t=2
t=3
*
Binomial Model
A single node at a random point in time t:
t
t+1
S
Su=u·S
Sd=d·S
t
t+1
c
cu
cd
*
Binomial Model
Formula to find c
Cox, Ross, and Rubinstein (1979)
*
Risk-Neutral Probabilities
The terms in brackets in the formula on the previous slide are called risk-neutral probabilities of stock price going up and down.
Example
Since the probabilities are risk-neutral we can discount at the risk-free rate!
Determining u and d
What if we have multiple intervals, say 90 dates over 3 months?
Let n=number of steps per year. So if we had 90 dates over 3 months, we will have 90*4 = 360 steps over a full year, then n=360.
Suppose we can estimate volatility of the underlying (annualized!) Say volatility is 30%.
Turns out the formulas for u and d are as follows:
u = e?/?n
d=1/u
Example: u = exp(.3/sqrt(360)) = 1.015937
And d = 1/1.015937 = 0.984313
Another Way to Find u and d
Alternatively, T = time to expiration in years, and k = number of steps before expiration. If we had 90 dates over 3 months, then T = 3/12 = 0.25 years, and k=90. Say annualized volatility is 30%. We can calculate
u = e?*?(T/k)
d=1/u
So u = EXP(.3*sqrt(0.25/90)) = 1.015937
And d = 1/1.015937 = 0.984313
Important
Note that in the binomial formula, volatility is annualized (per year); and risk-free rate is per period (step)!
Valuing a Gold Mine
Woe Is Me gold mine: founded in 1878
Played out by 1908 but is reopened occasionally depending on the price of gold
It owns land but worth very little
Cash and liquid securities $3 million
Not mining now
However, has market capitalization of over $100 million. Why?
Gold Mine (Continued)
Costs $0.5 million to reopen the mine, $0.25 million to shut it down (fixed cost)
Assume that for technical reasons the mine cannot be closed / reopened more frequently than twice a year
Production is 5,000 ounces per year (2,500 ounces per half year)
Extraction costs are $350 per ounce (variable cost)
According to terms of lease, cannot stockpile gold: must sell all gold produced.
Lease is for the next 100 years
Gold Mine (Continued)
We can view the mine as a package of call options on the price of gold.
What is (are) the exercise price(s)?
Is there a maturity date?
Gold Mine (Continued)
This is a real option problem.
Lets use common sense first. Should the mine be reopened if the price of gold is $350.15?
What if the price is $360? Remember volatility!
Suppose the mine is now open, and price drops to $345? Does it make sense to immediately shut down?
Need to find two threshold prices, popen and pclose, at which the open mine option is sufficiently in and out of the money, respectively. Volatility is key!
Gold Mine (Continued)
Common sense tells us that the threshold prices depend on volatility. How?
We also know that the threshold prices depend on the fixed costs of opening and closing the mine. How?
Next, we value the call option. Refer to the following diagram.
Gold Mine (Continued)
Mine is Open
Choices:
Mine is Closed
Keep Open
Close at $.25 million
Reopen at $.5 million
Keep Closed
Gold Mine (Continued)
Assume risk-free interest rate of 3.4% per six months
Assume volatility for the price of gold 20% per year
Suppose current price of gold is $350
Consider a 6-month-step binomial tree, 100 years long (200 total steps).
u = e0.20/?2 = 1.1519
d=1/u = 0.8681
Gold Mine (Continued)
Compute risk-neutral probabilities. Answer: about 58% of a rise in gold price. (Do yourself!)
Next, we need a computer simulation. Simulate, say 5000 paths for gold price
Next, pick some guesses for popen and pclose
For each chosen pair, and for each of the 5000 paths, calculate PV(cash flows) discounting at 3.4%. Average. Maximize with respect to popen and pclose.
The maximum value of PV(Cash Flows) = Value of Real Option
Gold Mine (Continued)
This example is not easy to implement
But it is widely used in practice to value real options
Better to use than Black-Scholes due to the complicated structure
Value PUD as a Call Option
S = DCF from the resource once developed
K = cost of developing the resource
r = risk-free interest rate, as usual
T = time to lease expiration
y = cost of delay, if any (analogous to dividend yield)
? = volatility of commodity price (???)
Call Option with Dividends
Development Lag
What if you exercise the option by developing the resource, but the cash flows dont start until N years after development?
Development lag = N
In that case, discount S back N years at the rate y before plugging in the formula on the previous slide.
Its like you incur additional cost of delay for N more years
Example
Source: Aswath Damodaran, Real Options
Oil property has an estimated reserve of 50 million bbls
The cost of developing the reserve is expected to be $600 million
The development lag is two years
Lease expires in 20 years
Oil margin (i.e., price production cost) = $12/bbl
Once developed, the net production revenue each year will be 5% of the value of the reserves.
The riskless rate is 8%
The variance in ln(oil prices) is 0.03
Value the real option to develop this reserve.
Cost of Delay
In this example, delaying exercise is costly because the lease clock is ticking. In the worst case scenario, if the company doesnt develop the reserve for 20 years, the rights to the property will be relinquished and the reserve will be worthless to the firm.
Once developed, each year the firm expects to net out 5% of the total value of reserve. Hence, y=5%. Every year of delaying development costs the company 5% of reserve life.
When Does y=0?
If the company were assumed to have indefinite rights to the property once its developed, then the company would waste no money by delaying development.
Suppose the company has 5 years to start drilling, or else it will lose the lease. However, once drilled, the property can be kept until the reserve is fully depleted, which is estimated to be 20 years after the production commences.
In this case, waiting to develop all the way until year 5 doesnt cost the company anything: it will still enjoy the same reserve life of 20 years whether it develops the reserve in year 1 or in year 5. Then y = 0.
Reserve Value Once Developed, S
Value of the reserve if developed today is $12*50 = $600 million
Alternatively, we could use the information about future cash flows from the reserve and find the value using DCF. Here, we are making a simplification.
Dont forget the development lag. Instead of $600 million, we have to discount = 600*e(-0.05*2) = $542.90
If we assume that all cash flows occur at year-end (instead of continuously), then we have to use 600/(1.05)2 = $544.22. Lets use that number for the Black-Scholes formula.
Notice the cost of developing today $600 is greater than the value of reserve, $544.22. The real option is out-of-the-money. But it still has value!!
Real Option
Plugging all inputs into the call option with dividend formula, we have (please verify yourself)
Even though this reserve costs more to develop than its currently worth under current oil prices, its not worthless to the company. It has the value of the real option because the company has the right to develop it once it becomes profitable. This right has value.
d1 1.035919
d2 0.261323
C 97.09589
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