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Black Scholes Monte Carlo Call Options Example Using Excel - YouTube
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we determine the European call option
numerically here
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in particular
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Using Monte Carlo this just means
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that we make multiple simulations so we model
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the share with a trend term
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and a noise term and so we see
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that if we simulate that many hundred
times
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what do we end
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up as. what would the call option be worth
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in each of those cases there is though
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the issue of risk neutral pricing though we only touch on that
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by that I mean that when you price using Monte Carlo
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we take the trend term
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as being the risk-free rate which is
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not obvious I mean if
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say you're trying to work out what the
call option is
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and you talk about a share which is
growing at
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20 percent a year for the last two or
three years
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then we're pricing that the same
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as if were talking about a share which
is growing at two percent
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or even declining so
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why are we allowed to do that that's a
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deeper question which
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will create content for further on down
the line
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but here we're just gonna use Monte
Carlo
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and show that we get the same answer
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as we do as we get when we do it
analytically
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so we saw
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that we could evaluate the price
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of a call option using black scholes in
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the first video where the result that we
got
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matched with a known correct answer
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where we know that if we
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set the five parameters to these values
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that the answer is equal to this from Hull
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so here
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we want to now evaluate the call option
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using Monte Carlo. in Monte Carlo what we
do
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is we stimulate the share price
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many many times so we go
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okay the share price starts at 42
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and then we know how it
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evolves me know that there's a drift term
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which is equal to 0.1 and we'll go into that, and there's
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a noise term like the weiner process
where sigma
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is equal to 0.2
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we price the option
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in a risk neutral world
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to what that means is that we set the
trend
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termed the growth to be 0.1
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which is the risk-free
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interest rate so you know you should be going
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wait...the share I'm talking about is
a high growth stock
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surely I should use mu based on what the
value
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on mu is four that
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share historically
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it turns out that
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you don't
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you basically use
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the you use the theorems of financial
maths
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which tell you that being a complete
market
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you want to use risk neutral pricing
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because that will give you the correct answer. essentially
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if you were to use the trend
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growth rate of your share that should
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equate to a higher risk level and that
high risk level
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should mean that you will be charged a
higher interest rate
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if the
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thing your trading with is being traded
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in a market that should be happening
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otherwise there's arbitrage so since
we're using the risk free
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interest rates to price things
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as opposed to
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risk-adjusted interest rate we have to use
or we're allowed to use
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mu equal to r
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also if he used our
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you get the correct if you don't want to
go into it
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in greater depth bodies
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be step
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in itself which'll copper in a later
video
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so what's done here
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well this is standard Excel we just
defined
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we write a function to see that you just
press
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Alt f11 and
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here's are function so
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inch or what it does is
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it breaks the time we're talking about
as into
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an steps so you know if your
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stimulating for one year I could have
say and as good a hundred
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and we move in increments of a hundred
to be here
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that's one of the primers we have in the
function otherwise
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wheat acres arguments the various
problem occurs
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Associated Wade the actual model
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so the initial the current share price
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to strike price the risk-free interest
ratio and Sigma
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that follows the time to Dan
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this is a standard programming so here
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we have a loop where we
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simulates one iteration
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so in the J loop
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where breaking the
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time that we're talking about automatons
a one-year will break that it and steps
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and each in each one we
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increment
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here's the share price s01e03 s1
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both s1 star mu
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to be the trend growth star delta T
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plus s1 star Sigma so BB
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the been a process term where you have
multiplied by the square root
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dirty and if you're familiar with beaner
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you would know that the other square
root for Delta term
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and if you go pee oases grow to Delta
illegal to
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the squarish opt out
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the reason I do dodge is so that I own
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you know I work out what the square root
adult is once
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at the beginning instead of doing it
every time again and again
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so that'll mean that I won't do as much
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computation I'm not wasting CPU time
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in this
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we see we are Aki have s1 and s2 are
going to the reason
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why we do the last in the next video but
basically we are recommending the share
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price
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say it began death said we're setting s1
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declare 0 the initial price and there
were implementing its by
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the standard not normal process
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whether the note random you and a beaner
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noise term dan after we've
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you im after we've simulated
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you know when time period so does a
one-year
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or 0.5 in disparate specific example
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damn we end up with up a particular
value of the share price
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what such a radical to after
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year or whatever now since where
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how to strike present K I have s1 is S
one minus K
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and if s1 is less than 0
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so if the share price is below the
strike price well then s01e02 20
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because in the final par
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s1 illegal to the value of the option
not
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great for a coat clarity but does the
job
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if the share price is higher than the
strike price
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ban s1 is s1 times
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well it's discounted I the risk-free
interest rate
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so in that one iteration we get older
share price for the quarter this
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and if it the share prices equal s1
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at the and after time t it is
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currently work s1 times exponential
minus
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RT so the total
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is able to
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s1 times and pods
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so it's divided it's divided by
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since we're gonna and open and parth
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we're gonna end up in s1 and pop the
different times
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so each other different simulations we
get an S on
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book since we simulate and pop times
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will end up with and pop different s
ones
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so on average it's going to be s
Wonderstone DDS 13 guess
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divided by and pop I and a half
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warned over and party is a good one
divided by and part
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so that I don't have to do the division
again and again
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it's a small it's a habit of not
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doing access work again
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there's a little bit more structure here
a multiplying it by half
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there's s1 and s2 dots 22
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the the variance reduction technique
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that I'm using here which
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again the said well we'll look about in
the next video
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the ones who simulated all these times
and this game today and all that kind of
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stuff
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das is equal to
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the value all the call option using
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want the car
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and we can see das its
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pretty much equal to the body that we
got
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analytically to
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so just go back there so I dot 4.9
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while the correct answer analytically I
know again this up
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this value is also given in the book
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to the correct answers for point
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759 i'd of 4.9
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but this was doing
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one simulation if I do it again
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our gas a different buddy if
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by do many many simulations so instead i
doing
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you know here and doing 500 terminations
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and this time I got actually very close
answer
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but again that's just look as the sole
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the last time I call 4.9 studies the
thing with Monte Carlo
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there is a noise element there you'll
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roughly get it pretty quickly you know
we're getting four point seven ish
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four point something its its its kinda
backer
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but then you have to do a lot more work
to get it
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more and more actors so
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if I wanted it to be you know acker to
Dec 10 percent
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third its within like .0 on or
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but Dolby 1 percent but you know
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if you want higher accuracy you need to
do more and more work
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the convergence race is one over
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the square root event so to get a
10-time por ACA
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I need to do a hundred times more that's
why
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we look art variance reduction
techniques
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in the next video where were able to
guess
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greater accuracy
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