probability distribution formula?

I resurrected this thread for a specific reason. Then it got hijacked.

Let's try again:

Thanks in advance.

 

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Kut2k2 said,

Looks like an ordinary Black-Scholes calculator. What's so special about this one?

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It uses a Monte Carlo generator which tells if either limit (upside or downside) is ever touched.

Peace and gtty,

Lar

 

Hi Kut2k2,

Here is a more in depth description directly from the calculator's help link.


What is a Monte Carlo Simulation?

Monte Carlo is, of course, the famous gambling city in the small European country of Monaco. If you had an unlimited amount of money, you could go to Monte Carlo and play the same number on, say, a roulette wheel multiple times. After a large number of times (each time is called a “trial”), you would have a good estimate of how likely it is that your chosen number would appear.

The Monte Carlo Simulation technique can be thought of as spinning a computer-based roulette wheel, and counting how many times a certain outcome occurs. The secret to success is to program your “computer roulette wheel” to emulate whatever real-life process it is that you are trying to learn about.

For the Calc2000 program, the computer is emulating the daily changes in stock prices, based on the model developed by Black & Scholes in their Nobel Prize-winning work on options. For each trial, the computer will randomly emulate the changes in stock prices that could occur for a number of trading days. It uses the supplied volatility number to determine how far a stock might move on a particular trading day. It is very important to supply accurate volatility numbers.

The price is watched during each trial to see if it exceeds the upside price, the downside price, or both. Each trial uses a different set of random numbers, and so will have a different outcome. After running all the trials, the program shows the percentage of the number of times that each event occurred.

Another key to success is to run your emulation for a large number of trials. The “Law of Large Numbers,” from the field of Statistics, states that over a large number of trials, the relative frequency with which an event occurs will approach the probability of its occurrence for a single trial.

If the number of trials is large enough (we recommend at least 10,000 trials), and the volatility numbers are accurate, then the relative frequency with which prices are exceeded will be a good estimate of the probability for any one trial. Of course, this is only one of many pieces of information that a trader would use when determining whether to buy or sell stock, stock options, or combinations


Peace and gtty,

Lar

 

Hi, Lar

Thanks for replying, especially in such detail.

The reason why I asked is that Monte Carlo simulations (MCS) are generally used when the probability distribution is unknown. We can calculate the exact probability of an "ideal" roulette wheel. But if we wanted to know the pdf of a particular wheel in a particular casino, because we believe that wheel has a defective balance or the old gamblers' tale that red-painted grooves were "stickier" than black-painted grooves might apply or something else was at work, then applying MCS to hours of collected data on that one wheel might make sense: to detect any significant deviation from the ideal wheel. But using MCS to approximate a known probability distribution (e.g., Black-Scholes gaussian) doesn't make much sense to me.

 

Hi Kut2k2,

I don't know how to respond usefully so I went to McMillian's site and found a contact for questions.

QUESTIONS? Contact Nathan 973-362-4558 for assistance.


I personally like the fact that I can get some kind of quantitative idea of a strike being touched before expiry. Admittedly, the markets do not really behave in a gausian manner but part of the game is knowing the left tail dwindles to zero and is a bit fatter; and the right tail is fatter and can be much longer.

Peace and gtty,

Lar

(Let us know if your questions were answered usefully or not)

 

I'm still trying to understand what's the point of the whole discussion.

Probability of the option being in the money is your delta (in forward measure), that would be calculated as

p(ITM, +/-) = N(+/-d)

where

d= ln(F/K)/vol*sqrt(t) - vol/2*sqrt(t)

and +/- would be call/put respectively.

Probability of the option hitting a barrier before or at an expiry date is a bit more complex, but can be solved for analytically too. I can post VBA code for both, (these are also pricing formulas for european and american digitals).

 

probabilities for former geniuses who are now just everyday joes:

ok, can anyone recommend something?