stochastic vol presentation

8/14/2019 Stochastic Vol Presentation

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Literature

Research 101

Model

Solution for the Mature Firm (Firm with No Growth Option)

Debt Value

The Q Theory of Investment with Stochastic

Volatility

Franois GourioMichael Michaux

Finance DepartmentThe Wharton School

University of Pennsylvania

5th December 2008

Franois Gourio and Michael Michaux The Q Theory of Investment with Stochastic Volatility
http://goforward/http://find/http://goback/


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Literature

Research 101

Model


Debt Value

Empirical Performance of the Q-theory

Philippon (Forth. QJE)


Investment regressions stylized facts for large firms (top quartile offirms sorted by size of the capital stock in 1981) are taken fromEberly, Rebelo, and Vincent (2008).



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Literature

Research 101

Model


Debt Value



Bond Q

Philippon (Forth. QJE) uses a Hayashi model with costless default(MM world) and obtains a mapping from bond yields to Tobins Q.

Using a simple setup, he obtains the following characterization.

Aggregate and firm level investment regressions using numerically

constructed bond Q show much improved fit.Franois Gourio and Michael Michaux The Q Theory of Investment with Stochastic Volatility


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Literature

Research 101

Model


Debt Value



Aggregate Investment Regressions



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Literature

Research 101

Model


Debt Value



Usual Measure of Q and Bond Markets Q


Lit t


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Literature

Research 101

Model


Debt Value



Usual Measure of Q and Investment Rate (levels)


Literature


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Literature

Research 101

Model


Debt Value



Bond Markets Q and Investment Rate (levels)


Literature


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Literature

Research 101

Model


Debt Value



Results: Aggregate Data


Literature


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Literature

Research 101

Model


Debt Value



Firm Level Investment Regressions


Literature


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Literature

Research 101

Model


Debt Value



Results: Firm Level Data


Literature Step 1 - Contrasting the existing facts


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Research 101

Model


Debt Value

p g g

Step 2 - Framing the problem in terms of the literature

Step 3 - Proposing a different explanation

Why Do Standard Models Fails?

How is a Model with Stochastic Volatility Different?

Step 1 - Contrasting the existing results

The correlation between I/K and Q is empirically weak.

In the standard corporate finance/investment models1, thecorrelation between I/K and Q is strong.

This gap between theory and the empirics begs the question why.

1such as Hayashi (ECO 1982), Gomes (AER 2001), Cooley and Quadrini (AER

2001), and Hennessy and Whited (JF 2005)Franois Gourio and Michael Michaux The Q Theory of Investment with Stochastic Volatility



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Research 101

Model


Debt Value

p g g






Some papers have argued that the data is at fault. The mostnotable paper is from Erickson and Whited (JPE 2000), wherethey appeal to mismeasurements of Q using GMM estimation.

Some papers have argued that financing frictions areresponsible. For example, Hennessy, Levy, and Whited (JFE2007) develop a Q theory of investment under financingconstraints.

Eberly et al. (WP 2008) develop a Generalized Hayashi modelwith Regime Switching and replicates the investmentregressions stylized facts when adding measurement noise to Q.




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Research 101

Model


Debt Value






This research proposes an alternative theoretical explanation of theweak empirical correlation between I/K and Q.

The empirical stylized facts are assumed to be true. The data(especially Q) is not mismeasured.

The bare bones of the model will be Hayashi 1982. The model inthis paper will departs from Hayashi on the real side only in 2respects: (a) firms exhibit DRS, and (ii) experience stochastic

volatility in their productivity shock.

Intuition: The addition of stochastic volatility essentially weakens thecorrelation between investment and Q.




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Research 101

Model


Debt Value





Standard Models

The standard models feature(i) a productivity shock z,

(ii) capital accumulation k (CRS/DRS, with or without adjustment cost),

(iii) debt b (priced or with a collateral constraint/debt capacity),

(iv) equity issuance (from costless to infinite cost).

These models look something like that,

V(k, b, c, z) = maxk,b

d g(d) + E

max(0,V(k, b, c, z))

s.t. d = (1 )ezk + (1 )k k + b b(1 + c) + (k + cb),

log(z) = log(z) + .

The discount price satisfies the usual Euler equation,

b = E

1

1 b(1 + (1 )c)1{V(s)>0} + k

(1 1{V(s)>0})

!

