Introduction to Real Options
‘Keep your options open’
• Dixit and Pindyck (1994) developed real options method
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• Method of dynamic programming (Kennedy 1986;) to financial options and
then began adapting their ideas to the real world; hence “real” options.
• RO method allows analysis of decisions in complex, dynamic, and non- linear world, subject to thresholds and risks of irreversible damage.
• An option is a right, but not the obligation, to take action. Every action creates new rights and forecloses others.
Financial options and real options
• An option is a right but not an obligation to do something.
• A financial option is a contract between two people in which one
purchases an option contract from the other.
• The purchaser the option obtains the right and the person who sold the option has the obligation.
• Difference to a real option? There is no seller of a real option contract. It is just one person deciding whether or not to invest. A person always has the option to invest or not.
• A real option has a price because not investing may mean profits are foregone.
• Investing unwisely may mean losses are incurred.
• So the trick is to invest wisely.
Put and call options
• In finance:
• a call option gives the purchaser the right but not the obligation to buy a number of shares (at an exercise price, at or within a specified time).
• a put option gives the seller the right but not the obligation to sell a number of shares (at an exercise price, at or within a specified time).
Kidman (Source: Dobes 2012)
• “Abstract. There is little direct evidence about the business model used by the legendary cattle king, Sir Sidney Kidman. Kidman’s properties were invariably stocked at less than full capacity, and were generally contiguous, forming chains that straddled stock routes and watercourses in the most arid zone of central Australia. Railheads at the ends of the chains provided access to the main capital city markets, and Kidman’s drovers supplied a wealth of information on competing cattle movements. This combination of features effectively afforded strategic transport flexibility in the form of so-called ‘real options’, especially during severe region-wide droughts. Alternative perspectives, such as the vertical integration of Kidman’s operations, or spatial diversification of land holdings, offer only partial insights. Faced with a highly variable and unpredictable climate, combined with the onset of erosion and the spread of rabbits, Kidman exemplifies human ability to adapt creatively to exogenous environmental shocks such as climate change”.
Australia’s arid zones
Arid zone boundary
Kidman properties
-active buyer and seller of properties
-strategy of acquisition of linked properties in more arid regions.
1935 snapshot
1. Breeding properties are
scattered across the north – source of cattle for southern markets
2. Contiguous landholdings west of the
Characteristics of property holdings (Source: Dobes, 2012)
• 1. Situated in arid, marginal country (rainfall highly variable and unpredictable).
• 2. Property acquisitions usually adjoined other owned properties.
• 3. Properties either straddled or were close to stock routes or rivers.
• 4. Ends of his property chains close to distribution points.
• Competing explanations for 1-4 above: • Supply chain
• Spatial diversification • Real options
Supply chain (Source: Dobes, 2012)
• Kidman’s success attributed to vertically integrated business operations:
• Cattle breeding
• Cattle fattening
• Final sale operations
• Dismiss this theory on grounds that vertically-integrated business can be practiced without buying up contiguous properties.
Spatial diversification (Source: Dobes, 2012)
• Portfolio Theory- (we will cover diversification in the coming weeks in lectures).
• Risk associated with returns for all the assets in a portfolio is a function of the sum of the individual weighted variances and weighted covariances between them.
• “Kidman demonstrated a commercial ability to survive major regional droughts when spatial diversification in itself would have been largely ineffective”.
• That is diversification can’t reduce systemic risk (regional drought, market risk)
Real options (Source: Dobes, 2012)
• Because of uncertainty about future events, maintain a degree of flexibility –”wait and see” as information reveals itself over time.
• Decisions can be made progressively as information is realized.
• Options in financial markets (to do certain things like buy stock)
• Real options can be acquired over physical things.
• Example:
• “Sinking a borehole, for example, provides the right or opportunity, but no obligation, to water stock during a drought. The
option of a borehole is an alternative to sending cattle for agistment, or purchasing a more expensive property in a higher rainfall area. The cost of the borehole is the premium paid for the ‘real option’, or right to water stock in a drought. Where the expected value of the benefits of cattle surviving future droughts exceeds the cost of the borehole, or the alternative of agistment costs, drilling may be worthwhile”.
