Volatility Smileless
Wonyoung
Choi 2015-9-16
Volatility skew is a phenomenon that
implied volatility (IV) increases as option becomes increasingly in-the-money
(ITM) or out-of-the-money (OTM) (quoted from the figure below). It has a
parabolic shape and because of the shape it is called 'volatility smile' also
as shown below. I assume that readers of this paper are familiar with options
and the Black-Scholes option pricing model in theory and in practice.
The lowest point appears usually at-the-money
(ATM) but often happens off the ATM, which is called smirk by some people as
shown below.
Volatility smirks[3]
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An example of the volatility skew in the
real world is shown below. The chart shows smile, smirk, and call-put gap.
The smirk: this happens quite often and (I
think) is a very interesting phenomenon but will not be discussed in this
paper.
The smile (shape): Natenberg wrote "One
inference which most traders draw from this shape is that the marketplace
appears to believe there is a better chance for a large price move in the real
world than is predicted by the Black-Scholes model."[4] He
also wrote "Unfortunately, dealing with a volatility skew can be a problem
because there is no exact formula to use as guide." This will be studied
in this paper.
The call-put gap: this happens quite often
and (I believe) I have explained in my earlier paper[5] and
will be discussed in this paper.
Fig. 3 Volatility Skew in reality
In the real world example above, some
values are plotted with a smooth curve while others are very erratic. The
primary reason is that ITM options usually have ask-bid gaps. So, the data show
only the last transacted prices, not the (real) current prices. So, usually the
volatility skew or smile is plotted only with OTM options, puts and calls. Deep-OTM
(DOTM) options have very low prices. Once the price reaches the minimum
tradable price, 0.01 in Korea, it remains the same for all the other deeper OTM
options. Deeper strike prices (usually symbolized as "X") automatically
raise the IV which makes the smile shape. So, it is not meaningful to plot
options from and below the minimum price. Hereafter, I will plot only ATM and
OTM options above the minimum price, call and put, to chart the volatility
skew. Figure 3, then, looks like figure 4. Figure 4 shows an extreme reverse
skew.
Fig. 4 IV of ATM and OTM options over min price
The call-put gap
The call-put gap happens because of the
futures basis gap and can be eliminated by applying the adjusted underlying
asset (usually symbolized as "S") to the Black-Scholes model (Choi,
2012). In this example, S (K200 in Korea, the basket of 200 big companies) is
233.54, which is used in the computation of IV in the home trading system (HTS)
and by the regulating authorities. KOSPI200 futures is 233.55. The theoretical
value of KOSPI200 futures is 234.44 and so the futures gap is -0.89. The
adjusted S is calculated as K200+gap, in this case 233.54 - 0.89 = 232.65. IVs
calculated with this adjusted S are charted below, clearly showing the removal
of the gap.
Black-Scholes model requires the price of
the underlying asset S to calculate the price and all the Greeks. However, S,
KOSPI200 in this example, is not tradable practically. So the futures of
KOSPI200 is used in practice as a surrogate for option trading decision making
and hedging. The gap is, I think, solely due to this surrogation. The gap is
almost completely eliminated by applying the adjusted S. The gap is
proportional with the futures gap, i.e., the bigger the futures gap, the bigger
the option call-put gap. This is natural and purely computational because
arbitrage should make it happen: so-called compound futures made with some
combinations of calls and puts can be traded together with futures of the
opposite direction with some risk-free gains made instantly.
The smirk
The smirk (similar to skewness in
statistics but, to distinguish the smile or the shape, this word is used) is
surely related with the futures gap because the futures gap moves the ATM. In
this sense, the smirk is computational. However, it seems that market
psychology is also involved: first, the smirk seems to be too wide for the
futures gap; second, the Greeks seems to be distorted too much, which will be
mentioned in the shape issue.
The smile (shape)
I doubt what Natenberg mentioned about the
smile quoted above. I think that the smile (i.e., the further off the ATM, the
higher the IV) should reflect some cost or benefit. Natenberg's quote says that
the smile is basically psychological (he said "the marketplace appears to
believe") while I believe that there should be some computational or
mathematical factor(s). Among the Greeks, gamma and vega (kappa) are favourable
to options while theta is the opposite. So, it can be hypothesized that gamma
or vega (these two are highly correlated) demand higher prices for options
while theta accepts lower prices. The first step to take would be to check the
correlations of IV with Greeks. The results are very interesting. The correlation
coefficient between IV and gamma/vega are about -0.9 and +0.9 for theta in many
actual cases, if the strike prices of options do not include the lowest IV
point. However the smirk is a problem here too. Call or put that includes the
lowest IV have a parabolic shape while the Greeks are not, which makes the
correlation coefficient almost 0. Still, the correlation coefficient beyond the
lowest point is again about ±0.9. The smirk interval also has the correlation coefficient about 0.9
but with the opposite sign.
To neutralize the effect of smirk, instead
of removing the smirk range, I extended put plotting range into the ITM up to strike
price with the lowest IV and the result is charted below. The chart plots calls
the smirk range. The correlations are calculated
for puts with a range of X from 192.5 to 250, the lowest IV, and for calls from
252.5 to 270. The correlation coefficients are -0.94 for puts and -0.91 for
calls.
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Actual data
and chart (ATM 240, lowest IV 250)
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With adjusted
S and only over min. price. Call only OTM and ATM, put extended to the lowest
IV
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The following chart plots residuals after applying
linear regression with vega (same result with gamma). Please note that calls
are shifted upward to equalize the average of the residuals. The shape strongly
suggest (especially for puts) that these could be some other factor or variable
and the range of the residuals is too wide (out of the original IV range 23%
points, the regression eliminated only 14%, leaving the residuals with about 9%
range).
In conclusion, I hypothesize that the so-called
volatility smile does NOT exist if the IVs are compensated with the Greeks.
The cause/effect of the smirk needs further
study and tests.
End.
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