.


Literature

R h 101

Step 1 - Contrasting the existing facts

St 2 F i th bl i t f th lit t


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Research 101

Model


Debt Value





Intuition

Assume a good shock hits, i.e. z= zHIGH.

What happens?

1 MPK increases, which implies that capital investment increases,i.e. I/K .

2 Equity value V increases (as Vz > 0), thus making Q increase,i.e. Q.

3 This effect yields a POSITIVE correlation between I/K and Q.


Literature

Research 101


Step 2 Framing the problem in terms of the literature


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Research 101

Model


Debt Value





Model with Stochastic Volatility

Let b = (1 + (1 )c)b.

V(k, b, z, ) = maxk,b

d g(d) + Eh

max(0,V(k, b, z, ))i

s.t. d = (1 )(ezk f) + (1 (1 ))k k +b

1 + (1 )c b,

log(z) = log(z) + , is Markov.

The discount price satisfies the usual Euler equation,

b =E

h

1

1b1

{V(s

)>0} +k

(11{V(s

)>0})

i

1 + 1

E

1{V(s)>0}

.

The implied coupon rate is then,

c =1

1

b

b 1

.


Literature

Research 101


Step 2 Framing the problem in terms of the literature


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Research 101

Model


Debt Value





Intuition

Assume a bad volatility shock hits, i.e. = HIGH.

What happens?

Effect on equity:1 The continuation value max(0, V(s)) increases, due to the call

option feature of equity.2 This will increase V, and thus Q.

Effect on investment:1 The pricing schedule c() increases, due to the put option feature

of debt.

2 This will (i) decrease debt b and/or (ii) increase yields c. This can

potentially reduce capital investment I/K

This effect yields a NEGATIVE correlation between I/K and Q.


LiteratureResearch 101

Model with Growth Options and Stochastic Volatility


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Research 101

Model


Debt Value

System of PDEs

Region 1

Region 2


Let the productivity (or demand function) be given by,

dXt

Xt= dt + jdwt, j = L,H.

The Bellman equation is given by,

V(X, j; cij) = (1)(XtKi cij)dt+(1+rdt)

1E[V(Xt+dXt, j+jj dt(jj); cij)].

Apply Itos Lemma,

rV =1

2VXX

2j X

2t + VXXt + jj [V(X, j ; cij ) V(X, j; cij)] + (1 )(XtK

i cij).




S f


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Research 101

Model


Debt Value

System of PDEs

Region 1

Region 2

System of PDEs

The system of PDEs to solve is,

0 =1

22HX

2t V

HXX + XtV

HX rV

H + H(VL VH) + (1 )(XtK

i ciH),

0 =1

2

2L X2t V

LXX + XtV

LX rV

L + L(VH VL) + (1 )(XtK

i ciL).

The boundary conditions are given by,

(Default) V(XDj , j) = 0, j = L,H

(Default) VX(XD

j , j) = 0, j = L,H

(Boundedness) limX

V(X, j)X

XDH

. Thus we need topartition X into regions over which the system of PDEs will be solved.




S t f PDE


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Model


Debt Value

System of PDEs

Region 1

Region 2

System of PDEs

The system of PDEs can be partitioned over 2 regions:

On the region XDL X,

0 = 122HX2t VHXX + XtVHX rVH + H(VL VH) + (1 )(XtKi ciH),

0 =1

22L X

2t V

LXX + XtV

LX rV

L + L(VH VL) + (1 )(XtK

i ciL).

On the region XDH X XD

L,

0 = 122HX

2t V

HXX + XtV

HX (r + H)V

H + (1 )(XtKi ciH).




System of PDEs


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Model


Debt Value

System of PDE s

Region 1

Region 2

Region 1: [XDL ,+)

The system of homogeneous PDEs to solve is,

0 =1

22HX

2t V

HXX + XtV

HX rV

H + H(VL VH),

0 =1

22L X

2t V

LXX + XtV

LX rV

L + L(VH VL).

Conjecture the solution to be,

VH = AHX , and VL = ALX

.

The system of PDEs translates into a system of algebraic equations,

0 =

1

22H( 1) + r H

AHX + HALX

,

0 =

1

22L( 1) + r L

ALX + LAHX

.




