Download A Real Option Perspective on the Future of the Euro

A Real Option Perspective on the Future of the Euro
Fernando Alvarez
University of Chicago
Avinash Dixit
Princeton
BFI – September 2013
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
1 / 40
Option Value of Abandoning Euro
Collapse of the euro has shifted from unthinkable to a real possibility.
Large literature on costs and benefits of permanently having either:
Currency union vs. Independent monetary policy.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
2 / 40
Option Value of Abandoning Euro
Collapse of the euro has shifted from unthinkable to a real possibility.
Large literature on costs and benefits of permanently having either:
Currency union vs. Independent monetary policy.
Our contribution: Optimal timing of abandoning euro.
Novelty: abandoning euro is almost irreversible and has large fixed costs.
Thus it fits the set-up from “real option valuation".
We compute the option value of (delaying) abandonment of the euro.
Should the Euro be maintained even at extremely high current costs?
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
2 / 40
Option Value of Abandoning Euro
Insights from real option literature
Abandoning irreversible investment project (or subject to fixed cost).
Examples: closing a mine, discontinuing a product line, etc.
Suppose project net present value (NPV) fluctuates randomly,
NPV can become very large or very small.
Optimal abandonment: NPV < fixed cost. Why?
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
3 / 40
Option Value of Abandoning Euro
Insights from real option literature
Abandoning irreversible investment project (or subject to fixed cost).
Examples: closing a mine, discontinuing a product line, etc.
Suppose project net present value (NPV) fluctuates randomly,
NPV can become very large or very small.
Optimal abandonment: NPV < fixed cost. Why?
As with a financial option:
Abandoning project gives protection to large downside.
Continuing project allows to enjoy large upside.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
3 / 40
Option Value of Abandoning Euro
Insights from real option literature
Abandoning irreversible investment project (or subject to fixed cost).
Examples: closing a mine, discontinuing a product line, etc.
Suppose project net present value (NPV) fluctuates randomly,
NPV can become very large or very small.
Optimal abandonment: NPV < fixed cost. Why?
As with a financial option:
Abandoning project gives protection to large downside.
Continuing project allows to enjoy large upside.
Replace abandoning Project by abandoning the Euro !
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
3 / 40
Option Value of Abandoning Euro
Cost & benefits of Eurozone (common currency area)
Large literature on this topic, e.g. Mundell-Fleming.
Cost of union of n countries: inability of country-specific monetary policy.
Country i nominal misalignment Xi , a measure of PPP (or wage differential)
that with independent monetary policy will (should) be “corrected".
Common union monetary policy Z .
Each country xi = Xi − Z and dXi = −µ Xi dt + σc dWc + σ dWi .
Size of idiosyncratic shock σ, which self-correct at speed µ .
Effect on each country (welfare, output) of misalignment ("β (xi )2 ").
Flow benefit, unrelated to monetary policy ("α"):
↓ transaction cost and ↑ trade, both due to common currency.
Fixed cost of abandoning union ("Φ"), stands from crisis.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
even more wonkish
4 / 40
Option Value of Abandoning Euro
Abandoning the Euro as investment project
Modeling euro as “union" deciding when to break it up complete or not at
all.
Optimal abandonment when combination of misalignment is large:
Abandon first time that
Pn
i=1
2
(xi ) = Y .
(Wonkish: to compute Y solve a perpetual multidimensional option,
w/quadratic payment. Also new result on non-monotone effect of σ on Y ).
Compare with now-or-never policy of abandonment when cost = NPV.
Also defined as threshold of misalignments
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
Pn
i=1
2
b.
(xi ) = Y
5 / 40
Option Value of Abandoning Euro
Abandoning the Euro as investment project
Modeling euro as “union" deciding when to break it up complete or not at
all.
Optimal abandonment when combination of misalignment is large:
Abandon first time that
Pn
i=1
2
(xi ) = Y .
(Wonkish: to compute Y solve a perpetual multidimensional option,
w/quadratic payment. Also new result on non-monotone effect of σ on Y ).
Compare with now-or-never policy of abandonment when cost = NPV.
Also defined as threshold of misalignments
Pn
i=1
2
b.
(xi ) = Y
How much extra misalignment (pain) should be tolerated?