Real options (Source: Dobes, 2012)
• Normal seasons- ‘Call’ options exercised when cattle were moved from a drought-stricken property to properties with available feed and water.
Real options (Source: Dobes, 2012)
• Peacocking stock routes- generated a ‘real option’ that could be exercised in times of severe drought, because it afforded the opportunity, but not the obligation to use strategically located properties to get cattle to market.
• A ‘call’ option on market prices: information combined with transport flexibility. Deal with price uncertainty by combining the availability of properties near railheads with an information network (drovers, camel drivers, aborigines, dingo-trappers) on competitor’s cattle movements. This allowed flexibility of choice over which market to deliver to.
Investment decision criteria
• Traditional measures
• Cost benefit analysis 𝑁 = σ𝑇 𝜌𝑡𝑁
• where 𝜌 = 1 and 𝛿 is the discount rate (1+𝛿)
•𝑁=𝐵−𝐶 𝑡𝑡𝑡
• N is net benefits, B is benefits and C is costs
0 -20 -40 -60 -80 -100 -120
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A time path for net benefits
Initial investment costs
Major maintenance
Decommissioning costs
• 𝑁 = σ𝑇 𝜌𝑡𝑁 = $77.58872 𝑡=0 𝑡
• Positive net benefit
• No identification of who
gets the benefits and who pays the costs
Investment decision criteria
• Traditional measures
• Benefit cost ratio:
• Projects should have Τ
σ𝑇 𝜌𝑡𝐵 𝑡=0 𝑡
σ𝑇 𝜌𝑡𝐶𝑡 𝑡=0
> 1 to be under consideration = 1.2236
• Distribution of benefits and costs
• For the data above Τ 𝐶
Investment decision criteria
• Traditional measures
• Internal rate of return (IRR) :σ𝑇 𝑁𝑡
𝑡=0 (1+𝑟)𝑡
• Project’s IRR is r, the rate which when used as discount rate ensures IRR=0
• Can use Solver to find r.
• r=0.112966 so decision rule is r > 𝛿 (hence construction today)
• Problems with IRR: if PV NB changes sign more than once, there may be more that one r, also positive balances prior to t=T may have reinvestment options.
Investment decision criteria
• Return on invested capital (RIC) used when comparing 2 or more projects. To solve for RIC, define a project’s balance at 𝑡 = 𝜏:
• 𝑃𝐵 = σ𝑡=𝜏 𝑁 (1 + 𝑖 )𝜏−1 i is risk-free discount rate (𝛿)or the RIC 𝜏 𝑡=0𝑡 𝑡
according to the rule:
• 𝑃𝐵 = 1 + 𝑅𝐼𝐶 𝑃𝐵 + 𝑁 if 𝑃𝐵 < 0
𝜏 𝜏−1𝜏 𝜏−1
• 𝑃𝐵 = 1 + 𝛿 𝑃𝐵 + 𝑁 if 𝑃𝐵 > 0 𝜏 𝜏−1 𝜏 𝜏−1
• Depending on whether PB > 0 or < 0 in 𝜏 −1, the PB will compound forward to 𝜏 by either the RIC or 𝛿. • RIC is the rate that drives the PB in the last period to 0. • RIC has “nice’ properties- it will be a unique value. • RIC=0.1118695 • Decision rule: invest if RIC > 𝛿
Real Options
• Uncertainty
• Irreversible?
• economically costly (SUNK COSTS) • ecologically irreversible
Irreversible decisions
• Some decisions may be irreversible and investment costs are likely sunk – for example:
• Undertaking an R&D programme
• Establishing irrigation canals
• Strip mining coal from the Liverpool plains
• Others may be reversible, even if the reversal incurs some cost – for example:
• Switching from extensive grazing enterprise to irrigated cotton enterprise
2 period model: old growth forest (Conrad 1999 Ch. 7)
• Possible actions:
• 1.Clear-cut forest today (t=0) yielding timber revenue (T0) and subsequent
agricultural production on land with net revenue in t=1 of D1.
