A Real Option Perspective on the Future of the Euro - University of

A Real Option Perspective on the Future of the Euro

Fernando Alvarez Avinash Dixit

University of Chicago 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.

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?



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 !



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 = −µXi 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. even more wonkish



Abandoning the Euro as investment project

Modeling euro as “union" deciding when to break it up complete or not atall.

Optimal abandonment when combination of misalignment is large:

Abandon first time that∑n

i=1 (xi )2 = 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∑n

i=1 (xi )2 = 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?


Quantifying the option value: depressing results

Borrow parameter values from empirical literature. parameters

Recall: optimal abandonment at Y > Y now-or-never abandonment.

Abandonment at either Y or Y implies correction in each country(depreciation or appreciations).

At optimal Y : average correction ≈ 25%.

At now-or-never Y : average correction ≈ 20%.

If maximally concentrated (Spain?), multiply each correction by 2.

Welfare loss of 4 % of GDP (once) if abandon Euro at Y now-or-neverinstead of following optimal policy.

Expected time, starting at Y to reach Y : more than 10 years! details


Many things left out ...

Model is symmetric. Are countries is Europe symmetric? symmetry .May not be important for option value. case of two countries

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 ondebt dynamics. Reinterpretation of Xi and recalibration.

Anticipatory dynamics as on Krugman’s style balanced of payment crises.


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.


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. other applications


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.


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

sup{τ,Z}

E

[ ∫ τ

0

n∑i=1

ui (Xi (t)− Z (t)) e−rt dt − e−r τ Φ | Xi (0) = Xi , i = 1, ...,n

]


Eurozone Problem, simplifications

Maximization over control Z ∗ static, depends only on current X1, ...,Xn

Z ∗ = arg maxz

n∑i=1

ui (Xi − z)

Quadratic utility: ui (x) = αi − 12βi x2 with positive αi , βi .

Almost all analysis for symmetric countries: βi ≡ β, µi ≡ µ, σi = σ

Quadratic utility w/same β =⇒ Z ∗ =∑n

i=11n Xi and

∑ni=1 xi = 0

Define Y =∑n

i=1 (Xi − Z ∗)2 . Using Itô’s Lemma and simplifying,

dY =[

(n − 1)σ2 − 2µY]

dt + 2σ 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.



Define flow utility of union:

U(Y ) ≡n∑

i=1

u(Xi − Z ∗) = nα− 12 β Y ,

Choose stopping time τ to solve the problem

V (Y ) = supτ

E[ ∫ τ

0U(Y (t) ) e−rt dt − e−r τ Φ

∣∣∣∣ Y (0) = Y]

where dY (t) =[

(n − 1)σ2 − 2µY (t)]

dt + 2σ Y (t)1/2 dW CS exit

Optimum policy has one dimensional threshold of abandonment Y .

In the region of inaction Y ∈ [0,Y ),

2σ2 Y V ′′(Y ) + [ (n − 1)σ2 − 2µY ] V ′(Y )− r V (Y ) + [ nα− 12 β Y ] = 0 .

At the threshold, value matching and smooth pasting conditions

V (Y ) = −Φ, V ′(Y ) = 0 .



0 2 4 6 8 10 12 14 16 18 20−120

−100

−80

−60

−40

−20

0

20

r = 0.05, αβ =0.5, Φ

β n =10 σ =0.3 µ = 0.025

Y =∑n

i=1 x2i

Value

func

tion

V(Y

)

Inact ion Region Abandon

Y

Fixed Cost −Φ

Figure : Graph of V (Y ) for illustrative parameter values.Alvarez, Dixit (UofC, Princeton ) Real Options Perspective on the Euro, Sept. 2013 14 / 40

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 > 0.

∂Y∂σ switches to negative as µ increases. Contrary to conventional result.


Option Value

Now or never problem: exit or stay now, not allowed to revise decision.

Now or never threshold Y is decreasing in σ: concavity of flow benefits.

Y = max{

2 (2µ

r+ 1)

[rΦ

β+ n

α

β

]− (n − 1)

σ2

r, 0}.

