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Maciej Trzetrzelewski
y
Abstract
Black-Scholes equation, after a certain coordinate transformation,
is equivalent to the heat equation. On the other hand the relativistic
extension of the latter, the telegraphers equation, can be derived from
the Euclidean version of the Dirac equation. Therefore the relativistic
extension of the Black-Scholes model follows from relativistic quantum
mechanics quite naturally.
We investigate this particular model for the case of European
vanilla options. Due to the notion of locality incorporated in this way
one nds that the volatility frown-like e
ect appears when comparing
to the original Black-Scholes model.
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] ?
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
2
1
went in a completely di
erent direction. Instead of changing W
, nancial
practitioners prefer to 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 [6]
d
t
=
(
t
; t)dt + (
t
; t)dW
t
(here
and are some deterministic functions).This generalization is
counter intuitive: the amplitude , that multiplies the random factor
dW
t
, is stochastic now, but shouldn't dW
contain all the randomness?
Moreover, stochastic volatility models also fail in certain situations
e.g. in the limit T ! 0 where T is the time to maturity [7]. This
could be a motivation to generalize further and introduce jumps i.e.
discontinuous moves of the underling S
t
t
[8].
1
It is clear that this way of making the models more general is
likely to have little explanation power. These models may t very
well to the market data but in say 10 years from now they will most
probably fail in some situations and one will have to make some other
generalizations to t the market data again. This implies that the
stochastic volatility models are non falsi able.
For example, if we agree on the fact that volatility is a stochastic
process and satis es (2) then there is a priori no reason not to go
further and assume that is also stochastic. This would make our
model even better calibrated to the market data. The possibilities are
quite frankly unlimited and if it weren't for the fact that Monte Carlo
simulations are time consuming, they would certainly be investigated.
Because one can always augment the model in such way that it will be
consistent with the data, it follows that the model cannot be falsi ed.
Nevertheless most nancial practitioners prefer stochastic volatility
models because then, one can still use Ito calculus and obtain
t
The reader will note that the line of reasoning presented here di
ers from the chronological
way these ideas were considered. Jumps were introduced in 1976, three years after
the Black-Scholes paper, stochastic volatility in 1993, local volatility in 1994.
3
t
(2)
2
some analytical, robust results (otherwise, when dW
is not a Wiener
process, little exact results/methods are known [9]). It may seem unusual,
from the scienti c point of view, that robustness of the model
is used as a criteria of its applicability. However quantitative nance,
unlike Physics, is not about predicting future events but about pricing
nancial instruments today. Therefore as long as our models are calibrated
to the market, minimize arbitrage opportunities and are stable
against small uctuations of the data, there is a priori no problem in
the existence of plethora of possible models in this subject.
t
In Physics the situation is much di
erent. There, we care about
predictions and recalibration is not allowed. A theory that contains
parameters and degrees of freedom in such amount that can explain
any experimental data, by just appropriately tting them, cannot be
falsi ed and hence is physically useless
2
. For every theory, it is absolutely
crucial to have an example of an experiment which outcome
may, in principle, disagree with the results of the theory. This way of
thinking is in fact opposite to the way one proceeds in nance.
Stochastic volatility models are clearly very successful but just like
in the case of fat-tail distributions they will not be able to explain why
before 1987 the Black-Scholes model was working well. In fact if one
assumes that volatility is stochastic then clearly it must have been
stochastic before 1987 - which seems not to be the case (one could
still object to this point by saying that before 1987 the volatility was
stochastic but with a tiny mean-reversion amplitude and hence the
model could be approximated by constant volatility).
In this paper we would like to approach these issues from a di
erent
perspective. It is well known that algorithmic trading became more
and more popular in the 80' - increasing the changes of the prices,
per second
3
. However there exists a concrete underlying limitation
for market movements: the change of any price S(t) cannot be arbitrary
large per unit of time i.e. there exist maximal speed c
such
that
_
S(t) < c
M
(market speed of light, [c
M
] = s
1
M
). An obvious proof
At this point it is worth noting that in theoretical physics there are constructions
(such as string theory) which su
er from making no predictions in this sense.