How large are the gains from the optimal relative to now-or-never policy?
How long can be the Eurozone between now-or-never and optimal policy?
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
5 / 40
Quantifying the option value: depressing results
Borrow parameter values from empirical literature.
parameters
b now-or-never abandonment.
Recall: optimal abandonment at Y > Y
b implies correction in each country
Abandonment at either Y or Y
(depreciation or appreciations).
At optimal Y : average correction ≈ 25%.
b : average correction ≈ 20%.
At now-or-never Y
If maximally concentrated (Spain?), multiply each correction by 2.
b now-or-never
Welfare loss of 4 % of GDP (once) if abandon Euro at Y
instead of following optimal policy.
b to reach Y : more than 10 years!
Expected time, starting at Y
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
details
6 / 40
Many things left out ...
Model is symmetric. Are countries is Europe symmetric?
May not be important for option value. case of two countries
symmetry
.
Eurozone acting optimally even at break-up. Important limitation.
Find that large extra cost is needed to deter individual members exit
.
Mundell-Fleming vs other models of cost and benefits, more emphasis on
debt dynamics. Reinterpretation of Xi and recalibration.
Anticipatory dynamics as on Krugman’s style balanced of payment crises.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
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Summary and Intro
Summary and Intro
Collapse of the euro has shifted from unthinkable to likely.
We model Eurozone as individual countries facing:
Flow benefits: independent of monetary policy, and
Flow costs: inability to correct country’s nominal “misalignments" shocks.
Beak-up: irreversible and subject to large fixed costs.
Whether/when to incur these fixed cost in face of ongoing uncertainty:
“Real Options” problem, abandonment beyond:
Fixed Cost > Net Present Value Benefits.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
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Summary and Intro
Common monetary policy (only) offsets Eurozone-wide shocks.
Mean reverting process for country’s misalignment (PPP deviation).
Use simple, reduced-form, static model of cost and benefits.
Benchmark parameters “calibrated" to macro-literature studies.
Problem of union w/ transfers & commitment (“Fiscal + Monetary Union").
Brief analysis of incentive of one deviant country.
Findings:
Small but not negligible option value.
Small extra cost can deter individual’s country exit.
Surprising theoretical results on option value.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
other applications
9 / 40
Literature Review
Large macro literature on cost/benefit currency union.
Large empirical literature on currency unions/outcomes.
Small macro literature on analysis of union considering breakup:
Main: Lippi-Fuchs (RES 06) repeated game between two countries.
Countries face Barro-Gordon type problem, Union solves that.
Full analysis of individual countries incentives to depart, and effect in policy.
Our work does not capture these features.
Instead more realistic stochastic model in multi-country environment.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
10 / 40
Union Problem, Set up
Countries i = 1, 2, . . . n “nominal" shocks Xi :
dXi = −µi Xi dt + σi dWi + σc dWc ,
standard BMs: Wi indepedent and Wc common
Eurozone common monetary policy Z .
Misalignment of country i, “real exchange deviation" given by:
xi ≡ Xi − Z
Utility flow of country i : ui (xi ) while in Eurozone and zero outside.
ui (·) decreasing in distance of xi from zero, and αi ≡ ui (0) > 0.
xi = 0 eliminate misalignment, in which case only gains from union.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
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Union Problem, Set up
Countries i = 1, 2, . . . n “nominal" shocks Xi :
dXi = −µi Xi dt + σi dWi + σc dWc ,
standard BMs: Wi indepedent and Wc common
Eurozone common monetary policy Z .
Misalignment of country i, “real exchange deviation" given by:
xi ≡ Xi − Z
Utility flow of country i : ui (xi ) while in Eurozone and zero outside.
ui (·) decreasing in distance of xi from zero, and αi ≡ ui (0) > 0.
xi = 0 eliminate misalignment, in which case only gains from union.
Fixed up-front cost of breaking Eurozone Φ > 0.
Eurozone stopping time τ (t) & monetary policy Z (t) solves
"Z
#
n
τ X
−rt
−r τ
ui (Xi (t) − Z (t)) e dt − e
Φ | Xi (0) = Xi , i = 1, ..., n
sup E
{τ,Z }
0
i=1
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
11 / 40
Eurozone Problem, simplifications
Maximization over control Z ∗ static, depends only on current X1 , ..., Xn
∗
Z = arg max
z
n
X
ui (Xi − z)
i=1
Quadratic utility: ui (x) = αi − 12 βi x 2 with positive αi , βi .
