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 <

[hidden email]> 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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