SØK460/ECON460 Finance Theory, Diderik Lund, 6 November 2002
Real options
• Real options are options within investment projects.
• New information arrives over time, e.g. on prices.
• Some decision may be made in future, after information known.
• Typically produce if high output price, not if low.
• Or invest if high output price, not if low.
• Or produce if low variable cost, not if high.
• Analogy to financial options: Avoid negative outcomes.
• Consider first production in one period.
• Quantity A, variable cost B, output price Pt.
• If no option: Production value APt − B.
• If option not to produce: Value of opportunity max(APt −B, 0).
• Difference important if Pr(APt < B) is non-negligible.
• Value at earlier date of having opportunity at later date?
• Analogous to option value problem.
• Underlying asset: Commodity with price Pt.
• Use McDonald and Siegel’s method, need assumptions.
• Assume Pt is geometric Brownian with drift.
• Assume existence of forward contracts, or sth. similar.
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SØK460/ECON460 Finance Theory, Diderik Lund, 6 November 2002
Real options: One-period example
• APt − B, assume A = 100, B = 1800.
• Assume output price known at t = 0: P0 = 28.
• Between t = 0 and t = 1, Pt is geom. Brownian w/drift.
• Need to know σ, assume σ = 0.29.
• No need to know drift parameter αP .
• (Bjerksund and Ekern call the drift parameter α.)
• Forward price at t = 0 for t = 1 output is F01 = 29.
• Risk-free interest rate is 10 percent per period.
• From these assumptions: Can calculate δ.
• P0e−δ = F01e−r ⇔ δ = r − ln(F01 /P0) = 0.065.
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SØK460/ECON460 Finance Theory, Diderik Lund, 6 November 2002
Real options: Example, contd.
Consider three alternatives, which may be realistic descriptions in
different situations.
(a) If produce at t = 0, value is AP0 − B = 28 · 100 − 1800 = 1000.
(b) Decide at t = 0 to produce at t = 1, unconditionally.
– Decision made before P1 is known.
– No option, but uncertain future value AP1 − B.
– Want to find value at t = 0 of a claim to this.
– Possible to use CAPM to find value of claim to P1.
– (Would need quadratic utility, since P1 lognormal.)
– CAPM requires E0(P1) and cov0(P1 , Rm).
– But market’s valuation already observable: Forward price.
– No need for expectation and covariance.
– Value is AF01e−rt − Be−rt = AF01 e−r − Be−r .
– Rewritten as (AF01 − B)e−r or AP0e−δ − Be−r .
– Given our numbers: = 1100 · 0.9048 = 995.3.
Comparison: Prefer to produce at t = 0.
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SØK460/ECON460 Finance Theory, Diderik Lund, 6 November 2002
Real options: Example contd.
(c) Have opportunity at t = 1 to produce or not (saving B).
– Can predict decision: Produce if and only if AP1 > B.
– Value at t = 1 is max(AP1 − B, 0).
– McDonald and Siegel option on AP1 with K = B.
– Equivalent: A options on P1 with K = B/A.
ln(AP0 e−δT /Be−rT ) 1 √
√
d=
+ σ T =
2
σ T
ln(AF01 /B) 1
ln(AP0 e−δ /Be−r ) 1
+ σ=
+ σ ≈ 1.7896,
σ
2
σ
2
√
d − σ T ≈ 1.4996,
N (d) ≈ 0.9633, N (d − σ) ≈ 0.9332,
W (AP0 , B, r, T, σ, δ) = AP0e−δ N (d) − Be−r N (d − σ)
= [AF01N (d) − BN (d − σ)]e−r ≈ 1007.8.
• Alternative (c) more valuable than alternative (b).
• Compare (a) to (c): If option at t = 1, then wait at t = 0.
• Compare (a) to (b): If no option, then produce at t = 0.
• In case (b) and (c): May compare value to investment cost now.
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SØK460/ECON460 Finance Theory, Diderik Lund, 6 November 2002
Real options: Bjerksund and Ekern’s models
• More complicated real options (—somewhat simplified here).
• Essential difference: More complicated timing decisions.
• Non-essential difference: Project lasts for long time.
• Non-essential: Known (cost) inflation, π. Set π = 0 here.
• Notation: Use α̂, α, S for McDonald and Siegel’s α, αP , P .
• Project produces q(τ ) at variable cost k(τ ).
• Important: These are fixed once project is started.
• τ measures “project time,” i.e., time from project start.
• Project lasts from date t to date t + t̄.
• Market value at start-up date is
Z t̄ ·
0
=
Et(S(t + τ ))e
Z t̄ ·
0
=
= S(t)
Z t̄
0
e
−α̂τ
ατ −α̂τ
S(t)e e
Z t̄ ·
0
−δτ
defining A and K.
S(t)e
−δτ
q(τ )dτ −
q(τ ) − k(τ )e
q(τ ) − k(τ )e
q(τ ) − k(τ )e
Z t̄
0
5
−rτ
−rτ
−rτ
¸
¸
¸
dτ
dτ
dτ
k(τ )e−rτ dτ = S(t)A − K,
SØK460/ECON460 Finance Theory, Diderik Lund, 6 November 2002
Bjerksund and Ekern’s models, contd.
• In sections II–IV: Always S(t)A − K if project starts.
• In section V also operating flexibility.
• Difference between models: Type of timing decision.
• Most flexible type: Any time between now and ∞.
• But project can only be done once.
B & E, section II
• Immediate “development” (= project start).
• A. Now or never.
– Yes if S(t)A > K, i.e., if S(t) > K/A ≡ SBE .
• B. Decide now when to develop, or to drop project.
– No decisions to be made after today.
– If decide at t to develop at T : Value seen from t:
Ct(T ) = e−δ(T −t) AS(t) − e−r(T −t) K
– Maximize w.r.t. T .
– Must solve 1st-order condition, check 2nd-order condition.
– Must check solution has T > t and Ct(T ) > 0.
– May perhaps find corner solution, T = t.
– Perhaps “corner solution” T → ∞, not if δ > 0.
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SØK460/ECON460 Finance Theory, Diderik Lund, 6 November 2002
Bjerksund and Ekern, model II B
dCt(T )
= −δe−δ(T −t) AS(t) + re−r(T −t) K.
dT
First-order condition: This is zero if and only if


