Hypothetical: Probabilities and price targets

M

MrAgi1

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.
 
MrAgi1 said:
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.
More...
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 :))
 
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Quanto said:
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 :))
More...

34% is about 1SD (see below), so now we can use also ND formulas to solve the Qs even more accurately!

ND.png
 
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 :-)
 
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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.
 
MrAgi1 said:
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.
More...

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.
 
panzerman said:
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.
More...
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.
 
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