Just curious about an hypothetical. Not an investing idea. The current stock price is $100. Within a certain time range, The probability of the stock price going up to reach $110 is 34% and the probability of the stock price going down to reach $80 is 20%. Note that the distance between $110 and $100 as compared to $80 and $100 is about 1/2. What is the probability that the stock price “within the given timeframe”: a). Does “not reach” either of $110 or $80. b) Reaches $110 "First" before it "May" reach $80. c) Reaches $80 “First” before it “May” reach $110. Assume random walk and normal distribution. Other assumptions that could give a rough estimate not a precise answer, just to simplify the problem. Like a regular maths/stats problem.
I could see someone (not me) working out the math for B or C but not A without a "... given timeframe"
IMO the given data is incomplete to answer the questions. B/c you gave probabilities for S from 100 to 80 and S from 100 to 110. But what is missing is how the remaining probability (100 - 34 - 20 = 46%) is distributed. IMO, w/o that information, answering the Qs is not possible (maybe except a). Update: given: p("100 to 80") = 34%, Srange=20 p("100 to 110) = 20%, Srange=10. But when we assume S=100 is the mean, then we can also assume symmetry (since ND), then: p("100 to 120")= 34%, Srange=20, pRest=32% --> div by 2 = 16% on each side, ie.: p("0 to 80") = 16%, and p("120 to +infinity") = 16% Now it should be possible to answer all Qs...: ...TBD... (to be done )
34% is about 1SD (see below), so now we can use also ND formulas to solve the Qs even more accurately!
Answers: a) p("Does not reach either of $110 or $80") = 100 - 34 - 20 = 46% (or (50 - 34) + (50 - 20) = 16 + 30 = 46%) Ie. it stays between 80 and 110, w/o touching, nor crossing, any of them. b) p("Reaches $110 First before it May reach $80") = 100 - 14 = 86% ("Within the given timeframe"). Hmm. or is it maybe just only 62.96% ? Or maybe 70% ? 'nuff done! I leave c for the others to solve
Solves for future price X: X = exp(sigma*t*x)*S where S = current price, sigma = volatility, t=sqrt(days/365), x = std dev. Solves for standard deviation x: x = log(X/S)/(sigma*t) Skew and kurtosis? Who needs them. Stochastic volatility? No problem.
a) is the price of a double-no-touch, formulas available online or in books (e.g. Haug) b) and c) are the ratios of two binaries, converted to percentages. You can approximate a binary with a very narrow vert, which will go to the binary price in the limit as the distance between strikes goes to zero. For a little extra accuracy, price the above undiscounted (that is assume RFR of 0%). Ignore any replies by Quanto, he's a multi-nick imbecile.
Got a question re volatility. Should volatility measure be the same length as in sqrt ( days/365 )? Not an ops guy if I sound ignorant. With this stuff I am.