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I have attempted to use the fact of having implied volatility, but have not been able to come up with a viable way to calculate the probability, any ideas? Suppose that a stock $S_t$ follows a lognormal model and that on May 29,2019 the closing price of the stock was $S_0$ was 61.5 and the implied volatility of the options with maturity T=0.4 was 120%. If we assume that the annual return required by the investors is 30% what is the probability $P(40\le S_t \le 55)$

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    $\begingroup$ On top of hypothesizing a (system of) equation(s) that describe the stock behavior, you could even follow a data-driven approach by plugging your implied volatility into some classification model to train and get a probability as output. But that would just be a toy model, don't expect to earn money with that. $\endgroup$
    – Lisa Ann
    Jun 11 '20 at 11:45
  • $\begingroup$ @lamdaepsilo: if the answers below are satisfactory, could you pls accept one of them (by clicking on the tick mark) so that your answer can be marked as "answered"? $\endgroup$ Jun 11 '20 at 19:35
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I assume you want to real-world probability, because the risk-neutral probability is not a probability in the 'likelihood' sense.

Under the real-world measure, we model the stock under the B-S model as:

$$X(t)=X(0)+\int^{t}_{0}\mu X(h)dh+\int^{t}_{0}\sigma X(h)dW(h)$$

If the market demands a 30% annual return, I will take that as the real-world rate $\mu$. Strictly speaking, we should also take the volatility estimated from a historical time series if we deal with the real-world measure, but I will just take your implied vol here for simplicity:

$$X(t)=61.5+\int^{t=0.4}_{0}0.3 X(h)dh+\int^{t=0.4}_{0} 1.2 X(h)dW(h) = \\ = 61.5exp \left( \left[ 0.3 - 0.5* 1.2^2 \right] 0.4 + 1.2 * \sqrt(0.4) Z \right) = \\ = 61.5exp\left( -0.456+0.759Z\right)$$

Therefore:

$$\mathbb{P}\left( 40<X_t<55\right)=\mathbb{P}\left(X_t<55\right)- \mathbb{P}\left( X_t<40\right)$$

Now:

$$\mathbb{P}\left(X_t<55\right)=\mathbb{P}\left(61.5exp\left( -0.456+0.759Z\right)<55\right) = \\= \mathbb{P}\left(ln(61.5) +\left( -0.456+0.759Z\right)<ln(55)\right) = \\= \mathbb{P} \left( Z< \frac{ln(\frac{55}{61.5})+0.456)}{0.759} \right) $$

You can do the same for $\mathbb{P}\left( X_t<40\right)$, work out the numbers yourself and you should get the answer.

Important Note: The above was just to demonstrate how real world probability could be calculated by blindly plugging numbers into the B-S model. However, pls note that if you want an actual real probability of a stock ending up within a specific range, the B-S model framework is not really suitable for that. Every market agent will have his or her (Bayesian) view of the state of the world and every market agent will view the probabilities differently. Even the choice of model that you will use to compute the probability is a Bayesian choice in itself. It's a really interesting problem, but its more of an "existentialistic" problem, rather than "practical" problem. High-frequency algo traders try to estimate probabilities all the time. They all use different models, different input data, etc...

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  • $\begingroup$ Hi there was a typo in my question I meant to say $P(40\le S_.4 \le55)$ it seems that you understood this. Just double checking that you accounted for this in the solution correct? $\endgroup$ Jun 11 '20 at 23:30
  • $\begingroup$ Also thank you sincerely I greatly appreciate it. $\endgroup$ Jun 11 '20 at 23:31
  • $\begingroup$ Yeh, I did the computation with $t=0.4$ in mind. Once you work out the probabilities, let me know what you get!!! :) $\endgroup$ Jun 12 '20 at 7:07
  • $\begingroup$ Did you arrive at this computation yourself, it does not make mathematical sense, in reality. I have never heard nor seen anyone use it. For real formula's regarding option derivations as a source please refer to Espen Haugs 2007 publication complete option priciing. $\endgroup$ Nov 11 '20 at 17:49
  • $\begingroup$ @nhoj: read the question carefully, it says "suppose the stock $S_t$ follows a lognormal model" => Lognormal model means Geometric Brownian Motion, and that is how I answered the question. As I point out in the Important note section of my answer, the GBM model is not suitable for inferring real-world probabilities (which I suppose is consistent with your criticism). $\endgroup$ Nov 12 '20 at 9:57
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Asset prices follow a random walk, so assuming probabilities and forecasting stock prices are not that accurate. Hence, investors try to project volatility rather than asset prices (i.e. implied vol) using GARCH, EWMA, or other vol forecasting models.

The optimal portfolio is to invest long term in a globally diversified portfolio with a focus on uncorrelated asset styles (growth assets, real assets, and hedge assets) or asset classes.

If you figure out a way to project asset prices, with somewhat strong accuracy. Please let me know so that we can set up an investment management firm 😗. Joking aside, I wish you the best of luck on figuring this out and I look forward for your findings.

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  • $\begingroup$ You are assuming the conventional wisdom as Gailbraith described worn out academic chestnuts, volatility arbitrage pairs trading and indeed volatilty pumping of stocks using Kelly criterion are far more profitable, and yes more obscure and out of the mainstream methods, and not on the MBA CFA curriculum, and they are less risky. And there is dozens of other methods besides the old hoary chestnut of buy and hold a diversified portfolio, of which Warren Buffet really never did or does, he is a full Kelly trader. These are all modern quantitative trading methods. $\endgroup$ Nov 11 '20 at 17:57
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There is a very simple method used by real traders going back to the sixties, it assumes as given or ceteris paribus as is the economics jargon, that you use an exponential growth formula, for eg. say volatility either Historical or implied of .3 or 30% then:

Expected move equals time 30 days / 252 trading days = .12 = sqrt(.12) = .364 Then volatility * adjusted time = .3*.364 = .10 then priceexp(.1) = say the price of $10 then 10exp(.10) = then the range of the price is 1/10 or 10% up or down over 30 days given the assumption of 1 standard deviation movement. It is a simple method used by real traders in the pits throughout the sixties, seventies, and eighties.

Similarly you can estimate the probability of a price reaching a threshold whether it be a stock price or option strike price by; ln(expected price/ Price)/ (Volatility*sqrt(time)) ln = naperian logarith ln not log 10 This can be modified with the above formula to estimate the future probability, rather than immediate. Why use implied volatility? Because it is the markets estimate of future volatility(movement- ie potential or risk up or down) as opposed to historical volatility which is the realized statistical variation around the mean.

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  • $\begingroup$ Garch and similar estimates are fine as a benchmark for comparison, and great for doctoral thesis and to fill up academic journals no real option trader uses it, or else you would go broke very quickly, it is useful as a benchmark against the implied volatility, implied volatility is where the money is. $\endgroup$ Nov 11 '20 at 17:44

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