• Present value of this is D
• 𝐷 = 𝑇 + 𝜌𝐷 where 𝜌 = 1 is the discount factor.
• 2. Leave the forest intact for amenity value (A0)
• Future values of A and T uncertain, suppose there are 2 possible states: • State 1( s=1) high timber prices (T1,1 > A1,1 )
• State 2( s=2) low timber prices (T1,2 < A1,2 )
2 period model: old growth forest
• Let 𝜋 be probability for s=1 and (1- 𝜋) be probability for s=2.
• Assume T0 > A0> 0
• Timber revenue can fund valuable government projects
• State-contingent decision rule: If timber not cut today government thinks it optimal to cut only if s=1.
• What is the expected present value of preservation today? •𝑃=𝐴 +𝜌𝜋𝑇 +(1−𝜋)𝐴
• D > P – cut timber today & grow crops
• D < P – don’t cut timber today & “wait & see”
2 period model: point of indifference?
•𝑇 +𝜌𝐷 −𝐴 =𝜌𝜋𝑇 +(1−𝜋)𝐴
0 1 0 1,1 1,2
Net value of cutting the forest today
Option value (OV) of preservation today (t=0) : discounted value of behaving optimally tomorrow
2 period model: comparative statics
• Increase in T1,1 or A1,2? OV increases, increased incentive to wait, preserve forest today.
• Increase in T0 or D1 or decrease A0 ? Increases value of cutting forest today.
• Increase in 𝛿 will RHS > LHS
• What is the sign of 𝑑𝜋 that preserves indifference?
𝑑𝜋 (𝑇 −𝐴 )
Recallassumed𝑇 >𝐴 00
• = 0 0 𝑑𝛿 (𝑇 −𝐴 )
• so if 𝑇 > 𝐴 1,1 1,2
and 𝑇 < 𝐴 1,1 1,2
Meaning? Increase discount rate decreases OV, we need to increase probability of high future timber price to maintain indifference.
OV: infinite horizon model.
• If cut forest in t or any time, the net agricultural; benefit is D1. over 𝑡. . 𝑡 + 2 ... ∞
• If forest not cut in t there are only 2 possible future states in t+1 and it’s optimal to cut the first time s=1.
• The probability of s=1 (𝜋) is stationary – constant
• Expected value of entering the next period with forest intact is
𝜋𝑇 + (1 − 𝜋)𝐴 - here T and A are stationary optimal net benefits in 1212
s=1 and 2 respectively-time subscripts suppressed.
OV: infinite horizon model
• NPV of decision to cut forest today?
•𝐷=𝑇 +𝜌𝐷 1+𝜌+𝜌2+⋯ =𝑇 +𝜌𝐷
OV: infinite horizon model
• NPV of decision to preserve forest today (allowing it to be cut any time into the infinite future)? P will comprise 2 expressions that converge.
• P = 𝐴 + 𝜌 𝜋𝑇 + (1 − 𝜋)𝐴 012
+𝜋𝜌2𝐷 1+𝜌+𝜌2 +⋯ +(1−𝜋)𝜌2 𝜋𝑇 +(1−𝜋)𝐴 112
+(1−𝜋)𝜋𝜌3𝐷 1+𝜌+𝜌2 +⋯ +(1−𝜋)2𝜌3 𝜋𝑇 +(1−𝜋)𝐴 112
+ 1−𝜋 2 𝜋𝜌4𝐷 1+𝜌+𝜌2 +⋯ +(1−𝜋)3𝜌4 𝜋𝑇 +(1−𝜋)𝐴 112
Meaning line by line:
• 𝐴 + 𝜌 𝜋𝑇 + (1 − 𝜋)𝐴 012
• is expected value in t=0 + discounted expected value in t=1
• 𝜋𝜌2𝐷 1+𝜌+𝜌2 +⋯ +(1−𝜋)2𝜌3 𝜋𝑇 +(1−𝜋)𝐴 112
• is discounted expected net present value in t=2 of a decision to cut or preserve in t=1. If forest cut in t=1,infinite flow of agricultural benefits (𝐷 ) plus the
expected value if timber hasn’t been cut (prob of not cutting is (1 − 𝜋))
In period t=3, the expected discounted NPV of a decision to cut or preserve in t=2
• +(1−𝜋)𝜋𝜌3𝐷 1+𝜌+𝜌2 +⋯ +(1−𝜋)2𝜌3 𝜋𝑇 +(1−𝜋)𝐴 112
• Probability of not cutting in =1
• Probability of cutting in t=2 probability of not cutting in
This process of computing joint probabilities and discounting will converge as 𝑡 → ∞
•𝑃=𝐴 +𝜌𝜋𝑇 +(1−𝜋)𝐴 1+(1−𝜋)+ (1−𝜋) 2+⋯..