Almost all analysis for symmetric countries: βi ≡ β, µi ≡ µ, σi = σ
Pn
Pn
Quadratic utility w/same β =⇒ Z ∗ = i=1 n1 Xi and
i=1 xi = 0
Define Y =
Pn
i=1
2
(Xi − Z ∗ ) . Using Itô’s Lemma and simplifying,
dY = (n − 1) σ 2 − 2 µ Y dt + 2 σ Y 1/2 dW ,
where W is a new standard Wiener process.
Observe σc cancels – ECB policy takes care of common monetary shock.
Same law of motion for Y as if n − 1 countries and Z = 0.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
12 / 40
Eurozone Problem, simplifications
Define flow utility of union:
U(Y ) ≡
n
X
u(Xi − Z ∗ ) = n α −
1
2
βY,
i=1
Choose stopping time τ to solve the problem
Z τ
−rt
−r τ
V (Y ) = sup E
U(Y (t) ) e dt − e
Φ Y (0) = Y
τ
0
CS
where dY (t) = (n − 1) σ 2 − 2 µ Y (t) dt + 2 σ Y (t)1/2 dW
exit
Optimum policy has one dimensional threshold of abandonment Y .
In the region of inaction Y ∈ [0, Y ),
2 σ 2 Y V 00 (Y ) + [ (n − 1) σ 2 − 2 µ Y ] V 0 (Y ) − r V (Y ) + [ n α −
1
2
β Y ] = 0.
At the threshold, value matching and smooth pasting conditions
V (Y ) = −Φ,
Alvarez, Dixit (UofC, Princeton )
V 0 (Y ) = 0 .
Real Options Perspective on the Euro, Sept. 2013
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Eurozone Problem, simplifications
r = 0. 05,
α
β
= 0. 5,
Φ
βn
= 10 σ = 0. 3 µ = 0. 025
20
Value function V (Y )
0
−20
Y
−40
−60
−80
−100
F i x e d C os t − Φ
I nac t i on Re gi on
−120
0
2
4
6
8
Y =
10
!n
12
i=1
14
Abandon
16
18
20
x 2i
Figure : Graph of V (Y ) for illustrative parameter values.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
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Option Value
Numerical Example: n = 5 , r = 0.05 , α = 1 , β = 2 , Φ = 100
Table : Abandonment threshold Y
σ
µ
0.0
0.2
0.3
0.0
10.00
12.85
13.58
0.0125
15.00
15.00
15.00
0.0250
20.00
18.57
17.32
Random walk case µ = 0 usual comparative static
∂Y
∂σ
∂Y
∂σ
> 0.
switches to negative as µ increases. Contrary to conventional result.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
15 / 40
Option Value
Now or never problem: exit or stay now, not allowed to revise decision.
b is decreasing in σ: concavity of flow benefits.
Now or never threshold Y
b = max
Y
2 (2
µ
rΦ
α
σ2
+ 1)
+n
− (n − 1)
, 0 .
r
β
β
r
b measures “pure option value".
Difference in thresholds Y − Y
b
Table : Option-inclusive versus now-or-never thresholds: Y − Y
µ
0.0
σ
0.2
0.3
0.00
10.00 − 10.00 = 0.00
12.85 − 6.80 = 6.05
13.58 − 2.80 = 10.78
0.0125
15.00 − 15.00 = 0.00
15.00 − 11.80 = 3.20
15.00 − 7.80 = 7.20
0.0250
20.00 − 20.00 = 0.00
18.57 − 16.80 = 1.77
17.32 − 12.80 = 4.52
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
16 / 40
Comparative Static of Threshold
problem
Normalization and homogeneity:
n α + r Φ µ σ2
Y =ϕ n,
,
,
β
r
r
Total cost of abandonment n α/r + Φ .
details
Objective h.o.d. 1 on : α , β , Φ.
Rates µ , r , σ 2 , α , β scale w/ units of time.