System of PDEs


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Model


Debt Value

System of PDE s

Region 1

Region 2

Region 1: [XDL ,+)

These equations must hold for all X, thus we have a system MA = 0, whereA (AH,AL)

, and,

M

4

122

H( 1) + r H H

L 122L( 1) + r L

5

.

This implies det(M) = 0, yielding a 4th order polynomial equation in ,

1

22H( 1) + r H

1

22L( 1) + r L

HL = 0.

Denote the 2 negative roots by 1 and 2, and the 2 positive roots by 3 and 4.




System of PDEs


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Model


Debt Value

System of PDE s

Region 1

Region 2

Region 2: [XDH, XDL ]

The homogeneous PDE to solve is,

0 =1

22HX

2t V

HXX + XtV

HX (r + H)V

H.

Conjecture the solution to be,

VH = CX .

The homogeneous PDE translates into an algebraic equation,

0 =1

22H

2 +

1

22H

(r + H).

The roots of that equation, denoted by 1 and 2, are given by,

{1, 2} =

2

2H

1

2

3

v

u

u

t

2

2H

1

2

3 2

+ 2r + H

2H

.



M d l

General Solutions

B d C di i


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Model


Debt Value

Boundary Conditions

Solution

General Solutions

The mature firm corresponds to the capital level i = 1. The value function for themature firm can be expressed as follows.


V(X, H; c1H) = A1HX1 + A2HX

2 + A3HX3 + A4HX

4 + (1 )r

XKi (1 )rciH,

V(X, L; c1L) = A1LX1 + A2LX

2 + A3LX3 + A4LX

4 +(1 )

r XKi

(1 )

rciL.


L,

V(X, j; c1H) = C1X1 + C2X

2 +(1 )

r + H XKi

(1 )

r + HciH.



Model

General Solutions

Bo ndar Conditions


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Model


Debt Value

Boundary Conditions

Solution

Boundary Conditions

The boundary conditions are,

(Boundedness) limX

V(X, j)

X


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Model


Debt Value

Boundary Conditions

Solution

Solution

The boundedness condition implies that A3H = A3L = A4H = A4L = 0.

The optimal default conditions are given by,

0 = C1(XDH)

1 + C2(X

DH)

2 +

(1 )

r + H XDHK

i

(1 )

r + H c1H,

0 = C11(XDH)

11 + C22(XDH)

21 +(1 )

r + H Ki ,

0 = A1L(XDL )

1 + A2L(XDL )

2 +(1 )XD

LK

1

r

(1 )c1Lr

,

0 = A1L1(XDL )

11 + A2L2(XD

j )21 + (1 )K

1

r .



Model

General Solutions

Boundary Conditions


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Model


Debt Value

Boundary Conditions

Solution

Solution

The continuity and smooth pasting conditions are given by,

0 = C1(XDL )

1 + C2(XDL )

2 A1H(XDL )

1 A2H(XDL )

2

H

(r + H )(r )(1 )XD

LK

1+

H

r(r + H)(1 )c

1H,

0 = C11(XDL )

11 + C22(XDL )

21 A1H1(XDL )

11 A2H2(XDL )

21

H

(r + H )(r )(1 )XDL K

1 .

The following 2 equations complete the system of 8 equations and 8 unknowns,A1L = L1A1H, A2L = L2A2H.



Model

General Solutions

Boundary Conditions


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Model


Debt Value

Boundary Conditions

Solution

System of equations

0 = C1(XDH)

1 + C2(XDH)

2 +(1 )

r + H XDHK

i

(1 )

r + Hc1H,

0 = C11(XDH)

11 + C22(XDH)

21 +(1 )

r + H Ki ,

0 = A1HL1(XDL )

1 + A2HL2(XDL )

2 + (1 )XDL K1r

(1 )c1Lr

,

0 = A1HL11(XDL )

11 + A2HL22(XDL )

21 +(1 )K

1

r ,

0 = C1(XDL )

1 + C2(XDL )

2 A1H(XDL )

1 A2H(XDL )

2

H(r + H )(r )

(1 )XDL K1 + Hr(r + H)(1 )c1H,

0 = C11(XDL )

11 + C22(XDL )

21 A1H1(XDL )

11 A2H2(XDL )

21

H

(r + H )(r )(1 )XDL K

1 .



