Volatility surfaces for equity indices, single name futures and USDZAR FX options. All from Bloomberg data.

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VOLATILITY SURFACES
June 2013
Presented by:
Dr Antonie Kotzé
Quantitative consultant to the JSE

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Some members are lost…..
INTRODUCTION

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Volatility is a measure of risk or uncertainty
Volatility is defined as the variation of an asset's returns – it indicates
the range of a return's movement. Large values of volatility mean that
returns fluctuate in a wide range – in statistical terms, the standard
deviation is such a measure and offers an indication of the dispersion or
spread of the data
VOLATILITY
The Volatility Surface: What is Volatility?
Volatility has peculiar dynamics:
− It increases when uncertainty increases
− Volatility is mean reverting - high volatilities
eventually decrease and low ones will likely rise to
some long term mean
− Volatility is often negatively correlated to the stock or
index level
− Volatility clusters - it is statistically persistent, i.e., if it
is volatile today, it should continue to be volatile
tomorrow

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Top40 Returns
Volatility
-0.15
-0.1
-0.05
0
0.05
0.1
Return
Top 40 Daily Logarithmic Returns
27 October 1997 October -
December
2008

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Top40 Volatility
Volatility is not Constant
5%
15%
25%
35%
45%
55%
65%
Oct-95
Apr-96
Oct-96
Apr-97
Oct-97
Apr-98
Oct-98
Apr-99
Oct-99
Apr-00
Oct-00
Apr-01
Oct-01
Apr-02
Oct-02
Apr-03
Oct-03
Apr-04
Oct-04
Apr-05
Oct-05
Apr-06
Oct-06
Apr-07
Oct-07
Apr-08
Oct-08
Apr-09
Oct-09
Apr-10
Oct-10
Apr-11
Oct-11
Apr-12
Oct-12
Apr-13
Volatility(%)
Date
Top 40 Volatiliy: 3 Month Rolling Historical Volatility: Mean Reversion
Long term
mean
Long term
Median
Asian Crisis
1997/98
Credit Crisis
Oct 2008

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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 3D or in
2 dimensions (strike versus volatility).
VOLATILITY
The Volatility Surface: What is Volatility?

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Top40
The Volatility Skew

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Empirical tests show that the quadratic model is best suited to SA
equities market – SVI for commodities and currencies
Safex implemented the following model during October 2010
With the moneyness NOT the strike
The volatility term structure is modeled by
Optimisation is performed using TRADED DATA only in obtaining the 5
parameters.
One condition to take care of is to ensure that calendar-spread arbitrage
is minimised
Volatility Surface: Functional Deterministic Form
Model Implemented
K

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The parameters are
Constant volatility (shift or trend) parameter.
correlation (slope) term. This parameter accounts for the negative
correlation between the underlying index and volatility. The no-
spread-arbitrage condition requires that
is the volatility of volatility (`vol of vol' or curvature/convexity)
parameter. The no-calendar-spread arbitrage convexity condition
requires that
Functional Form: meaning of parameters
Model Implemented
0 00 
1
01 1  
2
02 

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Deterministic Form
Why?
After much research, Safex decided on the deterministic route
Reasons:
− keep things as simple as possible - replicability
− A road trip to London confirmed the deterministic route
− Jim Gatheral’s book and course notes (AIMS Feb 2009)
− Two seminal research papers:
− Dumas, Fleming and Whaley – Implied volatility Functions:
Empirical Tests
− Tompkins – Implied Volatility Surfaces: Uncovering Regularities for
Options on Financial Futures

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Skew Comparison
Tests

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Term Structure Comparison
BB vs Safex

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Skew Comparison
BB vs Safex

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Deterministic Solution
Stability of Surfaces
Tompkins studies the implied volatility surfaces across different
markets
Motivation: assign economic significance to the functional form of the
smile patterns
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!
Tompkins stated: Given the primary objective of this research is to
assign economic significance to the estimation of equation (3), the fact
that the statistical significance of the coefficients (and the sign of the
impact) are retained once alternative regression approaches are used,
leads us to conclude the model is robust to the method of estimation.
Further to this: If the sole objective was to fit a curved line, this has
been achieved.

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Deterministic Volatility Surface
ATM Model Volatility

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Functional Form
Pricing Exotics
Why is a functional form so useful?
Can easily implement local volatility models used in pricing exotic
options like barriers
Used in implied binomial or trinomial trees
Or Dupire’s transformations used in Finite Difference Methods or
Monte Carlo simulation

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Volatility Surface
Stability
2012 study published in the North American Journal of Economics and
Finance

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Volatility Surface
Scatter Graph

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Volatility Surface
Top40 Surface

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Volatility Surface
BIL Scatter Graph

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Volatility Surface
BIL Surface

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Volatility Surface
Further Research
Wings are a problem – we know that. Not enough trades especially
above 110% moneyness
Playing with different deterministic functions like Gatheral’s SVI model
Currency skews
Currently generated by Super Derivatives in London
Very illiquid at the moment

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Volatility Surface: 18 June 2013
Further Research

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Volatility Surface: 18 June 2013
Further Research

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Questions?

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