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/biz/ - Business & Finance

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290011 No.290011[DELETED]  [Reply] [Original]

Any quants want to entertain a math major for a few minutes?

I took a mathematical statistics (400-level class, or 4th year) class and though we went pretty in-depth with binomial distributions, I didn't see any application problems involving finance. I was told/read that you can use it to determine stock prices, but wasn't entirely sure how to go about this due to lack of understanding about financing.

So anyone have any good materials for this? Or can you briefly explain the abstracts/applications in finance of it? I'd greatly appreciate it.

>> No.290041

bump

>> No.290048

>>290011
in finance, it's used especially in option pricing

there aren't that many direct applications of it

>> No.290080

>>290011

Sure, I can explain this to you.

On a very, very short timescale (think fractions of a second), almost* all of the time, a stock will do one of two** things- it can move up in value by a cent, or down in value by a cent. Let's call the probability of a up move by u% p and a down move d% 1-p, and we'll further assume that the stock price is large enough that u = 1/d. We'll call this process X[i]. Already the outline of a binomial distribution is beginning to become clear- we have a Bernoulli distribution.

Now, let's assume we wish to forcast the stock price one minute into the future. Then that's just S[T]=sum(X[i],i=0,600) (since we're dealing with 100 ms intervals and assuming p=1/2). This stock price is then a RV with distribution equal to S[0]+0.01*(binomialpdf(x,600,0.5)-3.0).

Nobody really does this, because knowing the future distribution of stock prices alone generally is unprofitable, since you can only bet if they go up or down. But that's what options are for! In case you haven't covered them, an option is a financial contract that give you the right, but not the obligation, to force me to buy or sell something from you at a specified price. For instance, if I sell you a call option on stock XYZ struck at $50 that expires in a month, then at any time in the next month, you can come to me and force me to sell 100 shares of XYZ from you for $50- even if it's trading at $120 in the open market.

Continued!

*Sometimes, thanks to HFT, these jumps can be much bigger as sudden effects are priced in. Think when the Fed raises rates/there's a terrorist attack/etc.

**It can also, obviously, stay the same. This three-pronged model is unimaginatively called the trinomial model. It's more complex, but also generally more accurate.

>> No.290086
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290086

Since you know that, in a month, this option will be worth max(current price - $50, $0), you can draw up a binomal tree like in pic related, only with a few hundred or thousand rows instead of three, and fill in the last column on the far right using that formula. Then, you can do some fancy math and figure out a recurrence relation- knowing the values in the i-th and i+1-th row in column t will allow you to figure out the value in the i-th row in column t-1 (that's the formula in the pic). Recursing this a bunch will eventually get you the value of the option at time 0, ie right now. This can be used to buy and price the option- if it's on sale for cheaper than the model, buy a bunch and you've earned a profit, and if it's more expensive, then sell a bunch and likewise earn a profit.

This model is generally of more historical interest or for accounting/tax purposes than live trading, though. As you're no doubt aware, after enough steps the binomial model converges to the normal distribution, and one of the most established facts of empirical finance is that stock returns do NOT follow a normal distribution- the chances of large jumps are much higher than a normal/binomial distribution predicts, and they get more erratic the finer your timescale predicts- annual and quarterly returns are pretty close to normal, daily returns are closer to Laplace, and intraday returns are almost power-law they're so pathological.

Any other questions?

>> No.290098

>>290086
nope, i understand the process, just needed to see an example. thank you very much, your posts were highly informative!

>> No.290115

>>290080
actually, now that I read a bit more into it, do we assume both u%/d% have equal probability of occurring?

>> No.290117

>>290115

Generally, no. You do assume that, in the long run, the expected value of the stock grows at the risk-free interest rate. This gives (p*u+(1-p)*d) = 1+r as a condition that you need to apply, and you usually require u and d are spaced far enough apart that the standard deviation (aka the volatility) matches the observed one of the stock. That's what the formula in the pictures is.

>> No.290125

>>290086

hey man, just corollary question here, what is your education history ?

I ask because I am doing an undergrad in Finance and Statistics at a top 5 global university and Im in my first year, I imagine ill be doing this kinda stuff at some point but when should I expect to be meeting this level of shit, its still way above me by the looks of it.

>> No.290128

>>290117
ahhh ok, I need to review and read further in my stats book, but you've given me an excellent jump-off point. thanks again!

>> No.290138

>>290125

I've answered this question before. BS/MS in mathematics at a technical university (think like Caltech, MIT, etc.). I worked for a DoE lab for a few years, and then I got hired away by a friend from uni working at a boutique hedge fund. Salary is highly variable and generally in the low to mid six figures. No, I won't hire you.