Difference in thresholds Y − Y measures “pure option value".

Table : Option-inclusive versus now-or-never thresholds: Y − Y

σµ 0.0 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


Comparative Static of Threshold

Normalization and homogeneity: problem

Y = ϕ

(n ,

nα + r Φ

β,µ

r,σ2

r

)

Total cost of abandonment nα/r + Φ . details

Objective h.o.d. 1 on : α , β , Φ.

Rates µ , r , σ2 , α , β scale w/ units of time. heterogeneity

Keeping r ,n fixed Y is increasing in nα+ Φ rβ , 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 :

Y ≈ 2n + 1n − 1

(nα + r Φ

β

)+ 16

n + 1(n + 3)(n − 1)

(α

β

)2 [(n − 1)µ− r

σ2

]

Approx. ∂Y∂σ T 0 ⇐⇒ r − (n − 1)µ T 0. We extended to any r , µ ≥ 0 .


Comparative Static of Threshold

Y ≈ 2n + 1n − 1

(nα + r Φ

β

)+ 16

n + 1(n + 3)(n − 1)

(α

β

)2 [(n − 1)µ− r

σ2

]and in general

∂Y∂σT 0 ⇐⇒ r − (n − 1)µ T 0

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


Two measures of Option Value:

Difference in thresholds: Y − Y . Use

Difference of typical misalignment = (Y/n)1/2 − (Y/n)1/2

- units of Xi as if all countries have same misalignment (except sign).

- largest misalignment in one country:(Y n−1

n

)1/2or twice the typical one.

Optimal value function evaluated at Y , per country:

Forgone gains =(

V(

Y)− V

(Y))/n =

(V(

Y)

+ Φ)/n

- losses, as a one time payment per country.

- extra value gained if threshold Y is used instead of Y .


Parameter Values

n = 5 regions of similar GDP size (Germany, France+Belgium, ....).

µ , σ2: 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) back-to-quantifying


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.

Diff. of Y with normalized now-never threshold Y ≈ 5% (increasing in σ)

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.



0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10.2

0.22

0.24

0.26

0.28

0.3

0.32

r = 0.05, αβ =0.0067, Φ

β n =0.067

volat il ity σ

norm

alized

thre

shold(

Y/n

)1 2

µ = 0.00µ = r

n−1=0.0125µ = 0.06µ = 0.1

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



Normalized threshold Y : change in exchange rate at exit. X

At benchmark values ≈ 25%, and decreasing in σ. X

Diff. of Y with normalized now-never threshold Y ≈ 5% (increasing in σ)

Benefit of option value ≈ 4% annual GDP (once).






0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10

0.05

0.1

0.15

0.2

0.25

0.3

r = 0.05, αβ=0.01, Φ

β n =0.1, µ =0.1

volat il ity σ

(

Y/n

)1 2an

d(

Y/n

)1 2

YY

Figure : Normalized thresholds as function of σ .Alvarez, Dixit (UofC, Princeton ) Real Options Perspective on the Euro, Sept. 2013 24 / 40


0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.1050

0.1

0.2

0.3r = 0.05, α

β =0.01, Φβ n =0.1, µ =0.1

volat il ity σ

0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.1050

0.05

0.1

0.15

(

Y /n)12 −

(

Y /n)

12

(V ( Y ) + Φ )/n

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





Diff. of Y with normalized now-never threshold Y ≈ 5% (increasing in σ)X

Benefit of option value ≈ 4% annual GDP (once). X






1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

r = 0.05, µ =0.1, Φαn =10, σ =0.08

β : sensit ivity to x 2/2

norm

alze

dth

resh

olds

(

Y/n

)1 2an

d(

Y/n

)1 2

Y

Y

Figure : Normalized thresholds as function of β .Alvarez, Dixit (UofC, Princeton ) Real Options Perspective on the Euro, Sept. 2013 27 / 40