3
There is a common belief that the crash in the 80' was due to algorithmic trading. We
do not share this point of view. Following R. Roll's argument: if the algorithmic trading
was to blame the crash would not have started in Hong-Kong where program trading was
not allowed yet.
4
of this assertion comes from the fact that the speed of information
exchange is limited by the speed of light. It seems that this limitation
should not be very restrictive since light travels about 30cm per
nano-second(ns). Assuming that servers of two counter parties are,
say, 30cm from each other, it takes at least 1ns to send an order.
Therefore we should not see any relativistic e
ects, unless we are considering
situation in which there are at least billions (10
) orders per
second, sent to a single server. At this point it is clear that future
development of high frequency trading may in principle inuence the
situation considerably.
However there is one feature of every liquid market whose consequences
are seen already and hence we would like to discuss it in more
details. Any price S(t) going (say) up from S(t) to S(t + t) > S(t),
must overcome all the o
ers made in the interval [S(t); S(t + t)].
This introduces a natural concept of friction/resistance in the markets
simply because there is always somebody who thinks that the
price is too high. This situation is similar to what happens in physical
systems e.g. electrons in conductors. An electron can a priori move
with arbitrary (but less than c - the speed of light) velocity. However
due to constant collisions with atoms of the conductor the maximal
velocity is in fact bounded even more. The drift velocity of electrons
can be as small as e.g. 1m=h. Perhaps a better physical example is
light traveling in a dense media where the e
ective speed of light is
c=n where n is the refractive index (e.g. n = 1:3; 1:5; 2:4 for water,
glass and diamond respectively). In extreme situations, when light
travels through the Bose-Einstein condensate, the e
ective speed of
light can be as small as 1m=s [10].
To see that this resistance e
ect is big in the markets let us consider
the logarithm x(t) = ln S(t) and the corresponding bound on the
derivative of x(t)
j _x(t)j = j
_
Sj=S < c
=S:
If we assume that the order of the underlying is about 100$ and that
c
M
is at least 10
9
s
1
M
then we obtain j _xj < 10
7
s
1
. On a daily basis
this implies that the di
erence
x := jx(day) x(previous day)j
can a priori be as big as 10
7
3600 24 = 8:64 10
12
. However at the
same time nothing alike is observed in the market. The value of x
for any asset was, to our knowledge, never bigger than 1. We have
5
9
analyzed top 100 companies (considering their market capitalization
as of March 2012) of the SP500 index. We order them w.r.t. decreasing
maximal absolute value of their log-returns. The list of rst 15 of
them is presented below.
Company log-return market move (close) date
WMT -0.735707 0.0192 ! 0.0092 Dec 1974
AAPL -0.730867 26.18 ! 12.60 Sep 2000
INTC 0.698627 0.0091 ! 0.0183 Jan 1972
C -0.494691 24.53 ! 14.96 Feb 2009
ORCL -0.382345 0.61 ! 0.42 Mar 1990
PG -0.360296 40.43 ! 28.20 Mar 2000
MSFT -0.356939 0.37 ! 0.26 Oct 1987
BAC -0.34205 7.10 ! 5.04 Jan 2009
QCOM 0.327329 8.47 ! 11.76 Apr 1999
JPM -0.3241 6.03 ! 4.36 Oct 1987
MRK -0.311709 40.90 ! 29.95 Sep 2004
AMZN -0.296181 16.03 ! 12.06 Jul 2001
KO 0-.283731 2.37 ! 1.78 Oct 1987
WFC 0.283415 19.37 ! 25.72 Jul 2008
IBM -0.268241 32.35 ! 24.74 Oct 1987
In the table we also added the corresponding movement of the stock
price (close) and the date for reference. The biggest change of the logreturn
is due to Walmart and Apple (note that the historical data we
use are subject to adjustments for stock splits) whose shares dropped
resulting in almost the same loss (in terms of log-returns). In any case
we see that the magnitude of log-returns may be of order of 10
, not
10
12
.