Keeping r , n fixed Y is increasing in
Alvarez, Dixit (UofC, Princeton )
nα+Φr
,
β
heterogeneity
and Y is increasing in µ.
Real Options Perspective on the Euro, Sept. 2013
17 / 40
Comparative Static of Threshold
problem
Normalization and homogeneity:
n α + r Φ µ σ2
Y =ϕ n,
,
,
β
r
r
Total cost of abandonment n α/r + Φ .
details
Objective h.o.d. 1 on : α , β , Φ.
Rates µ , r , σ 2 , α , β scale w/ units of time.
Keeping r , n fixed Y is increasing in
nα+Φr
,
β
heterogeneity
and Y is increasing in µ.
Well defined undiscounted ( r = 0 ) problem, as long as optimal τ is finite.
Taylor approximation around µ = r = 0 of Y :
Ȳ ≈ 2
Approx.
n+1
n−1
∂Y
∂σ
nα + r Φ
β
+ 16
n+1
(n + 3)(n − 1)
2 α
(n − 1)µ − r
β
σ2
T 0 ⇐⇒ r − (n − 1)µ T 0. We extended to any r , µ ≥ 0 .
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
17 / 40
Comparative Static of Threshold
n+1
Ȳ ≈ 2
n−1
nα + r Φ
β
and in general
n+1
+ 16
(n + 3)(n − 1)
2 α
(n − 1)µ − r
β
σ2
∂Y
T 0 ⇐⇒ r − (n − 1)µ T 0
∂σ
Result hold for any n ≥ 1 (properly interpreted).
Two impacts of a change in σ: option value vs concavity of flow benefits.
Extreme case: µ = 0, random walk and standard results.
Intuition: at the threshold an increase in volatility increases option value.
Extreme case: µ → ∞, so that each Xi becomes i.i.d.
Intuition: future distribution independent of current value,
thus higher volatility only decreases expected values.
other applications
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
18 / 40
Two measures of Option Value:
b . Use
Difference in thresholds: Y − Y
b /n)1/2
Difference of typical misalignment = (Y /n)1/2 − (Y
- units of Xi as if all countries have same misalignment (except sign).
1/2
or twice the typical one.
- largest misalignment in one country: Y n−1
n
b , per country:
Optimal value function evaluated at Y
b − V Y /n = V Y
b + Φ /n
Forgone gains = V Y
- losses, as a one time payment per country.
b .
- extra value gained if threshold Y is used instead of Y
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
19 / 40
Parameter Values
n = 5 regions of similar GDP size (Germany, France+Belgium, ....).
µ , σ 2 : real exchange rates for developed countries.
variance of year-to-year changes, σ ≈ 0.08
half-life real exchange rates about 3-5 years µ ≈ 0.1
α ≈ 0.02 annual GDP: international trade (IT) + transaction costs (TC)
IT: increases on trade due to union + welfare of increased trade (0.015)
TC: reduction of transaction cost of exchanging currencies (0.005)
β ≈ 2 per year: correcting a 10% misalignment increases GDP by 1%
As in sticky price models, consider deviations as a "wedge".
Use CES with elasticity of subs. 4 and tradable share 1/3.
Φ/n ≈ 0.2: cost of reintroducing new currency.
details
Multiyear drop GDP after balance of payment crises w/fixed exch. rates.
(expressed as fraction of one year GDP)
Alvarez, Dixit (UofC, Princeton )
back-to-quantifying
Real Options Perspective on the Euro, Sept. 2013
20 / 40
Numerical Evaluation of Option Value
Summary of numerical results
Normalized threshold Y : change in exchange rate at exit.
At benchmark values ≈ 25%, and decreasing in σ.
b ≈ 5% (increasing in σ)
Diff. of Y with normalized now-never threshold Y
Benefit of option value ≈ 4% annual GDP (once).
Very large σ or very small β threshold difference ≈ 20% .
Very large σ or very small β value function difference ≈ 10% GDP
Smaller difference w/changes in cost Φ on value functions & threshold.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
21 / 40
Numerical Evaluation of Option Value
α
β
r = 0. 05,
= 0. 0067,
Φ
βn
= 0. 067
µ
µ
µ
µ
!
