Generating volatility surfaces and skews and smiles for single name futures

1. Arbitrage-free Volatility Surfaces
for Equity Futures
Dr A. A. Kotzé
Financial Chaos Theory
March 2012
Saggitarius A*: supermassive black
hole at the Milky Way’s center

2. But the creative principle resides in
mathematics. In a certain sense,
therefore, I hold it true that pure
thought can grasp reality, as the
ancients dreamed.
Albert Einstein (1879-1955)
Niels Bohr
and
Albert Einstein

3. The Options Market
Myron Scholes (1941 - )
Robert Merton (1944 - ) Fischer Black (1938 -
1995)
When it is not in
our power to
determine what is
true, we ought to
follow what is
most probable
Rene Descartes (1596 - 1650)

4. The Volatility Skew
• The Black and Scholes model assumes that volatility is
constant.
• However, traders know that the formula misprices deep in-the-
money and deep out-the-money options.
• The mispricing is rectified when options (on the same
underlying with the same expiry date) with different strike prices
trade at different volatilities
• We say volatilities are skewed when options of a given asset
trade at increasing or decreasing levels of implied volatility as you
move through the strikes.
• The empirical relation between implied volatilities and exercise
prices is known as the “volatility skew/smile”.
•The volatility skew can be represented graphically in
2 dimensions (strike versus volatility).

5. Exchanges and Market Makers
• In liquid markets, exchanges value all options off their
MtM levels
• Exchanges like CME employ market makers who will
supply a MtM level for every option with open interest
• Safex has a different model. There are no market makers
• Every option with open interest has to be valued at the
end of every day for variation and initial margin purposes;
even if that option did not trade during the day
• Exotic option trade on Safex as Can-Do options. Volatility
surfaces are extremely important for local volatility models
• How does Safex do that?
• Risk management becomes an issue!

6. First Skews: ALSI from Polled Data

7. Generating a Skew
• Question: can a skew be generated from traded data?
• At the end of July 2009, everything changed
• Nutron forces option traders to supply 2 of the 3
quantities: premium, volatility, futures level traded at

8. Tompkins Research
• Studies the implied volatility surfaces across different
markets
•Claim to have “all publically available data” between dates
as shown
• Data obtained from relevant exchanges
• Number of option prices examined across the 16 markets
was 1,862,473
• Most comprehensive empirical study to date!
Motivation:
 Assign economic significance to the
functional form of the smile patterns
 Determine stability of skews across
time

9. Tompkins Results for Equity Derivatives

10. The Safex Functional Form for each Expiry
Arbitrage……
Arbitrage……

11. Linking across Time: the volatility term structure

12. ATM Term Structure: Alsi for 22 Mar 2012
Note: used to be mostly downward sloping before 2008 crisis

13. Alsi Volatility Surface: 22Mar 2012
No-Spread
arbitrage: skews do
not cross one
another

14. DTOP: Volatility Surface
• Principle Component Analysis (PCA) of historical Alsi
volatility surfaces
• First 3 components describe about 95% of skew changes:
tilt/slope; shift/trend curvature/vol of vol
• South African study by Bonney, Shannon and Uys in 2008
• Obtain time series for these components
• Calculate correlation between components and absolute
daily Dtop changes
• Extract components for Dtop
• Use components as parameters in quadratic skew
equation
• Follow methodology as described by Carol Alexander

15. Enhancements: further work
• Wings
• Traded data shows our market does
not trade far away from the ATM:
strike spread limited
• Skews published from 70% - 130%.
Usually have to extrapolate
• Var futures (swaps) and exotic
options need more accurate wings
• Model volatility can be negative:
wrong
• Dtop PCA/correlation model needs to
be enhanced

16. Volatility Surfaces for Single Stock Futures
• Market not very liquid
• Options do trade on a skew
• Safex introduced linear skews just over a year ago.
Floating skews the same for all SSFs

17. Wits Honours Project
• Follow the Tompkins methodology in establishing the
shape of the skew using traded data
• Fit data using the quadratic model similar to ALSI
• Captured all trades from 31 July 2009 until 14 October
2011
• Look at most liquid contracts
Perfect numbers
like perfect men
are very rare
Rene Descartes (1596 - 1650)

18. Results: AGLQ

19. Results: AGLQ empirical vs Safex – 2 months till expiry

20. Results: BILQ

21. Results: MTNQ

22. Remarks
• Near dated skews are curved and not linear
• Skews become linear for far dated contracts
• This study did not take no-arbitrage principles into
account
• Data is an issue

23. Another Honours Project
• Using non-parametric methods in estimating the skew
• Study followed the canonical valuation method performed
by Stutzer and a similar method of Duan
• This methodology determines the implied volatility skew
purely based on the historical prices of an underlying
asset without needing option prices.
• A study on SA data was undertaken in 2005 by Mark de
Araújo and Eben Maré

24. Methodology
• Compute empirical distribution
• Require risk-neutral distribution f(x)
• Solve for f(x) that minimizes the relative entropy between
itself and normalised empirical distribution
• Such that E(f(x)) = r.
• Simulate asset price at maturity with Monte Carlo
• Compute expected value of the option
• Repeat pricing procedure
• Vary maturity (T) and strike (K)
• Convert option values to Implied Volatilities (IV)

25. de Araújo and Maré Results for ALSI

26. ALSI Skews: new study 2011

27. Final Remarks
• Empirical methods give good results for fairly liquid option
contracts
• Problematic when data is a problem: illiquid contracts and
a limited strike range
• Nonparametric methods give results for any instrument
where one can find closing prices for the underlying linear
instrument
• Nonparametric skews are more curved than the market
skews
Estimating skews for single stock options only
partially solved

28. Contact
Dr Antonie Kotzé
Email: consultant@quantonline.co.za
Phone: 082 924-7162
http://www.quantonline.co.za
Disclaimer
This article is published for general information and is not intended
as advice of any nature. The viewpoints expressed are not
necessarily that of Financial Chaos Theory Pty Ltd. As every
situation depends on its own facts and circumstances, only specific
advice should be relied upon.