This implies that there is a huge resistance in the market for the
price to move up or down (notice that some of the log-returns in our
table are positive, e.g. for Wells Fargo). Therefore one may conclude
that the e
ective maximal velocity of S(t) is much smaller than c
.
For completeness we also performed the same analysis for other markets.
Below we present the results for Forex majors, some precious
metals and major indices.
6
0
M
Forex major log-return market move (close) date
AUDUSD -0.192451 1.2304 ! 1.015 Nov 1976
UDSCHF 0.103529 1.9949 ! 2.2125 Dec 1982
EURUSD 0.0619917 0.6192 ! 0.6588 Feb 1973
GBPUSD 0.0459699 1.1819 ! 1.2375 Mar 1985
USDCAD -0.0388492 1.2688 ! 1.2218 Oct 2008
USDJPY -0.0950101 293.26 ! 266.68 Feb 1973
Commodity
XAGUSD -0.222755 12.87 ! 10.20 Feb 1983
XPTUSD -0.221841 594.6 ! 476.3 Mar 1980
XAUUSD -0.203157 809.9 ! 661 Jan 1980
Index
DJI -0.256325 2246.7 ! 1738.7 Oct 1987
SPX -0.228997 282.7 ! 224.84 Oct 1987
NKX 0.200503 135.89 ! 166.06 Aug 1951
NDX 0.17203 2128.78 ! 2528.38 Jan 2001
DAX -0.137061 1589.28 ! 1385.72 Oct 1989
FTM -0.119613 2528.55 ! 2243.49 Oct 1987
Again, all the log-returns are small. This con rms our claim that
the maximal value of j _xj is smaller than 1 per day. We will use the
notation c
for the upper bound of j _xj.
In the next section we present a basic idea investigated in this
m
paper - the existence of the bound on log-returns implies that the corresponding
PDF, p(x; t), cannot be positive everywhere but must be
0 for jxj > x
max
:= c
t. This generically introduces a skew/smirk of
the volatility when comparing to the Gaussian distribution. Based on
the market data analysed above we claim that this e
ect can in fact
be noticeable. The main question is then, in what way we can generalize
the Black-Scholes model so that the niteness of c
m
is taken
into account. Towards this direction it seems natural we study the
relativistic generalization of the di
usion equation. One could object
that such relativistic extension is a bit arti cial. After all, using the
analogy of an electron in the conductor, the electron is only slowed
down to drift velocity and no relativistic e
ects occur at this speed.
This argument is of course true in generic cases. However there are
examples of conductors (graphene surfaces, for a review see e.g. [11])
for which description of electrons is e
ectively given by the massless
Dirac equation i.e the description is relativistic even though the elec-
7
m
tron's speed is still not even close to the speed of light c (it is about
1% of c). This is due to a particular honey-comb lattice structure of
the graphene. It is therefore a physical example of the non-relativistic
processes whose e
ective description nevertheless requires relativistic
equations due to the speci c structure of the environment. We see no
reason why a phenomena of this kind could not take place in nancial
markets.
In Section 3 we review the correspondence between the relativistic
di
usion equation, the telegraphers equation and the Dirac equation
found a few decades ago [12, 13, 14, 15, 16, 17]. The di
usion equation
can be obtained from the telegrapher equation in the limit v ! 1,
where v is the velocity of a particle. Since the Black-Scholes equation
is equivalent to the di
usion equation in the x = ln S variables,
we make a proposal that its proper relativistic extension is given by
the telegrapher equation with v replaced by c
(Section 4). As a
result, in Section 5, we arrive at a pricing formula for options and
present numerical analysis for option prices, put-call parity and implied
volatility. In particular we nd that the proposed formula allows
for arbitrage opportunities. In the region of parameters where put-call
parity is not violated signi cantly we calculated the implied volatility
and nd a volatility-frown like e
ect. Lastly we perform the 1=c
expansions and nd exact formula for 1=c
2
m
m
corrections (1=c
terms
give no contribution). This result can then be used to evaluate the
implied volatility exactly when c
2 Basic idea
m
is large
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