[R-SIG-Finance] Option pricing, basic question
Adam Ginensky
adamno227 at gmail.com
Thu Jun 9 17:07:12 CEST 2016
The comments below address the issue of drift and \sigma . As for the
interest rate used, I think you will find that the difference in the
values obtained by using different interest rates is on the order of
magnitude of rounding error.
There is a famous quote of Box - " All models are wrong, some are
useful ". This applies to Black-Scholes in spades. The B-S model is
almost universally used to price options, but there are a number of
important caveats about using B-S to price options. I would summarize
my comment by saying that the volatility of the underlying is merely
a guideline as to what volatility to use in pricing options. For at
the money options, the implied volatility ( the \sigma that will give
the value of the option) is a market consensus of what volatility will
be going forward. In times of high volatility, the implied volatility
will always be lower (and I say always in an informed, not theoretical
way) than the realized volatility and similarly in times of low
volatility the implied volatility will typically be higher than
realized volatility - both situations reflecting 'regression to the
mean' nature of market movement. When I was an active floor trader,
this was called 'tilt' by many people. I would sum this up by saying
that the implied volatility is the market consensus as to what
realized volatility will be going forward.
As another exercise, get the prices of relatively short term options
mid-day on the Friday before a 3 day weekend. The implied volatility
will typically seem absurdly low but will seem reasonable if the
volatility is computed with the date being 3 days forward. That is the
implied volatility will 'price in' the three day weekend . This is
also true for most ordinary weekends. Further, this is just a piece
of the issue. The Black-Scholes model significantly underprices out
of the money options. This is because the real world distribution of
log returns has 'fat tails'.
Finally, there is no such thing as 'the volatility'. For a given
option class, implied volatility is usually different at every strike.
This is in fundamental contradiction to the assumptions of B-S, but
reflects the reality of what happens in the real world. For stocks,
volatility of out of the money puts is almost always higher than the
volatility for out of the money calls. This reflects the fact that
markets usually go down faster than they go up as well as the reality
that people usually want insurance against stock prices going down,
not insurance against stock prices going up. This volatility curve is
dynamic. As the market goes higher, the difference between the
volatility of out of the money puts and the volatility of out of the
money calls will usually increase. When the market goes back down,
the volatility difference usually decreases.
The point is that modeling implied volatility is a complicated and not
well understood item. hth.
On Thu, Jun 9, 2016 at 1:02 AM, thp <thp at 2pimail.com> wrote:
> Hello,
>
> I have a question regarding option pricing. In advance:
> thank you for the patience.
>
> I am trying to replay the calculation of plain
> vanilla option prices using the Black-Scholes model
> (the one leading to the analytic solution seen for
> example on the wikipedia page [1]).
>
> Using numerical values as simply obtained from
> an arbitrary broker, I am surprised to see that
> the formula values and quoted prices mismatch
> a lot. (seems cannot all be explained by spread
> or dividend details)
>
> My question: What values for r (drift) and \sigma^2
> are usually to be used, in which units?
>
> If numerical values are chosen to be given "per year",
> then I would expect r to be chosen as \ln(1+i),
> where i is the yearly interest rate of the risk-free
> portfolio and \ln is the natural logarithm. Would the
> risk-free rate currently be chosen as zero?
>
> The \sigma^2 one would accordingly have to choose
> as the variance of the underlying security over
> a one year period. Should this come out equal in
> numerical value to the implied volatility, which is
> 0.2 to 0.4 for the majority of options?
>
> Tom
>
> [1] https://de.wikipedia.org/wiki/Black-Scholes-Modell
>
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