Structural GARCH: The Volatility-Leverage
Connection
Robert Engle1
Emil Siriwardane1,2
1 NYU Stern School of Business
2 U.S. Treasury, Office of Financial Research (OFR)
WFA Annual Meeting: 6/16/2014
Introduction
100
2.4
90
2.2
2
80
1.8
70
1.6
60
1.4
1.2
50
1
40
0.8
30
0.6
20
0.4
10
0.2
0
1998
2000
2002
2004
2006
Dat e
2008
2010
0
2012
De bt t o Eq uit y
1-Mont h R e aliz e d ( Annualiz e d) Volat ility
BAC Leverage and Realized Volatility
Leverage and Equity Volatility
œ
Crisis highlighted how leverage and equity volatility are tightly
linked
œ
“Leverage Effect” has been around - e.g. Black (1976), Christie
(1982) - but...
œ
A dynamic volatility model that incorporates leverage directly
has remained elusive
This Paper
œ
GARCH-type model where equity volatility is amplified
(non-linearly) by leverage as in structural models of credit
œ
Asset return series from observed equity series
œ
Assets have time-varying volatility at high frequencies
œ
Statistical test of how leverage affects volatility
œ
Two applications:
1. Systemic Risk: SRISK and Precautionary Capital (today)
2. Leverage Effect (in the paper)
Theoretical Foundation
Structural Models of Credit
œ
Under relatively weak assumptions on the vol process, structural
models say Et = f (At , Dt , æA,t , ø, rt )
œ
œ
œ
œ
At = market value of assets
Dt = book value of debt
æA,t = stochastic asset volatility
Generic dynamics for assets and asset variance (allow for jumps later):
dAt
= µA (t)dt + æA,t dBA (t)
At
d æ2A,t = µv (t , æA,t )dt + æv (t , æA,t )dBv (t)
œ
BA (t) and Bv (t) potentially correlated
Equity Returns and Equity Volatility
Introducing the Leverage Multiplier
œ
Apply Itō Lemma and ignore drift (our model is daily, and daily equity
returns º 0):
dEt
∫t æv (t , æA,t )
= LMt æA,t dBA (t) +
dBv (t)
Et
Et 2æA,t
º LMt £ æA,t £ dBA (t)
µ
∂
dEt
volt
º LMt £ æA,t
Et
œ
œ
œ
LMt = LM (Et /Dt , 1, æA,t , ø, rt ) is the “leverage multiplier”
LMt amplifies asset shocks and volatility
Two questions:
1. How much does the higher order term contribute? Not Much
2. What does LMt look like? Robust shape across models
The Leverage Multiplier: Three Basic Properties
Discrete Time: GARCH Option Pricing
5
4
B SM
3
2
1
0
0
20
40
7
6
5
4
2
1
5
4
H e st on
2
1
0
20
40
D eb t t o E q u i t y
20
40
60
D eb t t o E q u i t y
Lev er a g e M u l t i p l i er
Lev er a g e M u l t i p l i er
D eb t t o E q u i t y
6
3
MJ D
3
0
0
60
15
60
10
8
Lev e r age Mult iplie r
Lev er a g e M u l t i p l i er
Lev er a g e M u l t i p l i er
Popular Continuous Time Option Pricing Models
6
10
5
BSM
G ARC H-N
G ARC H-t
G J R-N
G J R-t
6
4
SVJ
2
0
0
20
40
60
0
0
5
10
D eb t t o E q u i t y
15
20
25
30
35
40
45
De bt t o Eq uit y
1. LM(0) = 1. Mechanical, since assets = equity
2. Monotonically increasing. More leverage means more risk
3. Concave. Reducing leverage more powerful than increasing leverage
50
Structural GARCH
Our Specification
œ
The challenge is choosing the right functional form for LMt
œ
We use simple transformations of Black-Scholes-Merton (BSM)
functions:
∑
≥
¥ D ∏¡
t
f
BSM
BSM
f
LMt (Dt /Et , æA,t , ø) = 4t
£g
Et /Dt , 1, æA,t , ø £
Et
œ
œ
g BSM (·) is inverse BSM call function. ¢BSM
is BSM delta
t
¡ 6= specific option pricing model
Our parametrization preserves necessary properties of LM , but
still allows us to change its scale
The Full Recursive Model
Structural GARCH
rE ,t = LMt °1 £ rA,t
= LMt °1 £
q
h A ,t £ " A ,t
hA,t ª GJR(!, Æ, ∞, Ø)
∑
≥
¥ D ∏¡
t °1
BSM
BSM
f
LMt °1 = 4t °1 £ g
Et °1 /Dt °1 , 1, æA,t °1 , ø £
Et °1
s
∑t +ø
∏
X
æfA,t °1 = Et °1
h A ,s
s =t
So parameter set is £ = (!, Æ, ∞, Ø, ¡)
Estimation Results
Estimation Details
œ
œ
œ
Estimate for 82 financials via QMLE; iterate over ø 2 [1, 30]
Equity returns and balance sheet information from Bloomberg
Dt is exponentially smoothed book value of debt
œ
œ
smoothing parameter = 0.01, so half-life of weights º 70 days
We estimate the model using two approaches for æfA,t °1 , then
use the highest likelihood:
1. A dynamic forecast for asset volatility over life of the option
2. The unconditional volatility of the asset GJR process
Bank of America: Structural GARCH Estimation
Annualiz e d Volat ility
¡ = 1.4 (t = 11.4)
3
2
1.5
1
0.5
0
1998
Le ve r age Mult iplie r
Asse t s
E q u i ty
2.5
2000
2002
2004
2000
2002
2004
2006
2008
2010
2012
2006
2008
2010
2012
20
15
10
5
0
1998
Dat e
Parameter Values
Cross-Sectional Summary of Estimated Parameters
Parameter
!
