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Re: Black-Sholes ?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg15151] Re: [mg15104] Black-Sholes ?
*From*: Brian Boonstra <boonstb at cmg.FCNBD.COM>
*Date*: Wed, 16 Dec 1998 03:11:24 -0500
*References*: <199812120859.DAA04911@smc.vnet.net.>
*Sender*: owner-wri-mathgroup at wolfram.com
Hi Yves
What you have written there looks like a (slightly wrong) version of
the Black-Scholes SDE/PDE, whose solution can be found in the book
"Options, Futures, and Other Derivative Securities", by John Hull.
To answer your question about the assumption that logarithms of price
returns are normally distributed: the Black-Scholes PDE arises from
this assumption, and can be solved in closed form (see below) only for
particular payoff conditions, such as the European call. Other payoffs
(like American options) give rise to boundary conditions that force
you to integrate the PDE numerically, a more complicated matter.
If you wanted to consider different distributions, you should pay
attention to the fact that the Black-Scholes analysis makes the
lognormal assumption for infinitesemal time increments dt, which then
adds up (even for large time increments), to another lognormal
distribution. If you choose another time-independent distribution
for rates of return in infinitesemal times, you will still get
lognormal distributions over finite times because of the Central
Limit Theorem (some technicalities ignored).
If you want to make the assumption that the distribution is
non-lognormal over a period of, say, 1 year, then you must pay
attention to what that does to your distribution after two such
periods - 2 years. In general, they will no longer have the same
shape. And of course, the behavior in infinitesemal time would be
murky as well. So a rigorous analysis might escape you.
That said, there is a certain amount of literature out there
investigating different probability distributions, notably to explain
volatility "skew". I might start with the tree scheme found in RISK
magazine a few years ago (republished in their book "From
Black-Scholes to Black Holes").
We can find the value of an option only because there exists a
replication, hedging, and arbitrage strategy that forces its value to
be fixed relative to the prices of other securities (e.g. the stock),
independent of people's risk preferences. This strategy is what allows
you to go from the Black-Scholes SDE to the Black-Scholes PDE.
Without the strategy, there is no unique value to the option. The
value of an option in the presence of such a strategy is something
like
V = Expectation_{all paths} [ V(path) ]
which works out to be an integral in the case of vanilla options. This
is explained best by books like Baxter and Rennie's "Financial
Calculus".
The vanilla call:
\!\(\(-\(1\/2\)\)\ E\^\(\(-r\)\ \((\(-t\) + T)\)\)\ X\
\((1 + Erf[
\(\((c - sigma\^2\/2)\)\ \((\(-t\) + T)\) +
Log[S\/X]\)\/\(\ at 2\
sigma\ \ at \(\(-t\) + T\)\)])\) +
1\/2\ E\^\(\(-\((\(-c\) + r)\)\)\ \((\(-t\) + T)\)\)\ S\
\((1 + Erf[
\(\((c + sigma\^2\/2)\)\ \((\(-t\) + T)\) +
Log[S\/X]\)\/\(\ at 2\
sigma\ \ at \(\(-t\) + T\)\)])\)\)
The vanilla put:
\!\(1\/2\ E\^\(\(-r\)\ \((\(-t\) + T)\)\)\ X\
\((1 - Erf[
\(\((c - sigma\^2\/2)\)\ \((\(-t\) + T)\) +
Log[S\/X]\)\/\(\ at 2\
sigma\ \ at \(\(-t\) + T\)\)])\) -
1\/2\ E\^\(\(-\((\(-c\) + r)\)\)\ \((\(-t\) + T)\)\)\ S\
\((1 - Erf[
\(\((c + sigma\^2\/2)\)\ \((\(-t\) + T)\) +
Log[S\/X]\)\/\(\ at 2\
sigma\ \ at \(\(-t\) + T\)\)])\)\)
- Brian
"Yves Gauvreau" wrote:
> The famous Black-Sholes solution for pricing derivative is based on the
> assumption that the log of price returns are normally distributed. Now
> suppose that the distribution of stock price returns is not normaly
> distributed as many authors suggest. This would meen that we have to
> derive a new equation for the derivative taking into account this other
> distribution.
>
> Also suppose you have another distribution to investigate. How someone
> could approch this problem to find a solution ?
_______________________________
Dr Brian K Boonstra
Vice President, Quantitative Research
First National Bank of Chicago
1 First National Plaza
Chicago, Illinois 60670
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