1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.1

0.2r = 0.05, µ =0.1, Φ

αn =10, σ =0.08

β : sensit ivity to x 2/2

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

0.1

0.2

(

Y /n)12 −

(

Y /n)

12

(V ( Y ) + Φ )/n

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







Very large σ or very small β threshold difference ≈ 20% . X

Very large σ or very small β value function difference ≈ 10% GDP. X




0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.1

0.15

0.2

0.25

0.3

0.35

r = 0.05, αβ =0.01, µ =0.1, σ =0.08

Fixed cost per country Φ/n

norm

alze

dth

resh

olds

(

Y/n

)1 2an

d(

Y/n

)1 2

Y

Y

Figure : Normalized optimal threshold as function of Φ.Alvarez, Dixit (UofC, Princeton ) Real Options Perspective on the Euro, Sept. 2013 30 / 40


0 0.5 1 1.5 2 2.50

0.05

0.1

0.15r = 0.05, α

β =0.01, Φβ n =0.25, σ =0.08

Fixed cost Φ

0 0.5 1 1.5 2 2.50.02

0.04

0.06

0.08

(

Y /n)12 −

(

Y /n)

12

(V ( Y ) + Φ )/n

Figure : Two measures of the option value as a function of Φ .Alvarez, Dixit (UofC, Princeton ) Real Options Perspective on the Euro, Sept. 2013 31 / 40







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. XΦ vs n α

r



0.06 0.07 0.08 0.09 0.1 0.11 0.120

10

20

30

40

50

60

70

80

90

100r = 0.05, α

β=0.01, Φ

β n =0.1

volat il ity σ

Expe

cted

timeto

hitY

atY

(yea

rs)

µ = 0µ = r/(n − 1) =0.013µ = 0.056µ = 0.1

Figure : Expected Time until hitting Y from Y .Alvarez, Dixit (UofC, Princeton ) Real Options Perspective on the Euro, Sept. 2013 33 / 40


0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.1240

42

44

46

48

50

52

54

56

58

60r = 0.05, α

β=0.01, Φ

β n =0.1 σ√2µ

= 0.18

volat il ity σ

Expe

cted

timeto

hitY

atY

(yea

rs)

Figure : Expected Time until hitting Y from Y . Constant unconditional variance.Alvarez, Dixit (UofC, Princeton ) Real Options Perspective on the Euro, Sept. 2013 34 / 40

Individual Country Exit


φ smallest fixed cost that deters one country exit under collective policy.

Compute value of individual country assuming no transfers and optimalunion policy given by Y and corresponding τ .

State of problem (y ,Y ) with original dynamics for dY with BM dW and

dy =

[σ2 n − 1

n− 2µ y

]dt + 2σ

√y

n − 1n

dWy with y = (Xi − Z )2,

E [ dy dY ] = 4 σ2 y dt and E [ dWy dW ] =

[yY

nn − 1

]1/2

dt .

value function (nothing to optimize!) on 0 ≤ Y ≤ Y , 0 ≤ y ≤ n−1n Y :

v(Y , y) = E[ ∫ τ

0

(α− 1

2 β y)

e−rt dt − e−r τ Φ

n|Y (0) = Y , y(0) = y

]φ ≡ min

Y ,yv(Y , y)

which we show >Φ

nfor n > 2




φ smallest fixed cost that deters one country exit under collective policy.

Compute value of individual country assuming no transfers and optimalunion policy given by Y and corresponding τ .

State of problem (y ,Y ) with original dynamics for dY with BM dW and

dy =

[σ2 n − 1

n− 2µ y

]dt + 2σ

√y

n − 1n

dWy with y = (Xi − Z )2,

E [ dy dY ] = 4 σ2 y dt and E [ dWy dW ] =

[yY

nn − 1

]1/2

dt .