Æ
∞
Ø
¡
œ
œ
œ
Mean
2.7e-06
0.0458
0.0721
0.9024
0.9834
Mean t-stat
1.70
3.07
2.91
80.08
4.00
% with |t | > 1.64
47.2
86.1
80.6
100
73.6
(!, Æ, ∞, Ø) are standard GJR parameters - for assets, not equity
Average ø = 8.34
Leverage matters
Application: SRISK
SRISK
How much would a financial firm need to function normally in another crisis?
œ
Acharya et. al (2012) and Brownlees and Engle (2012)
œ
Three steps:
1. GJR-DCC model using firm equity and market index returns
2. Expected firm equity return if market falls by 40% over 6 months
¥ LRMES
3. Combine LRMES with book value of debt to determine capital
shortfall in a crisis
œ
The crisis in this case is a 40% drop in the stock market index
over 6 months
The Role of Leverage?
Thought Experiment with Structural GARCH
œ
Firm experiences sequence of negative equity (asset) shocks
œ
Level of leverage goes up rapidly
œ
Leverage multiplier increases, equity vol amplification higher
œ
Painfully obvious in the crisis, so build into SRISK
Bank of America
Capital Shortfall: 2006-2011
Precautionary Capital
Defining Precautionary Capital
e.g. How much additional equity would a bank need, today, to be 90% sure they
won’t need bailout money in a future crisis?
œ
SRISK: how much capital would a firm need in a financial crisis
to return to a equity/asset ratio of k %?
œ
Precautionary Capital: How much capital do we have to add to
the firm today so that we can have a level of certainty, Æ, that
the firm meets a capital requirement of k % in a crisis?
œ
Uses the quantiles of the future return distribution
œ
We set k = 2% and vary Æ
Primary Takeaway in a Nutshell
œ
Standard volatility models don’t have a channel for leverage, so
adding equity to the firm today won’t reduce future risk
œ
Structural GARCH: reducing leverage today reduces future risk
œ
œ
The effect is further enhanced by the concavity of the LM
Engle and Siriwardane (2014) use this idea to suggest a
risk-based total leverage capital requirement
Precautionary Capital: BAC
BAC on 10/1/2008: E0 = 173.9 bn; D0 = 1, 670.1 bn
k=0.02
Confidence that Firm Meets Capital Requirement in Crisis (%)
100
90
80
GJR
70
Structural GARCH
60
50
40
30
20
10
0
−200
−100
0
100
200
300
400
Size of Equity Injection (bn)
500
600
700
What’s Next
Other Applications
œ
Endogenous Crisis Probability with Structural GARCH
œ
Estimation of Distance to Crisis
œ
Endogenous Capital Structure and Leverage Cycles
œ
Counter-cyclical Capital Regulation
œ
Model of CDS Volatility
Appendix
Ignore Higher Order Terms
∫t æv (t , æA,t )
dEt
= LMt æA,t dBA (t) +
dBv (t)
Et
Et 2æA,t
How much do the higher order terms contribute?
œ
Not much. Simple intuition...
œ
Volatility mean reversion speed ø typical debt maturities, so ...
œ
œ
Back
Total volatility over option is effectively constant
We verify in paper for variety of option pricing models
Dynamic Forecast vs Constant Forecast
œ
Estimate two types of models:
1. Using a dynamic forecast for asset volatility over life of the
option
2. Using unconditional volatility of GJR process
œ
Then take the model that delivers the highest likelihood
œ
A few outliers where ¡ hits lower bound (exclude from
subsequent analysis):
œ
SCHW, JNS, LM, BK, BLK, NTRS, CME, CINF, TMK, UNH