+𝜌𝜋𝐷1+𝜌+𝜌+⋯ 1+ 1
(1−𝜋) 2 + ⋯ . .
Now 1 + 𝜌 + 𝜌2 + ⋯
converges to
1 + (1−𝜋) +
converges to
So, P can be written:
𝜋𝑇 + 1−𝜋 𝐴
•𝑃=𝐴+ 1 2+𝜋𝐷/𝛿𝛿+𝜋 0 𝛿+𝜋 1
• Expected present value of not cutting in t=0.
• As 𝜋 → 1 then probability of cutting in t=1 → 1 and present value of
not cutting in t=0 should be :
• 𝑃 = 𝐴 + 𝜌 𝑇 + 𝐷 /𝛿 011
infinite period model: point of indifference?
•𝑇+1−𝐴=1 2+1
𝐷 𝜋𝑇 +(1−𝜋)𝐴 𝜋𝐷 0𝛿0 𝛿+𝜋 𝛿𝛿+𝜋
PV of cutting & conversion to agriculture less loss of amenity
Infinite horizon OV of not cutting in t=0
Numerical example to illustrate the magnitude and potential importance of the OV
• 𝑇 = $20𝑚; 𝐴 =$3m; 𝐷 =$5m per period forever 001
• 𝛿=0.05 ; 𝜋=0.5
• 𝑇 = $25𝑚 in state 1; 𝐴 =$3.7m in state 2 12
• From 𝑇 + 1 =
𝜋𝑇 +(1−𝜋)𝐴 1
we find = $120-3 = $117 (97.5% of P)
• OV = 𝑇 0𝛿0
• This case is point of indifference between cutting today in t=0 or preserving
What happens if 𝜋 =0.75?
• Scale tips towards preservation (D unchanged, P increases to $121.34347m)- higher probability of larger net revenues for timber encourages delay
• What happens if 𝜋 =0.25? P decreases to $116.416677m- decision would be to cut today
What happens if 𝛿=0.1?
• Scale tips against preservation (D decreases to $70m, P falls to
$68.583333m)- cut today is optimal
• What happens if 𝐷 =$2.5m? D decreases to $70m- P falls to
$74.5454545m decision would be to delay cutting at least one period
Trigger (threshold) values for irreversible decisions
• The investment has uncertain future net benefits and is irreversible • Continuous time model
• Some math results will be stated without derivation
• Nevertheless, trigger values will make sense intuitively
• Cost of project is K ($m) can be built at time 𝜏 • Net benefits from the project defined as N
Trigger values for irreversible decisions
• Net benefits N are unknown at t > 𝜏 but evolve as GBM:
• 𝑑𝑁=𝛼𝑁𝑑𝑡+𝜎𝑁𝑑𝑧
• 𝑑𝑁/𝑁 = 𝛼𝑑𝑡 + 𝜎𝑑𝑧
• 𝛼𝑑𝑡 is the mean percentage change
• Random component (𝜎𝑑𝑧) – 𝑑𝑧 = 𝜀(𝑡) 𝑑𝑡. Where 𝜀(𝑡) is Niid (0,1)
• 𝑁 = (1 + 𝛼)𝑁 + 𝜎𝑁 𝜀 (discrete time version of GBM)
𝑡+1 𝑡 𝑡 𝑡+1
• From a known N0 we can generate time paths for net benefits.
