alright I’m reading Paul wilmut on quantitative Finance 2nd edition chapter 1.2 this is one of the things I learned was the time value of money which is pretty
cool and basically the thing this is the first real math in the book it’s a way of deriving the continuous calm before mule ofor continuous compounding from discrete compounding time value of money so discrete compounding if you have an interest rate R and it’s compounded M times per
year the formulas pretty simple right you have one plus R over m to the power M and that’s the the factor that your money grows every year the continuous case is the limit as M goes to infinity of the same formula 1 plus R over m to the M so what is the limit that’s the question so how
do you find this so the the trick is you can rewrite 1 plus R over m to the M as e to the M times e to the M times log of 1 plus M why is this this is a just a property of logarithms so as an example you can do 5 to the 7 equals e to the 7 times log of 5 so you can check that that actually works
out this is just a another way of writing it then this log of 1 plus R over m you can do a Taylor series expansion of this so don’t remember Taylor series you have to break out your calculus book start getting your memory flowing so here’s the formula for Taylor series f of X plus Delta
X equals f of X plus Delta x times Delta X times DF DX evaluated at X plus 1/2 Delta x squared second derivative f respect to X also at X plus higher order stuff alright so this is just taking a function evaluated at a point and perturbing it and how does the perturbation affect the function well
it’s it’s the base plus the linear part of the derivative so this is the linear perturbation and here is the second derivative this is the quadratic you know extrapolation from how far you are from the point and you know it keeps going to the higher dimensions or higher powers so
that’s the Taylor series expansion at X so what can we do with log of 1 plus R over m so R over m is the Delta factor and our function is just log and so what do we get we get log of 1 plus R over m equals log of 1 plus times and what’s the derivative of log is one over the argument so
we’re evaluating it at one so we actually get 1 over 1 plus stuff so this is actually equal to log of 1 is 0 so just R over m plus the higher-order stuff so what does that mean that means that limit as R over m goes to 0 of 1 plus R over m to the M equals this e to the M log 1 plus over m so
i equals e to the M times R over m and we can ignore the higher order stuff because this is going to be powers of R over m which are going to 0 and so that will be the are there you go so in one year with continuous compounding your money grows by a factor of e to the R and if you don’t you
multi-year over T years it’ll be e to the RT pretty obviously so that’s it that’s pretty exciting this is a fun application of Taylor series expansion and some simple facts about logs and exponents and you can derive the time value of money you
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For anyone who is watching the video recently.
You can do the same derivation using L'Hopitals rule using only highschool level calculus.
Regardless, the content in these videos is excellent. Thank you for uploading ☺️
that's really helpful
And how can I get the formula of forward price when there's a dividend pay in between?
Anyone has any other book recomendations on same topic?
Thanks! Although, where are Chps 1.1, 1.3-1.8, 2.1 – 2.11, and so on?
Why would you explain such a simple thing in such a complicated way. The limit of (1+1/n)^n = e then you just need a simple substitution?!
Is this from the book Paul Wilmott Introduces Quantitative Finance 2nd Edition ISBN 978-0470319581?
Thank you may God bless you abundantly
Can you suggest some books?
The log should be ln (which is log with base e). If you just write log without specifying the base its assumed to be base 10.
As Allan pointed out, you can shorten this video to 1 min by just stating that (1+1/m)^m
> e when m>infty (this is the pure definition of e. It then follows that (1+r/m)^m -> e^r.Great video Nathan! Have you completed a mathematical finance degree?
From the value ln(1 + r/m)^m (it is a known limit), I think we can see that it is already convergent to e^r…………
welp. got an interview for a financial quant job. Hope this makes for good revision!
That might work but there are some tricky issues with m changing as you take the limit. If you have a working derivation let me know.
Thanks Nathan! But also, can't you just binomially expand from the first expression and then take the limit?
Yes, I'm using log for the base e logarithm. In general I use subscripts after the log to indicate the base unless it's a natural log, which gets no subscript. This is the mathematics convention. Engineers use ln for base e and log for base 10. Computer scientists often use log for base 2.
so in this case "log" is what other people would say is "ln" ?
yeeeeah…cool stuff
Thanks – that had me confused!
Thanks Nathan, you teach really well
The Wacom Capture would work fine. My only complaint when I tried it was that it felt a bit cramped, I'm used to the larger area of the one I have. But for "virtual lecturing" that's not a big deal, it's more important for graphics design work.
To get a demo account for the web-based recording system, send me an email at: nathan (at) lessontastic (dot) com
There should be something to try by early next week.
Also anyone else reading this, you're also invited to sign up!
Thanks for the quick reply Nathan!
Don't want to turn this into a hardware discussion, but I was under the impression that a "simple" Wacom Capture would do. If not, I would appreciate a heads up before I actually buy one.
A browser based system sounds great. A demo account would be appreciated!
Carl
Thahks! I'm using a Wacom Intuous3 graphics tablet and a cheap XLR microphone for audio. For software I used to use MyPaint and a screen capture program, but I'm actually working on a new web-based system that does everything from within the browser. Let me know if you want a demo account (once it's ready).
Nice work Nathan!
Thinking about doing something similar in another field. What kind of equipment are you using?
The rest of the world uses "ln" to mean base e logarithm.
you can take logarithm of the expression and use l'Hopital's rule to get r. then the expression itself will be e^r
Just a matter of how to denote one thing or another.
Exactly the same issue between "e" and "exp".
Plus, Nathan is right. Only physicists use log as base-10 logarithms.
pv table
great explanation! clear and simple~ thanks
Method short cutting taylor series. m*ln(1+r/m) =r*[ln(1+r/m) – ln(1)]/(r/m) as m -> inf the rhs is just r*derivative of ln(x) evaluated at x= 1, which is just r.
@papipoet I'm a mathematician, I use "log" to mean base e logarithm. Down with the ln heretics!