http://arxiv.org/pdf/1307.5122.pdf
1 Introduction
Among many unrealistic assumptions made in the Black-Scholes model
[1], one is particularly problematic - constant volatility . When the
current market data are used against the Black-Scholes formula one
nds that must in fact depend on the strike K, and time to expiry
T, in order to make the pricing formula work. Therefore the market
data imply that is not constant but a function
(K; T) - called implied
volatility. The shape of the curve
I
I
(K; T) with T xed, is often
U shaped so that it became a standard practice to call it a volatility
smile. However that shape can also look more like a skew (a smirk)
or a frown depending on the data/market one is considering.
Clearly, the fact that
(K; T) is not constant falsi es the BlackScholes
model. However, it is also well known that this situation was
I
CRISIL Global Research and Analytics, Av. Libertador 1969, Olivos, Buenos Aires,
Argentina, e-mail: trzetrzelewski@myopera.com
y
Opinions expressed in this document are only personal views of the author.
1
Out[121]=
NORMALIZED FREQUENCY
60
50
40
30
20
10
completely di
erent before the market crash in late 80'. In the equity
market before 1987, the implied volatility was indeed fairly constant
- why it is not constant nowadays [2] ?
leave W
unchanged and assume that
is a function = (S; t) - called local volatility [4]. Then the smile
is explained by assuming that increases for large j ln S
t
j - if this is
the case then the tails of the Gaussian distribution will become fatter
leave W
unchanged and assume that
is a function = (S; t) - called local volatility [4]. Then the smile
is explained by assuming that increases for large j ln S
t
j - if this is
the case then the tails of the Gaussian distribution will become fatter.
There exists a way to nd the function = (S; t) directly using the
market data [5]. However it turns out that this model also has its
drawbacks i.e. while the smile can be accommodated, its dynamics
(the dynamics of the smile when the strike changes) is not captured
correctly. This brings us to further generalization by assuming that
itself is a stochastic process
One could explain this problem by blaming everything on yet another
unrealistic assumption of the Black-Scholes model - that the
underlier S
(where W
t
t
undergoes the geometric Brownian motion
dS
t
=S
t
= dt + dW
t
; 2 R; > 0 (1)
is a Wiener process). It follows form (1) that log-returns
(i.e. returns of ln S
) have Gaussian distribution. However it is very
well known [3] that the actual log-returns are not distributed like that
- instead they exhibit fat tails (Figure 1a). Therefore a rather nat-
t
aL REALHDJIL VS. GAUSSIAN DISTRIBUTIONS
-0.04 -0.02 0.00 0.02 0.04
0
DAILY LOG RETURNS UNTIL 2013
NORMALIZED FREQUENCY
60
50
40
30
20
10
bL REALHDJIL VS. GAUSSIAN DISTRIBUTIONS
-0.04 -0.02 0.00 0.02 0.04
0
DAILY LOG RETURNS UNTIL 1987
Figure 1: Distribution of daily log returns for Dow Jones (dotted) and the
corresponding Gaussian distribution (continuous). a) since 27 May 1896 to
10 May 2013, b) since 27 May 1896 to 2 Jan 1987. The mean and the height
of the Gaussian distribution are adjusted accordingly.
ural way to generalize (1) is to replace W
with the process whose
PDF exhibits fat tails corresponding to the ones observed in the markets.
However a careful inspection shows that this cannot be the main
reason of the volatility smile observed today. The point is that even
before 1987 the log-return distribution revealed fat tails (see Figure
1b; note that Mandelbrot's paper [3] was published in 1963) but at the
same time the Black-Scholes model was working well. This is clearly
an issue. If fat tails are the reason of all these discrepancies then why
the constant volatility assumption was correct before 1987?
t
Because of practical reasons the models that consider generalizations
of W
t
are not very popular and the development in this subject
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