• Parameters: N0 is $5m, 𝛼=0.04, 𝜎=0.1, 𝜀(𝑡) generated by Excel’s random number generator (N distribution)
Trigger value for N
• Expected net benefit in t : 𝐸 𝑁 = 1 + 𝛼 𝑡𝑁 𝑡0
• 3 generated time 14 paths (realisations) 12
for net benefits (Nt)
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Nt1 Nt2 Nt3
What is the trigger for investment (N*)?
• N evolve according to GBM process, what value of N triggers investment in this project that we would never abandon?
• Net benefits are log-normally distributed, the expected net benefit:
𝑡𝛼𝑒𝑁 = )𝑡(𝑁 𝐸 •
• Analytic expression for the PV of 𝐸 𝑁(𝑡) :
𝑁 =𝑡𝑑𝑡𝛼−𝛿−𝑒∞𝑁=𝑡𝛿−𝑒𝑡𝛼𝑒𝑁∞•
• In general, 𝑉 = 𝑁 where 𝛿 − 𝛼> 0 (if 𝛿 − 𝛼 < 0 )?
What is the trigger for investment (N*)?
• Assume as yet no investment, but option to invest is live
• What is the value of the option to invest at the trigger value N*?
• 𝑉 denotes the value of waiting. 𝑉 = 𝛾𝑁𝛽 assumes N< N* 𝑊𝑊
•𝛽=0.5−𝛼 + 𝛼−0.52+2𝛿>1 𝜎2 𝜎2 𝜎2
• As 𝑁 → 0,𝑉 → 0 ( net benefits fall, value of option falls). As N 𝑊
increases value of option increases
Trigger for investment N* and 𝛾 can be determined by 2 conditions:
• 1. value matching condition requires 𝑉 = 𝑉 − 𝐾 𝑊𝐼
• Implies indifference between waiting and the discounted net benefit less construction cost. 𝛾𝑁𝛽= 𝑁 – K
• 2. smooth pasting condition requires equality of 1st derivatives of value function at N*: 𝑉′𝑊 = 𝑉′𝐼
𝛽𝛾𝑁𝛽−1= 1 (𝛿−𝛼)
• 2 eqs, 2 unknowns, solving: 𝑁∗= 𝛽 𝛿−𝛼 𝐾 and (𝛽−1)
𝛾 = 𝑁𝛽−1/ 𝛽 (𝛿−𝛼)
Trigger value. 𝑁∗= 𝛽 𝛿−𝛼 𝐾 (𝛽−1)
• N* is trigger value that stochastically evolving N much reach before we invest • ∆= 𝛽 𝛿−𝛼 is critical coefficient for irreversibility & uncertainty.
• If ∆ > 𝛿. More conservative investment rule.
• Compare to simple cost benefit criteria: invest if N> 𝛿 K • Assume K=100 and 𝛿 = 0.1.
• We can plot N* for realisations of N
N* and 3 realisations for Nt
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Nt1 Nt2 Nt3 N*
• Only one realisation reaches the trigger value -Nt1
• It also falls below N* in t=29
• This is an inherent risk with irreversibility : even though the drift
component 𝛼 =0.04 indicated N will drift upward because of
𝜎, once the investment is triggered there is still the possibility for the investment outcome to sour- returns may not cover interest payments on K
References
• Conrad (1999) Resource Economics Ch. 7
• Hardaker, Huirne and Anderson (2004) Coping with Risk in
Agriculture, CAB International.
• Chapter 11: Risky decision making and time
• Dobes, L. (2012) Sir Sidney Kidman: Australia’s cattle king as a pioneer
• of adaptation to climatic uncertainty The Rangeland Journal, 34, 1- 15.
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