1. Kian Guan LIM and Christopher TING Singapore Management University
On the Term Structure of Model-Free Volatilities and Volatility Risk Premium
Kian Guan LIM and Christopher TING
Singapore Management University

2. Motivations on Volatility
Volatility forecast is critical to stock and option pricing
Historical volatilities
Implied volatilities
Model-free volatility
Term Structures
Volatility Risk
Is it priced? Term structure?
Industry Development
CBOE VIX “Fear” gauge
Variance and volatility swaps
Options and futures on volatility indexes

3. 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

4. Theory
Expected variance from time 0 (today) to T under risk-neutral measure Q
The volatility s is the forward-looking volatility forecast
Proposition 1
Put price
Call price
Risk-free rate r assumed constant

5. Generalized Diffusion Process
m and n are not necessarily constant, but may depend on
other adapted state variables
Their difference is second-order

6. Integrated Variance
Total variance from time 0 to T

7. Derivation
Log(1+z) < z
Necessarily positive

8. A Pictorial Presentation
Area under the two curves defined by the intersection point
Implied forward price F0 determined by put-call parity F0 X
Option price
Put
Call
D is the sum of dividends

9. some technical problems
Discrete price grid
Cannot be smaller than $0.01
Discrete strike price interval
Minimum interval
Limited range of strike prices
Bid-offer spread
Use a correct price to avoid arbitrage
Thin volume
Far term options

10. Discretized Approximation
Average of Riemann upper sum and lower sum

11. 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

12. Range of Strike Prices

13. 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

14. Descriptive Statistics
Daily average

15. 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

16. Exact versus Approximate
The difference d = a - e*
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

17. Volatility Term Structure
Constant maturity
Range of values over the sample period is smaller the longer is the constant maturity

18. Term Structures of Volatility Changes
Absolute changes of daily volatility less drastic the longer the constant maturity
Run Table3.m

19. Term Structure of Fear Gauges
Daily change in model-free volatility ti 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 ti
Run Table6.m

20. Asymmetric Correlations: Lt = c0+c1t++c2t-+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

21. 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

22. Slope Estimate An example of a single day regression of (T) on T.
61.6% of all daily regressions produces positive slopes

23. Term Structure of Slope Estimates
upward sloping volatility term structure corresponds with positive stock returns & lower risk
Upward sloping (737 estimates)
downward sloping volatility term structure corresponds with negative stock returns & higher risk
Downward sloping (460 estimates)

24. 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

25. variance risk premium variance swap (a forward contract) buyer’s
payoff is (R2 - 2) x Notional Principal
where R2 is realized annualized volatility computed over contract maturity [0,T] from daily sample return volatility and model-free 2 over [0,T] is the strike under fair valuation at start of contract t=0.
Mean excess return (R2 - 2)/ 2 may be construed as volatility risk premium that the buyer is compensated for taking the risk.

26. 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

27. 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