Using VIX Futures as
Option Market-Makers’
Short Hedge
10 December 2010
Yueh-Neng Lin, Anchor Y. Lin
National Chung Hsing University
ynlin@dragon.nchu.edu.tw
+886-4-22857043
1
Volatility Risk
• Given the large and frequent shifts in volatility
in the recent past, especially in periods like
Black Monday of Stock Market Crash in 1987,
Gulf War I in 1990, Asian Currency Crises in
1997, Russian Default and LTCM in 1998,
Bursting of Dot-Com Bubble in 2000, 9-11 in
2001, Gulf War II in 2003, Subprime Mortgage
Crises of 2007, and Leman Brothers bankruptcy
in September 2008, there is a growing need for
instruments to hedge volatility risk.
2
Volatility Risk Management
• Financial instruments to hedge volatility risk:
1. Traditional Call and Put options
• also have delta and gamma risk
2. Volatility/Variance Swaps
• A forward contract on realized or implied
volatility/variance
• Only available at OTC
3. Strategies
• Options on a straddle (Carr and Madan, 1998)
• Options on at-the-money forward-start straddle (Brenner,
Ou and Zhang, 2006)
• Forward-start strangle (Rebonato, 1999)
3
Volatility Risk Management
(continued..)
4. Futures or options on a volatility index
• Brenner and Galai (1989) believe that different volatility
measures that are highly correlated to the volatility
exposures of all market participants are different.
• Hence, the volatility index may be based on standard
deviation of historical data, implied volatility of options, or
the combination of historical and option data.
• Then, the futures or options on alternate volatility index
may satisfy the hedging demand of different participants.
• VIX futures and VIX options
4
VIX
• The Chicago Board Options Exchange (CBOE)
introduces the VIX (volatility index) in 2003,
which is the sentiment volatility implicit in
S&P 500 index (SPX) option prices.
• VIX squared is the forward price of annualized total
variance of S&P 500 index returns over 30 days
(τ=30/365), implied in SPX option prices:
Pt
var
(T ) /   E [ 

F
t
2

K0

0
t 
t
( u du  J u dN u )] / 
 K Pt (T , K i )
K
 VIX
2
t
2
i
B (t , T )
2

2



K0

1  Ft (T )
 
 1 
B (t , T )
  K0

 K C t (T , K i )
2
Ki
5
2
VIX & S&P 500
• VIX tends to rise when the S&P 500 falls.
• VIX tends to decline or remain constant when
•
•
the S&P 500 rises.
Historically, VIX hits its highest levels during
times of financial turmoil and investor fear.
Hence, VIX is also known as the investor “Fear
Index” (Whaley, 2000) in the marketplace.
6
7
VIX & VIX Derivatives
•
•
•
•
VIX futures are launched on March 26, 2004.
VIX options are launched on February 24, 2006.
VIX term structure are built in December 2008.
Negative Correlation between S&P 500 Index
and VIX
•
VIX futures and VIX options could be a
“catastrophe hedging” tool for stock portfolios.
8
VIX Futures
•
Since VIX futures settle to the forward 30-day
implied volatility of the S&P 500 (forward VIX),
they are natural to hedge the “forward vega”
risk of S&P 500 options.
9
Hedging a Futures Call
Option’s Forward Vega Risk
• A short position on S&P 500 futures call
options is chosen as our target instrument,
since it consists of the volatility risk between
option’s expiry and its underlying futures’
expiry, or equivalently the “forward vega”
risk.
10
The aim of this paper is to
investigate whether and
by how much VIX futures
improve hedging a
futures call option’s
vega risk
11
Hedge a Call Option’s Spot Vega Risk
• Psychoyios and Skiadopoulos (2006) use the
short position on a standard European-style
call option as the target instrument.
• Two hedging schemes with two traditional option
models and three volatility option models to
investigate which combination gets the best hedging
performance.
• They find that the traditional option is a more
efficient instrument than the volatility option for
hedging a traditional option.
12
American-Style Futures Option
•
•
For options with early exercise potential,
compute the BAW (Barone-Adesi and Whaley,
1987) early-exercise premium, treating it as if
the stock volatility.
Adding this early-exercise adjustment
component to the European futures option price
in the pricing formula should result in a
reasonable approximation of the corresponding
American futures option price.
13
Implicit Features of S&P 500 Index Prices
1. Stochastic volatility (SV)
•
•
Heston (1993), Cox, Ross and Ingersoll (1985)
The volatility is allowed to be mean-reverting
and stochastic.
2. Price jumps (SVJ)
•
•
Bates (1996), Bakshi, Cao and Chen (1997)
To internalize more skewness and kurtosis.
14
American-Style Futures Options
(continued..)
•
For stochastic volatility and jump-diffusions, the
constant-volatility analytic approximation results
in Bates’ (1996) American currency futures
options.
15
VIX Futures Valuation Formula
16
17
18
VIX - VIX Futures
•
What is the relationship?
• Futures prices can swing from a premium to a
discount to VIX.
• VIX futures prices are based purely on
expectations.
• No cost of carry between spot VIX and VIX futures.
19
Fair Price of VIX Futures (continued..)
• Carr and Wu (2006) present that the price of the VIX
futures has a lower bound and an upper bound.
• Dupire (2006) finds the concavity adjustment which
needs to be subtracted from the price of forward variance
to arrive at the fair value of VIX futures.
• Zhang and Zhu (2006) choose a Heston-type stochasticvariance model and developed a numerical expression for
VIX futures.
• Lin (2007) drives VIX futures price formulas under affine
stochastic-volatility index processes with the convexity
adjustment.
20
Fair Price of VIX Futures
• By Martingale pricing theory, the current
VIX futures prices expiring at t can be
computed under a risk-neutral probability
measure Q as,
Q
Ft
VIX
(T )  E t ( VIX T ) 
Q
E t ( VIX T ) 
Q
2
2
var t ( VIX T )
8  [ E t ( VIX T )]
Q
2
21
3/2
Strategies for Hedging
Forward Volatility Risk
22
Hedging Strategies
• This study uses a short position on the futures
option as the target portfolio at time t, i.e.
TARt=－Ct(F)
• The expiration dates of SPX futures options and SPX
futures are respectively T1 and T2 , where t <T1<T2 .
• T2 － T1 ≡ 30 days
• The hedged portfolio consists of
• (1) N1,t shares of underlying SPX futures Ft , and
• (2) N2,t shares of either forward-start strangle portfolios
(HS1) or VIX futures (HS2), both denoted as INSTt .
23
Hedging Strategies (continued..)
• The profit of this hedging portfolio is defined
as
 t  N 1, t Ft  N 2 , t INST t  C t ( F )
• Further add the constraints of delta-neutral
and vega-neutral by
 t
 Ft
 t
 t
 N 1, t  N 2 , t
 N 2 ,t
 INST
 t
 INST
 Ft
t