"1
normalized threshold Y /n 2
0.32
= 0.00
= n r− 1 =0.0125
= 0. 06
= 0. 1
0.3
0.28
0.26
0.24
0.22
0.2
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
vol at i l i t y σ
Figure : Normalized optimal threshold as function of σ for selected µ.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
22 / 40
Numerical Evaluation of Option Value
Summary of numerical results
Normalized threshold Y : change in exchange rate at exit. X
At benchmark values ≈ 25%, and decreasing in σ. X
b ≈ 5% (increasing in σ)
Diff. of Y with normalized now-never threshold Y
Benefit of option value ≈ 4% annual GDP (once).
Very large σ or very small β threshold difference ≈ 20% .
Very large σ or very small β value function difference ≈ 10% GDP
Smaller difference w/changes in cost Φ on value functions & threshold.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
23 / 40
Numerical Evaluation of Option Value
r = 0. 05,
α
β
= 0. 01,
Φ
βn
= 0. 1, µ = 0. 1
Ȳ
b
Y
0.3
0.2
0.15
!
Y /n
"12
1
221
and Yb /n
0.25
0.1
0.05
0
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
vol at i l i t y σ
Figure : Normalized thresholds as function of σ .
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
24 / 40
Numerical Evaluation of Option Value
r = 0. 05,
α
β
= 0. 01,
Φ
βn
= 0. 1, µ = 0. 1
0.3
0.15
!
Y /n
"12
−
#
$1
! /n 2
Y
! ) + Φ ) /n
(V (Y
0.2
0.1
0.1
0.05
0
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0
0.105
vol at i l i t y σ
Figure : Two measures of the option value as a function of σ .
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
25 / 40
Numerical Evaluation of Option Value
Summary of numerical results
Normalized threshold Y : change in exchange rate at exit. X
At benchmark values ≈ 25%, and decreasing in σ. X
b ≈ 5% (increasing in σ)X
Diff. of Y with normalized now-never threshold Y
Benefit of option value ≈ 4% annual GDP (once). X
Very large σ or very small β threshold difference ≈ 20% .
Very large σ or very small β value function difference ≈ 10% GDP
Smaller difference w/changes in cost Φ on value functions & threshold.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
26 / 40
Numerical Evaluation of Option Value
Φ
αn
# $12
! "1
normalzed thresholds Y /n 2 and Y! /n
r = 0. 05, µ = 0. 1,
0.45
= 10, σ = 0. 08
Ȳ
!
Y
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
β : s e ns i t i v i t y t o x 2 /2
Figure : Normalized thresholds as function of β .
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
27 / 40
Numerical Evaluation of Option Value
r = 0. 05, µ = 0. 1,
0.2
!
Y /n
"12
−
Φ
αn
= 10, σ = 0. 08
0.2
#
$1
! /n 2
Y
! ) + Φ ) /n
(V (Y
0.1
0
0.1
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
0
2
β : s e ns i t i v i t y t o x /2
Figure : Two measures of the option value as a function of β .
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
28 / 40
Numerical Evaluation of Option Value
Summary of numerical results
Normalized threshold Y : change in exchange rate at exit. X
At benchmark values ≈ 25%, and decreasing in σ. X
b ≈ 5% (increasing in σ)X
Diff. of Y with normalized now-never threshold Y
Benefit of option value ≈ 4% annual GDP (once). X
Very large σ or very small β threshold difference ≈ 20% . X
Very large σ or very small β value function difference ≈ 10% GDP. X
Smaller difference w/changes in cost Φ on value functions & threshold.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
29 / 40
Numerical Evaluation of Option Value
# $12
! "1
normalzed thresholds Y /n 2 and Y! /n
r = 0. 05,
α
β
= 0. 01, µ = 0. 1, σ = 0. 08
Ȳ
0.35
!
Y
0.3
0.25
0.2
0.15
0.1
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
F i x e d c os t p e r c ount r y Φ /n
Figure : Normalized optimal threshold as function of Φ.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
30 / 40
Numerical Evaluation of Option Value
r = 0. 05,
α
β
= 0. 01,
Φ
βn
= 0. 25, σ = 0. 08
0.15
0.08
0.1
0.06
!