Model

General Solutions

Boundary Conditions


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ode


Debt Value

ou da y Co d t o s

Solution

X

The coefficients can be expressed as a function of the default thresholds,

C1C2

=

(XDH

)1 (XDH

)2

1(XDH

)11 2(XDH

)21

! 1

2 (1)r+H

XDH

Ki (1)

r+Hc1H

(1)r+H

Ki

3

,

A1HA2H

=

l1(XDL )1 l2(XDL )

2

l11(XDL

)11 l22(XDL

)21

!

1

H

d

(1)XD

L K

1r (1)c

1Lr

(1)K1r

I

e .

0 = C1(XDL )

1 + C2(XDL )

2 A1H(XDL )

1 A2H(XDL )

2

H

(r + H )(r ) (1 )X

D

L K

1 +

H

r(r + H) (1 )c1H,

0 = C11(XDL )

11 + C22(XDL )

21 A1H1(XDL )

11 A2H2(XDL )

21

H

(r + H )(r )(1 )XDL K

1 .



Model

System of PDEs

General Solutions

Boundary Conditions


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Debt Value

Boundary Conditions

Solution

System of PDEs

Although debt can only be issued when an investment is made, the market value ofdebt is calculated for each capital level Ki and volatility level j. The debt valuesatisfies the Bellman equation,

B(X, j; ci) = cidt + (1 + rdt)1E[B(Xt + dXt, j + jj dt(j j); ci)].

Apply Itos Lemma,

0 =1

2BXX

2j X

2t + BXXt rB+ jj [B(X, j ; ci) B(X, j; ci)] + ci.

The system of PDEs to solve is,

0 =

1

2

2

HX

2

t B

H

XX + XtBH

X rBH

+ H(BL

BH

) + ci,

0 =1

22L X

2t B

LXX + XtB

LX rB

L + L(BH BL) + ci.

Again, we need to partition X into regions over which the system of PDEs will besolved.



Model

System of PDEs

General Solutions

Boundary Conditions


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Debt Value

Boundary Conditions

Solution

System of PDEs

The system of PDEs can be partitioned over 2 regions:


0 =1

22HX2t BHXX + XtBHX rBH + H(BL BH) + ci,

0 =1

22L X

2t B

LXX + XtB

LX rB

L + L(BH BL) + ci.


L,

0 = 122HX

2t B

HXX + XtB

HX (r + H)B

H + ci.



Model

System of PDEs

General Solutions

Boundary Conditions


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Debt Value

Boundary Conditions

Solution

General Solutions

The debt value function can be expressed as follows.


B(X, H; ci) = B1HX

1 + B2HX

2 + B3HX

3 + B4HX

4 +ci

r ,

B(X, L; ci) = B1LX1 + B2LX

2 + B3LX3 + B4LX

4 +ci

r.


L,

B(X, H; ci) = D1X1 + D2X2 + cir + H

.



Model

S l i f h M Fi (Fi i h N G h O i )

System of PDEs

General Solutions

Boundary Conditions


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Debt Value

Boundary Conditions

Solution

Boundary Conditions

The boundary conditions are,

(Boundedness) limX

B(X, j; ci)

X


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Debt Value

y

Solution

Solution

The boundedness condition implies that B3H = B3L = B4H = B4L = 0.

The optimal default conditions are given by,

0 = D1(XDH)

1 + D2(XDH)

2 +ci

r + H,

0 = B1L

(XD

L)1 + B

2L(XD

L)2 +

ci

r,

The continuity and smooth pasting conditions are given by,

0 = D1(XDL )

1 + D2(XDL )

2 B1H(XDL )

1 B2H(XDL )

2 H

r(r + H)ci,

0 = D11(XDL )

11 + D22(XDL )

21 B1H1(XDL )

11 B2H2(XDL )

21.

The following 2 equations complete the system of 8 equations and 8 unknowns,

B1L = L1B1H, B2L = L2B2H.



Model


System of PDEs

General Solutions

Boundary Conditions


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Debt Value

y

Solution

System of equations

0 = D1(XDH)

1 + D2(XDH)

2 +ci

r + H

,

0 = B1HL1(XDL )

1 + B2HL2(XDL )

2 +ci

r,

0 = D1(XDL )

1 + D2(XDL )

2 B1H(XDL )

1 B2H(XDL )

2 H

r(r + H)ci,

0 = D11(XDL )

11 + D22(XDL )

21 B1H1(XDL )

11 B2H2(XDL )

21.


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