t

C t ( F )
C t ( F )
 t
 Ft
0
0
24
Two Hedging Schemes
• Hedging Scheme 1 (HS1):
• The hedged portfolio consists of N1,t shares of
underlying futures, and N2,t shares of forward-start
strangle portfolios.
• The forward-start strangle portfolio consists of
• (1) a short position of short-term strangle (the expiration
date of the standard call and standard put is T1 ; the strikes
of the call and put are K2 and K1 with K1 < K2 ) ; and
• (2) a long position of long-term strangle (the expiration
date of the standard call and standard put are T2 ; the strikes
of the standard call and put are K2 and K1 ).
25
Two Hedging Schemes (continued..)
• Hedging Scheme 1 (HS1):
INST t   C t (T1 , K 2 )  Pt (T1 , K 1 )  C t (T2 , K 2 )  Pt (T2 , K 1 )
26
Two Hedging Schemes (continued..)
• Hedging Scheme 2 (HS2):
• The hedged portfolio consists of N1,t shares of
underlying futures, and N2,t shares of the VIX
futures, i.e.
INST t  Ft
VIX
(T )
27
Hedging Error
• The hedging error is defined as the additional
profit (loss) over the risk-free return :
H E t (t   t )    t  t   t ( e
r t
 1)
 ( C t   t  C t )  N 1, t ( Ft   t  Ft )  N 2 , t ( IN STt   t  IN STt )
 [  C t  N 1, t Ft  N 2 , t IN STt ]( e
r t
 1)
28
Absolute Hedging Error
• The absolute hedging error through a hedging
period (T1 － t) is calculated as
M
TDHE ( t , T1 ) 
 | HE
t  ( l 1)  t
(t  l t ) e
r t ( M  l )
|
l 1
where
M  (T1  t ) /  t
29
Empirical Data
• In order to assess the hedging performance of forward
•
•
•
•
•
volatility risk, we use the SPX futures, the SPX options
and the VIX futures to hedge a short position on the
SPX futures call option.
The data period: from July 3, 2006 to June 30, 2009.
The daily prices for the SPX futures and the SPX futures
option are obtained from the CME.
The daily prices for the SPX options are obtained from
CBOE, and the daily prices for the VIX futures are
captured from CBOE.
The interest rate data are daily annualized Treasury-bill
rates obtained from DataStream Database.
The daily dividend-yield ratio data are obtained from the
S&P Corporation.
30
Empirical Data (continued..)
1. SPX futures that expire in March, June,
September, and December.
2. SPX futures call options that expire in February,
May, August, and November.
3. Forward-start strangle portfolio involves in two
strangles.
• 1ST strangle: SPX options that expire in February, May,
August, and November.
• 2nd strangle: SPX options that expire in March, June,
September, and December.
4. VIX futures that expire in February, May, August,
and November.
31
Empirical Results
• Our findings show that VIX futures outperform
forward-start strangle strategies at most moneynessmaturity categories and the results are robust across
models and hedging strategy.
• Overall, VIX futures are a better hedging instrument
than standard options if the target option is a futures
option, or equivalently if the risk exposure is the
forward volatility risk.
• In particular, incorporating SV on average
outperforms in OTM hedging.
• However, adding price jumps further enhances the
hedging performance for SR options during the postcrash-relaxation period.
32
The End.
Thank you.
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