"12
−
#
$1
! /n 2
Y
! ) + Φ ) /n
(V (Y
0.05
0
Y /n
0
0.5
1
1.5
2
0.04
0.02
2.5
F i x e d c os t Φ
Figure : Two measures of the option value as a function of Φ .
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
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Numerical Evaluation of Option Value
Summary of numerical results
Normalized threshold Y : change in exchange rate at exit. X
At benchmark values ≈ 25%, and decreasing in σ. X
b ≈ 5% (increasing in σ)X
Diff. of Y with normalized now-never threshold Y
Benefit of option value ≈ 4% annual GDP (once). X
Very large σ or very small β threshold difference ≈ 20% . X
Very large σ or very small β value function difference ≈ 10% GDP. X
Smaller difference w/changes in cost Φ on value functions & threshold. X
Φ vs n α
r
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Real Options Perspective on the Euro, Sept. 2013
32 / 40
Numerical Evaluation of Option Value
r = 0. 05,
α
β
= 0. 01,
100
Expected time to hit Y at Y! (years)
µ
µ
µ
µ
90
Φ
βn
=
=
=
=
= 0. 1
0
r /( n − 1) = 0. 013
0. 056
0. 1
80
70
60
50
40
30
20
10
0
0.06
0.07
0.08
0.09
0.1
0.11
0.12
vol at i l i t y σ
b.
Figure : Expected Time until hitting Y from Y
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Real Options Perspective on the Euro, Sept. 2013
33 / 40
Numerical Evaluation of Option Value
r = 0. 05,
α
β
= 0. 01,
Φ
βn
= 0. 1
√σ
2µ
= 0. 18
Expected time to hit Y at Y! (years)
60
58
56
54
52
50
48
46
44
42
40
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
vol at i l i t y σ
b . Constant unconditional variance.
Figure : Expected Time until hitting Y from Y
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
34 / 40
Individual Country Exit
Individual Country Exit
φ smallest fixed cost that deters one country exit under collective policy.
Compute value of individual country assuming no transfers and optimal
union policy given by Y and corresponding τ .
State of problem (y , Y ) with original dynamics
for dY with BM dW and
r
n−1
n−1
2
dy = σ 2
− 2 µ y dt + 2 σ y
dWy with y = (Xi − Z ) ,
n
n
1/2
y n
2
dt .
E [ dy dY ] = 4 σ y dt and E [ dWy dW ] =
Y n−1
value function (nothing to optimize!) on 0 ≤ Y ≤ Y , 0 ≤ y ≤ n−1
n Y :
Z τ
Φ
α − 12 β y e−rt dt − e−r τ
v (Y , y ) = E
| Y (0) = Y , y (0) = y
n
0
φ
≡ min v (Y , y )
Y ,y
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
35 / 40
Individual Country Exit
Individual Country Exit
φ smallest fixed cost that deters one country exit under collective policy.
Compute value of individual country assuming no transfers and optimal
union policy given by Y and corresponding τ .
State of problem (y , Y ) with original dynamics
for dY with BM dW and
r
n−1
n−1
2
dy = σ 2
− 2 µ y dt + 2 σ y
dWy with y = (Xi − Z ) ,
n
n
1/2
y n
2
dt .
E [ dy dY ] = 4 σ y dt and E [ dWy dW ] =
Y n−1
value function (nothing to optimize!) on 0 ≤ Y ≤ Y , 0 ≤ y ≤ n−1
n Y :
Z τ
Φ
α − 12 β y e−rt dt − e−r τ
v (Y , y ) = E
| Y (0) = Y , y (0) = y
n
0
Φ
φ ≡ min v (Y , y ) which we show >
for n > 2
Y ,y
n
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
35 / 40
Individual Country Exit
r = 0. 05,
α
β
= 0. 01,
Φ
βn
= 0. 1 σ = 0. 08 µ = 0. 1
0.1
V /n
v
!