T = ln 

rK 
1
·
+ t,
δAS(t) r − δ
but a corner solution may be optimal if
¯
rK
dCt(T ) ¯¯¯
¯
<
0
⇔
< 1.
dT ¯¯T =t
δAS(t)
Second-order condition for maximum?
d2Ct(T )
= δ 2e−δ(T −t) AS(t) − r 2e−r(T −t) K < 0
2
dT
2
 r K

.
⇔ (r − δ)(T − t) < ln  2
δ AS(t)


Positive Ct(T ) requires


K 

.
e−δ(T −t) AS(t) − e−r(T −t) K > 0 ⇔ (r − δ)(T − t) > ln 
AS(t)
Summing up, an interior maximum with Ct(T ) > 0 requires
2
 K

 rK

 r K

 < (r − δ)(T − t) = ln 
 < ln 
.
ln 
AS(t)
δAS(t)
δ 2AS(t)






A necessary condition for this is r > δ, but even then, if rK <
δAS(t), a corner solution T = t is optimal.
Conclude: T = t if S(t) ≥
rK
δA
= δr SBE . If not, see f.o.c.
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SØK460/ECON460 Finance Theory, Diderik Lund, 6 November 2002
Bjerksund and Ekern, section III
• Development decision at fixed future date.
• European call option on AS(T ) with exercise price K.
• Use McD & S’s formula, rewritten to B & E’s notation:
W (AS(t), B, r, T, σ, δ)
= AS(t)e
where
−δ(T −t)
N (d) − Ke
−r(T −t)
√
N (d − σ T − t)
ln(AS(t)e−δ(T −t) /Ke−r(T −t) ) 1 r
√
d=
+ σ (T − t).
2
σ T −t
• Depending on circumstances, an extended problem may occur:
• Perhaps possible to choose T at t, but maintain option.
• In that case: Maximize W () value above w.r.t. T .
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SØK460/ECON460 Finance Theory, Diderik Lund, 6 November 2002
Bjerksund and Ekern, section IV
• Development decision at any time.
• American option: When to start?
• Optimal decision takes form of a strategy.
• Rule for how to decide depending on incoming info.
∗
• In this case: Critical price, SAP
.
∗
• Optimal strategy: Start the first time S(t) ≥ SAP
.
• Expiration date may be finite or infinite.
• For financial options: Most often some finite date.
• Real options: Often ∞, but licenses may expire.
∗
and option value.
• When ∞, formulae exist for SAP
• With finite expiration date: No formula. Numerical methods.
• Will not derive formulae here. Diffcult math.
With
v
u
u
u
u
t
1 r−δ
r − δ 1 2
r
ε= − 2 +
−
+
2
2
σ
σ2
2
σ2
the trigger price is
ε
ε K
∗
SAP
=
SBE =
,
ε−1
ε−1A
and the option’s value is
WAP = K


ε
1  ε − 1 AS(t) 

 .
·
ε−1
ε
K
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