[R-SIG-Finance] options/BS/MC

[rex]

neal smith <nealsmith314 at gmail.com> [2011-05-14 09:23]:
>However we are told that the option price doesn't depend on the mu of
>the stock, and there are stocks which have consistently higher mu's
>(BRK, GOOG) than other stocks (YHOO) for a similar level of vol.  So
>let's say I have a high-mu stock:
>
>mustocks<-exp(rnorm(10000, mean=.15, sd=vol))
>mucallvals<-pmax(0.,mustocks-strike)
>print(summary(mustocks))
>   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
> 0.8155  1.0850  1.1630  1.1680  1.2440  1.7400
>
>print(summary(exp(-.02)*mucallvals))
>   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
>0.00000 0.08153 0.15670 0.16590 0.23860 0.73860
>
>so i can buy this call for the BS price of $.05, which equals its
>discounted payoff in RN, but now on this high-mu underlying i get a mean
>discounted value of .165, ie. 200% mean roi, with the lower quartile
>giving a paltry 60% return.
>
>what am i doing wrong?

Nothing, unless you expect the values to be equal. The BSM value is
based on a hypothetical riskless hedge, but buying the option is
anything but riskless, and the expected future price of a high-mu
stock (and hence the call option) will be far different from the
expected future price of a stock with a mu equal to the riskless rate.

I found the Wilmott thread mostly unhelpful, and so I've put up a page
using R to illustrate the issues. I hope it's useful.

http://www.nosyntax.net/cfwiki/index.php/Trend_vs_option_value

Anyone may edit the page or the discussion page, and I encourage
comments on the discussion page. There may well be typos, but I hope
there are no conceptual blunders.

-rex
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