"
V (Y )/n and v Y , Y ( n−1
n )
0
ï0.1
ï0.2
− Φ /n
Ȳ
ï0.3
ï0.4
ï0.5
ï0.6
−φ
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Y
Figure : One country exit decision: V /n vs v and Φ/n vs. φ
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
36 / 40
Individual Country Exit
Table : Minimum fixed cost to deter individual country’s exit: φ
Φ
n
φ
0.10
0.28
0.15
0.39
0.20
0.52
0.25
0.65
0.30
0.80
β
φ
1.0
1.09
1.5
0.75
2
0.52
2.5
0.40
3.0
0.34
σ
φ
0.06
1.00
0.07
0.73
0.08
0.52
0.09
0.40
0.10
0.33
µ
φ
0.00
0.31
0.06
0.38
0.10
0.52
0.12
0.65
0.14
0.80
For α = 0.02, β = 2, µ = 0.1, σ = 0.08, and
Φ
n
= 0.2 .
Middle column has benchmark values, for which φ ≈ 52% annual GDP..
“Extra" cost to stop deviant ≈ 32% annual GDP.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
back to things...
37 / 40
Heterogeneity: case of n = 2.
The case of two (n = 2) heterogenous countries
Are countries symmetric?
Allow σ12 6= σ 2 , β1 6= β2 and w.l.o.g. α1 6= α2 .
1-dimensional problem since X1 − Z = −(X2 − Z ), thus Y ≡ 21 (X1 − X2 )2 .
Threshold is the same as if:
α
=
β
=
+ 12 α2 , σ 2 = 12 σ12 + 12 σ22 , and
β1 β2
2
≡ harmonic mean ≤ 12 β1 + 12 β2
β1 + β2
1
2 α1
Optimal union-wide policy Z more potent, respond more to higher β
(take extreme case β1 = 0, then Y = ∞).
Y : increases with dispersion on β’s and depend on average of (σ 2 , β, α) .
b , so no change on option value.
Effect on Y is the same as on Y
back CS
back things left out
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
38 / 40
Dynamics
Private Sector Model (or lack of thereof)
No (serious) model of private sector, just AR(1)’s for misalignment.
No role for expectations:
As Y reaches Y prob. of jump (on exchange rate?) goes to one.
Large capital flows as prob. of jump goes to one.
Likely policy changes around that time -as in Krugman’s B. of P. crises.
Conjecture: exit will be even sooner.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
39 / 40
Dynamics
Private Sector Model (or lack of thereof)
No (serious) model of private sector, just AR(1)’s for misalignment.
No role for expectations:
As Y reaches Y prob. of jump (on exchange rate?) goes to one.
Large capital flows as prob. of jump goes to one.
Likely policy changes around that time -as in Krugman’s B. of P. crises.
Conjecture: exit will be even sooner.
On the other hand, actual B. of P. crises have been widely anticipated.
Fixed cost Φ, in a reduced form, may be able to capture that.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
39 / 40
Dynamics
Private Sector Model (or lack of thereof)
No (serious) model of private sector, just AR(1)’s for misalignment.
No role for expectations:
As Y reaches Y prob. of jump (on exchange rate?) goes to one.
Large capital flows as prob. of jump goes to one.
Likely policy changes around that time -as in Krugman’s B. of P. crises.
Conjecture: exit will be even sooner.
On the other hand, actual B. of P. crises have been widely anticipated.
Fixed cost Φ, in a reduced form, may be able to capture that.
"Model" completely ignores dynamics of sovereign debt.
Alternatively: “powerful" Fiscal Union as in Werning and Fahri
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
fiscal unions
39 / 40
Conclusions
Conclusions
Surprising theoretical results on option value:
Optimal threshold decreasing in volatility σ.
Yet option value, properly defined, increasing in volatility.
other applications
Small but not negligible option value.
b instead of optimal Y :
If break-up occurs at now or never Y
One time loss ≈ 4% GDP.
Tolerate cumulative inflation mis-alignment ≈ 5 − 10% smaller.
Medium-large extra cost can deter individual’s country exit.
Requires about one time cost ≈ 30% of yearly GDP.
Alvarez, Dixit (UofC, Princeton )
Real Options Perspective on the Euro, Sept. 2013
40 / 40
APPENDICES
Figure : Are countries symmetric?
back
back things...
