Wednesday, February 12, 2014

bs01 local volatility is a function = (S; t), increases for large ln S(t)

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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