A Guided Walk Down Wall Street: An Introduction to Econophysics

1056 Giovani L. Vasconcelos
We then have
where
dZ = −re −rt Sdt + e −rt dS
= (µ − r)Zdt + σZdW
= σZd ˜W ,
˜W (t) = [(µ − r)/σ] t + W (t). (116)
Now recall that in the Black-Scholes model the stock price
follows a geometric Brownian motion
dS = µSdt + σSdW, (117)
which in terms of the process ˜W (t) given in (116) reads
dS = rSdt + σSd ˜W . (118)
From Girsanov theorem we know that there is a equivalent
martingale measure Q that turns ˜W (t) into a Brownian
motion. Equation (118) then shows that w.r.t the measure
Q the price S(t) follows a geometric Brownian motion
with mean rate of return equal to r. The probability density
f Q (S ′ , T ; S, t) can now be obtained directly from (54), by
simply setting µ = r and S 0 = S. One then gets
f Q (S ′ , T ; S, t) =
where τ = T − t. Inserting (119) into (114) we obtain
F (t, S) =
⌋
1
S ′√ 2σ 2 τ exp ⎧
⎪⎨
⎪ ⎩
−
−r(T −t)
e
√
2πσ2 (T − t)
∫ ∞
0
[ (
ln
)
S ′
S
− (r − 1 2 σ2 )τ
2σ 2 τ
] 2
⎫
⎪⎬
⎪⎭ , (119)
Φ(S ′ )e [ln(S′ /S)−(r− 1 2 σ2 )(T −t)] 2 dS′
S ′ , (120)
⌈
which is precisely the expression obtained for the generic
solution of the Black-Scholes equation given in (92). In the
case of a European call option we have Φ(S ′ ) = max(S ′ −
K, 0), which inserted into (120) yields, after some algebra,
the Black-Scholes formula (84).
It is interesting to notice that under the equivalent martingale
measure Q the stock price in the Black-Scholes
model follows a geometric Brownian motion with the mean
rate of return equal to the risk-free interest rate r; see (118).
It is as if all investors were risk neutral, in the sense they
would be willing to invest on a risky stock even though its
expected return is just what a risk-free bank account would
yield. For this reason, the pricing method based on the
equivalent martingale measure is commonly referred to as
risk neutral valuation. Of course, actual investors are not
risk neutral. However, in an efficient market there is a ‘special
reference frame’ where investors can be treated as if
they were indeed insensitive to risk.
6 Beyond the Standard Model of Finance
I: Non-Gaussian Distributions
We have seen above that the Black-Scholes model is an elegant
and powerful theoretical construct: it is complete, efficient,
arbitrage-free, and Gaussian (in the sense that the
stock returns are normally distributed). It is thus important
to ask whether real markets actually fit into this nice framework.
There are two main ways in which real markets may deviate
from the standard Black-Scholes model: i) the returns
may not be normally distributed or ii) there may exist longmemory
effects on the time series. In this Section we will
discuss the possibility that asset prices may follow a non-
Gaussian stable Lévy process, while in the next section we
investigate whether financial data might exhibit long-term
memory.
Mandelbrot [22] in 1963 was perhaps the first person
to challenge the paradigm that returns are normally distributed.
He analyzed cotton prices on various exchanges
in the United States and found evidences that their distribution
of returns decays as a power law and hence much slower
than a Gaussian. An example of this ‘fat tail’ behavior can
be seen in Fig. 8, where I plot the distribution for the returns
of the Ibovespa index. In this figure, it is also shown a Gaussian
distribution with the variance of the data, which appears
as a parabola in the linear-log scale of the graph. One clearly
sees that the empirical distribution does indeed have ‘fatter
tails’ when compared with the Gaussian distribution.
The Gaussian distribution is special for two main reasons.
The first one is the Central Limit Theorem [21] that
states that the sum of infinitely many independent random
variables (with finite variance) will converge to a Gaussian
variable. The second one is the fact that it is a stable distribution,
in the sense that the sum of two independent Gaussian
random variables is also a Gaussian variable. It is thus
natural to ask whether there are other stable distributions.
The French mathematician Paul Lévy showed that there is
indeed a whole family of stable distributions of which the
Gaussian is but one particular case. In what follows, I will