APPENDICES
Total Cost of Break Up
Integrate (path by path) terms involving α and Φ:
Z τ
h α
i
α
e−rt n α dt − e−r τ Φ = n − e−r τ n + Φ
r
r
0
Present value of gains unrelated to monetary policy
+
Fixed cost of abandonment Φ .
back normalization
nα
r
≡
Pn
i=1
r
ui (0)
APPENDICES
Total Cost of Break Up
Integrate (path by path) terms involving α and Φ:
Z τ
h α
i
α
e−rt n α dt − e−r τ Φ = n − e−r τ n + Φ
r
r
0
Present value of gains unrelated to monetary policy
nα
r
≡
Pn
i=1
ui (0)
r
+
Fixed cost of abandonment Φ .
back normalization
Parameter values n = 5 , α = 0.02 , r = 0.05 ,
Total Cost =
nα
r
+ Φ = 2 + 1 = 3 , so
2
3
Φ
n
=
1
5
from loss of flow benefits.
back to summary
Total Cost per country =
back parameters
α
r
+
Φ
n
= 0.2 :
= 0.60 yearly GDP.
APPENDICES
Werning and Fahri’s “Fiscal Union"
Alternative derivation of flow benefit
P
i
ui (xi ) for collective.
Based on static Obstfeld and Rogoff’s model in “Redux".
Tradeable : endowments each country. Flexible prices.
Non-tradeable: CRTS labor only. Nominal prices set in advance.
Policy instrument for Fiscal-Monetary Union:
common monetary policy.
country (and state) specific: labor, portfolio taxes, and lump-sum taxes.
Shocks to tradeable’s endowment and non-tradeable’s productivity.
Quadratic approximation: labor wedge in each country.
back to private sector
back to fiscal union
APPENDICES
Now-or-never problem
Define the present discounted value of staying forever in the union as:
"Z
#
n
∞ X
−rt
sup E
ui (Xi (t) − Z (t)) e dt | Xi (0) = Xi , i = 1, ..., n
{Z }
0
i=1
"Z
∞
≡ VE (Y ) = E
U (Y (t)) e
−rt
n
X
dt | Y (0) = Y ≡
(Xi − Z )2
0
=
nα−
(n − 1) β σ
4µ
i=1
2
1
−
r
1
2
β
Y−
(n − 1) σ
2µ
2
1
.
2µ + r
Note that VE is linear on Y , which is sum of squares of the Xi0 s.
Note that σ 2 has a level effect on VE , due to concavity of U.
b ≥ 0 is solution VE Y
b + Φ = 0, or zero.
Now-or-never threshold Y
back to option value
#
APPENDICES
Related literature on “Option Value"
Threshold for inaction: random walk shock + fixed cost of action
Comparative static of threshold w.r.t. volatility
Exit from industry (Dixit, Hopenhayn)
Labor reallocation (Caballero-Bertola)
Lumpy Physical Investment (Abel-Eberly, Khan-Thomas, Bloom,...)
Durable purchases (Grossman-Laroque, Eberly, ...)
Price Adjustment (Caplin-Lehay, Golosov-Lucas, Bavra)
back to intro
back to CS
back to conclusions
APPENDICES
Eurozone chooses stopping time τ (t) & monetary policy Z (t) to maximize
back
"Z
E
0
τ
n X
i=1
#
β
2
α − (Xi (t) − Z (t)) e−rt dt − e−r τ Φ | Xi (0) = Xi , i = 1, ..., n
2
Countries i = 1, 2, . . . n “nominal" shocks Xi :
dXi = −µi Xi dt + σi dWi + σc dWc ,
standard BMs: Wi indepedent and Wc common
APPENDICES
Eurozone chooses stopping time τ (t) & monetary policy Z (t) to maximize
back
"Z
E
0
τ
n X
i=1
#
β
2
α − (Xi (t) − Z (t)) e−rt dt − e−r τ Φ | Xi (0) = Xi , i = 1, ..., n
2
Countries i = 1, 2, . . . n “nominal" shocks Xi :
dXi = −µi Xi dt + σi dWi + σc dWc ,
standard BMs: Wi indepedent and Wc common
Eurozone common monetary policy Z .
Misalignment of country i, “real exchange deviation" given by:
xi ≡ Xi − Z
xi = 0 eliminate misalignment, in which case only gains from union.
Value of not-being in the union (with optimal policy) normalized to 0.
Fixed up-front cost of breaking Eurozone Φ > 0.