Comment on the Black-Scholes Pricing Problem

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Comment on the Black-Scholes Pricing Problem

Lev Mikheev

To cite this version:

Lev Mikheev. Comment on the Black-Scholes Pricing Problem. Journal de Physique I, EDP Sciences,

1995, 5 (2), pp.217-218. �10.1051/jp1:1995122�. �jpa-00247051�

J.

Phys.

I France 5

(1995)

217-218 _FEBRUARY 1995, PAGE 217

Classification

Physics

Abstracts

ol.75 02.50 ol.90

Comment

_on

the Black-Scholes Pricing Problem

Lev V.

Mikheev(*)

NORDITA, Blegdamsvej 17, DK-2100, Copenhagen kl,

Denrnark

(Received

31

August

1994, received in final for~n 4 October 1994,

accepted

17 October

1994)

In _a recent paper [1] Bouchaud and Sornette

presented

an

interesting analysis

of the Black-

Scholes derivative

pricing

model.

They

made _an

attempt

to rederive the model in _{terms more} familiar to workers in statistical

physics

and to extend it _to the _cases in which the

price

of the stock

(or

_any other

security) underlying

the

given

^derivative is a random process distributed

differently

from the

log-normal

law

generally accepted

in mathematical finance

[2].

Here I would like to

point

^{that the}

simple

rederivation of the Black-Scholes model

given

in [1] overlooks _a

major

achievement of that

theory:

its

ability

to puce the derivatives without

making assumptions

^{about the} _auerage ret~lrn _on the stock. All

arguments

^of _{[1] assume} that both

parameters characteriz1~lg

trie

log-normal

distribution of the stock

price,

the

volatility

(1. ^e. the diffusion coefficient of the

logarithm

of the

price)

and the average return

(the

drift

velocity

of the

logarithm

of the

price),

_are

known;

the _average return is taken _to be

equal

to the risk-froc interest rate. The authors then define the

price

of the derivative

by calculating

its average over the

given

distribution of the stock

price

at the

expiration

of the

derivative,

and then

proceed

to _recover the Black-Scholes nsk-free

portfolio (combination

of

selling

the denvative with

buying

a proper number of the

shares)

_as the

"optimal portfolio"

which minimizes the risk to the seller of the derivative.

This

presentation

of the Black-Scholes

theory

inverts its

logic. Pricing

a derivative via

calculating

its

expectation

^with respect to the

imaginary log-normal

distribution of the stock puce, in which the average return _is set

equal

to the nsk-free interest rate, ^{is in} fact well known in mathematical finance under the _name of "risk-neutral valuation"

[2].

^Its

possibility

however

is a consequence of the existence of the risk-free

portfoho

constructed

by

Black and Scholes.

A crucial element of the real

market, incorporated

into modern mathematical

finance,

is the trade-off between nsk and return. Different securities _are

generally

characterized

by

very different _{average returns} and

nsks; higher

_{returns are}

expected

from

risky

securities

(risk

bears its

price

for the

seller).

While the return and the

volatility

_are

positively correlated,

no quan- titative

relationship

between them _can be established without

making

additional

assumptions [3], except

^that a risk-free

portfolio (combination

of

secunties)

must return the risk-free rate.

By construction,

_m the Black-Scholes risk-free

portfolio

both the noise and the _average drift

are

completely compensated. Consequently

the Green function for the linear

partial

differential

equation describing

the time evolution of the _puce of the denvative does not contain the

average ^return

parameter.

Convolution of the

boundary condition, representing

^the known

(*)

mikheev@nordita.dk.

©

Les Editions de

Physique

1995

218 JOURNAL DE

PHYSIQUE

I N°2

value of the denvative at

maturity,

with this Green function turns out to be

equivalent

to the risk-neutral valuation. Note that if the average ^return on the stock _were known

(in _practice

it

rarely is)

and the

expectation

value of the

price

of the derivative _were calculated with the real distribution of the future

price

of the

stock,

the method used

by

the autuors would lead to an

error (1. e.

arbitrage

would be

possible)

tue correct

price

of tue derivative is net

equal

to tue

current

expectation

of its value at

expiration. Correspondingly,

tue results obtained in [1] for

models with time-correlated fluctuations of tue stock

price,

m which _case

constructing

_a riskless

portfolio

tutus out to be

impossible, apply only

to the

imaginary

risk-neutral

world,

but not to the

real,

risk-averse market.

Also,

in view of this

dicussion, calling

the Black-Scholes

portfolio

"optimal"

_seems

misleading (f~llly hedged

would be _more

appropriate):

_zero risk is

optimal only

under condition of

equal

returns; m the market in which

taking higher

risks

generally

^leads to

higher

returns absolute

optimization

is

impossible.

In summary,

apphcability

of the results obtained in [1] is restricted to the

imagmary

risk- neutral

market;

the risk-return trade-off

underlying

the Black-Scholes

analysis requires

more

elaborate treatment of the

problem.

Acknowledgments

I thank Ashvin Chhabra for

drawing

_my attention to

[1],

^{Glen Swmdle for valuable discussion} and Santa Barbara ITP for

hospitality.

This work _was

supported

in

part by

the National

Science Foundation under Grant No. PHYB9-04035.

References

[1] ^{Bouchaud J.-P. and Sornette} D., J.

Phys.

I France 4

(1994)

863.

[2] ^Hull J.

C., Options, Futures,

and orner denvative securities

(Prentice-Hall, Englewood Clilfs,

N.

J.,

1993).

[3] ^The

capital

asset pmcing model _[2]

yields

linear

relationship

between trie _{ezcess average} return of

a secunty _over that of trie rnarket

as a whole and trie

susceptibility

of trie fluctuations _in trie puce of the security to those of the market

(known

_as beta of the

security).

It is _very hard to

predict

the _retum of trie market, however.