value function (nothing to optimize!) on 0 ≤ Y ≤ Y , 0 ≤ y ≤ n−1n Y :

v(Y , y) = E[ ∫ τ

0

(α− 1

2 β y)

e−rt dt − e−r τ Φ

n|Y (0) = Y , y(0) = y

]φ ≡ min

Y ,yv(Y , y) which we show >

Φ

nfor n > 2



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.6

0.5

0.4

0.3

0.2

0.1

0

0.1r = 0.05, α

β =0.01, Φβ n =0.1 σ =0.08 µ = 0.1

Y

V(Y

)/n

andv(

Y,Y

(n−1

n))

Y−Φ/n

−φ

V /nv

Figure : One country exit decision: V/n vs v and Φ/n vs. φAlvarez, Dixit (UofC, Princeton ) Real Options Perspective on the Euro, Sept. 2013 36 / 40


Table : Minimum fixed cost to deter individual country’s exit: φ

Φn 0.10 0.15 0.20 0.25 0.30φ 0.28 0.39 0.52 0.65 0.80

β 1.0 1.5 2 2.5 3.0φ 1.09 0.75 0.52 0.40 0.34

σ 0.06 0.07 0.08 0.09 0.10φ 1.00 0.73 0.52 0.40 0.33

µ 0.00 0.06 0.10 0.12 0.14φ 0.31 0.38 0.52 0.65 0.80

For α = 0.02, β = 2, µ = 0.1, σ = 0.08, and Φn = 0.2 .

Middle column has benchmark values, for which φ ≈ 52% annual GDP..

“Extra" cost to stop deviant ≈ 32% annual GDP. back to things...


Heterogeneity: case of n = 2.

The case of two (n = 2) heterogenous countries

Are countries symmetric?

Allow σ21 6= σ2 , β1 6= β2 and w.l.o.g. α1 6= α2.

1-dimensional problem since X1 − Z = −(X2 − Z ), thus Y ≡ 12 (X1 − X2)2.

Threshold is the same as if:

α = 12α1 + 1

2α2 , σ2 = 1

2σ21 + 1

2σ22 , and

β = 2β1 β2

β1 + β2≡ harmonic mean ≤ 1

2β1 + 12β2

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 (σ2, β, α) .

Effect on Y is the same as on Y , so no change on option value.

back CS back things left out


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 fiscal unions


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.

If break-up occurs at now or never Y instead of optimal 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.


APPENDICES

Figure : Are countries symmetric? back back things...

APPENDICES

Total Cost of Break Up

Integrate (path by path) terms involving α and Φ:∫ τ

0e−rtnα dt − e−rτΦ = n

α

r− e−rτ

[nα

r+ Φ

]Present value of gains unrelated to monetary policy nα

r ≡∑n

i=1 ui (0)r

+

Fixed cost of abandonment Φ . back normalization

Parameter values n = 5 , α = 0.02 , r = 0.05 , Φn = 1

5 = 0.2 :

Total Cost = nαr + Φ = 2 + 1 = 3 , so 2

3 from loss of flow benefits.back to summary

Total Cost per country = αr + Φ

n = 0.60 yearly GDP.back parameters

APPENDICES

Werning and Fahri’s “Fiscal Union"

Alternative derivation of flow benefit∑

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:

sup{Z}

E

[∫ ∞0

n∑i=1

ui (Xi (t)− Z (t)) e−rt dt | Xi (0) = Xi , i = 1, ...,n

]

≡ VE (Y ) = E

[∫ ∞0

U (Y (t)) e−rt dt | Y (0) = Y ≡n∑

i=1

(Xi − Z )2

]

=

[nα− (n − 1)β σ2

4µ

]1r− 1

2 β

[Y − (n − 1)σ2

2µ

]1

2µ+ r.

Note that VE is linear on Y , which is sum of squares of the X ′i s.

Note that σ2 has a level effect on VE , due to concavity of U.

Now-or-never threshold Y ≥ 0 is solution VE

(Y)

+ Φ = 0, or zero.

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 maximizeback

E

[∫ τ

0

n∑i=1

(α− β

2(Xi (t)− Z (t))2

)e−rtdt − e−r τΦ |Xi (0) = Xi , i = 1, ...,n

]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.

A Real Option Perspective on the Future of the Euro - University of

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