On the Term Structure
of Model-Free Volatilities
and Volatility Risk Premium
Kian Guan LIM and Christopher TING
Singapore Management University
1
Motivations on Volatility

Volatility forecast is critical to stock and option
pricing





Volatility Risk


Historical volatilities
Implied volatilities
Model-free volatility
Term Structures
Is it priced? Term structure?
Industry Development



CBOE VIX “Fear” gauge
Variance and volatility swaps
Options and futures on volatility indexes
2
Main ideas of paper




Present an enhanced method to compute model-free
volatility more accurately
Enable the construction of the term structure of
model-free volatilities from 30- to 450-day constant
maturities
Explore the longer term structure of model-free
volatility
Gain further insight into the volatility risk premium
3
Theory



Expected variance from time 0 (today) to T under
risk-neutral measure Q
The volatility s is the forward-looking volatility
forecast
Call price
Put price
Proposition 1
Risk-free rate r assumed constant
4
Generalized Diffusion Process
m and n are not necessarily constant, but may depend on
other adapted state variables

Their difference is second-order
5
Integrated Variance

Total variance from time 0 to T
6
Derivation
Log(1+z) < z
Necessarily positive
7
A Pictorial Presentation

Area under the two curves defined by the intersection
point
Implied forward price F0 determined by put-call parity

Option price
Call
Put
F0
X
D is the sum of dividends
8
some technical problems

Discrete price grid


Discrete strike price interval



Minimum interval
Limited range of strike prices
Bid-offer spread


Cannot be smaller than $0.01
Use a correct price to avoid arbitrage
Thin volume

Far term options
9
Discretized Approximation

Average of Riemann upper sum and lower sum
10
In overcoming approx due to
discrete strike price

Cubic Splines

No risk-free arbitrage conditions as constraints
Volume as weight
Piece-wise integration - Closed-form


11
Range of Strike Prices
300
250
Option Price
200
Put
Call
150
100
50
0
300
400
500
600
Strike Price
700
800
900
12
Data from Optionmetrics

CBOE S&P 100 index option









European style
Re-introduced on July 23, 2001
March quarterly cycle
Minimum strike interval 5 points
Near term, out-of-the money, 10 or 20 points strike intervals
Far term, 20 points strike intervals
7 to 8 terms for any given trading day
Weekly options excluded
Sample period

July 23, 2001 through April 30, 2006
13
Descriptive Statistics

Daily average
14
Interpolation and Extrapolation are critical



The discretized approximation formula leads to
an upward estimate of model-free volatility
The market volatility indexes might have an
upward bias, especially for those that have a
larger strike price interval
Variance swap buyers may be paying for the
bias
15
Exact versus Approximate


The difference d = sa - se*
Relative size of strike price interval



Relative size: 5 points/S&P 100 index level
d1: sub-sample defined by relative size smaller than median
value
d2: sub-sample defined by relative size larger than median
16
Volatility Term Structure


Constant maturity
Range of values over the sample period is smaller the
longer is the constant maturity
17
Term Structures of Volatility Changes

Absolute changes of daily volatility less drastic the
longer the constant maturity
Run Table3.m
18
Term Structure of Fear Gauges

Daily change in model-free volatility sti and daily change in S&P 100 index
level Lt are negatively correlated – volatility rises lead to stock market falls.
Second column is autocorrelation in sti
Run Table6.m
19
Asymmetric Correlations: Lt = c0+c1st++c2st-+t
larger magnitude of  equity when short-term volatility increases than when it decreases
larger magnitude of  equity when long-term volatility drops than when it increases
20
Change of Volatility Correlations
with Index Returns

Shorter constant maturity



Higher correlation with a market decline
Consistent with Whaley (JPM,2000) – short-term volatility
spike bad for short-term investors
Longer constant maturity


Higher correlation with a market rise
Quite surprising
21
Slope Estimate
0.039
s 2 = 0.032042+0.000013 x T
0.038
Model-Free Variance s 2
0.037
0.036
0.035
0.034
An example of a single day
regression of s(T) on T.
0.033
0.032
61.6% of all daily regressions
produces positive slopes
0.031
0.03
0
50
100
150
200
250
300
Days to Maturity T
350
400
450
500
22
Term Structure of Slope Estimates


Upward sloping (737 estimates)
Downward sloping (460 estimates)
upward sloping volatility term structure corresponds
with positive stock returns & lower risk
downward sloping volatility term structure corresponds
with negative stock returns & higher risk
23
This Figure plots the daily time series of S&P 100 index in Panel A and the slope
estimates of the model-free variance term structure in Panel B. The horizon axis shows
the dates in yymmdd. The sample period is from July 2001 through April 2006.
Change in slope estimates
is positively correlated with
the change in index level
24
variance risk premium


variance swap (a forward contract) buyer’s
payoff is (sR2 - s2) x Notional Principal
where sR2 is realized annualized volatility computed
over contract maturity [0,T] from daily sample return
volatility and model-free s2 over [0,T] is the strike
under fair valuation at start of contract t=0.
Mean excess return (sR2 - s2)/ s2 may be construed as
volatility risk premium that the buyer is compensated
for taking the risk.
25
Variance Swap Payoff and Return for Buyers

Volatility risk premium
$
%
asymmetric larger gains at right skew and lower loss at left skew up to 180 days
increasing maturity
26
Conclusions
An enhanced method to construct term structure
 Hopefully a better understanding of the behavior of a long term
structure of volatility
 Look at a number of asymmetric effects
 Explore term structure of variance risk premium
 Validation of mean-reverting stochastic volatility
Uncompleted Tasks
• how to check term structure using other long-life traded
instruments
• how to make sense of the explored term structure effects in
trading and hedging strategies
• more rigorous statistical confirmations of the term structure
results

27