>> No.290142

>>290138

hahhah no its cool, I wasnt looking for a job, more like you're where id like to be in a few years so I was interested in asking, how long have you been working at that hf, and whats your position ? are you taking any risk yourself or just analysis, also what products do you trade, hft algos ? oh and whats your AUM ?

>> No.290143

>>290125
OP here but throwing this out here: binomial distribution is typically covered in your first mathematical statistics class you will take, which will require calc III as a pre-req. at my public state university its a 400-level class (calc III is 200-level) so you won't see anything like this for another semester or 2, depending where you started in math. the person who replied sounds like they at least have a masters or is a VERY intelligent undergrad in mathematical finance or similar. Some of the concepts he talks about (trinomial distribution) i don't even think are covered by my school's stats department at undergrad level.

>> No.290144

>>290138

Re the math, this is easy. If you want to be a quant, you'll need to go way, way beyond this. Generally speaking, if we were hiring a new team member, I'd want him to know diff eq, stochastics, calc, theoretical statistics, programming in a couple different languages, a decent physics background, some applied math/math modelling, plus something unique to offer us we don't have.

That said, that isn't the norm. A firm grounding in statistics, some math modelling, and decent familiarity with higher math, and some programming will probably be fine if you aren't planning to jump into a quant role right away- ie work as an analyst somewhere.

>> No.290152

>>290143

oh ive done binomial, normal and poisson at high school and more now at undergrad, but this trinomial shit (although I got the gist I think of it representing the fat tail aspect of stock movement distributions) and the actual math behind it sounds hella hard

>> No.290156

>>290142

>how long have you been working at that hf, and what's your position?

Around five years, quantitative researcher (It's like twelve people, so our positions are weird). The firm has one account that trades according to algos I help write, so I don't know if you'd consider that "taking risk". We trade anything liquid and deep, mainly lower frequency. AUM is not supposed to be discussed..

>> No.290155

>>290143
cont'd: but what he's supplied is (what I think) a great explanation for someone who has only taken their first mathematical statistics class and has not seen the concepts applied in a financial setting. read the wikipedia article on binomial distribution if you're really interested, they give some examples using a coin toss

>> No.290159

>>290143
>>290143
wtf m8

binomial distribution is taught in highschool

a 2nd year math course goes through several distributions and moments and shit

>> No.290162

>>290144

yeah im looking to take stochastics and diff eq in my next years as optionals, we have alot of statistics as required courses, however I wont get any exposure to physics or I guess applied math modelling outside it being applied to finance problems, also we dont have any programing training in the course, what language should I be looking to learn if I should know atleast ONE for these kinda positions, also how important is the theoretical shit like real analysis and that kinda shit ?

>> No.290172

>>290144
is linear algebra highly recommended? i found the subject very interesting and useful, enough to be allowed to take the graduate level of advanced linear algebra and do well in the class, i just haven't seen lots of applications outside of solving diff. equations.

>> No.290173

>>290156

hahahaha you seriously cant through a ballpark AUM figure out there, youre a boutique so its not like ill be able to recognise your fund from the AUM like I would one of the big ones, when you say liquid and deep does that transfer across asset classes such as liquid bonds or etfs or currencies or are we talking just stocks ? although then again currencies arent exactly low frequency but bonds fit that bill, although outside the recent attempt to create electronic exchanges its all OTC isint it so you must be taking buy and sell signals from your algos ?

how many portofolio managers do you have ? If that position exists, I know it does in most funds but you said your positions are weird so it may be a flat heirarchy idk

>> No.290176

>>290152
>but this trinomial shit (although I got the gist I think of it representing the fat tail aspect of stock movement distributions)
kek

>> No.290180

>>290162

>what language should I be looking to learn

If one, go for Python. If you can learn a couple, which is easy after you pick up one, diversify a bit with Mathematica, Python, C/C++.

>how important is the theoretical shit

Depends on the theoretical shit. Stochastic processes, differential equations, theoretical statistics, etc., are pretty fucking important. Something like real analysis or number theory or abstract algebra is less directly applicable, but knowing it serves as a signal that you're smart as fuck and can handle anything we throw at you.

>> No.290183

>>290159
i was never interested in math in high school (until I discovered adderall in college) and at least in our department, a high school understanding of binomial distribution wasn't "complete" - that said, yes, math. stats is technically a 2nd year class since its only pre-req is calc III, but the concepts we were taught were difficult enough to deem it a 400-level class

>> No.290181

>>290176

haha correct me if im wrong im just throwing this out there

>> No.290190

>>290180

yeah fair enough I was talking about the real analysis and number theory and that kinda stuff, tbh I think we may get a bit of MATLAB and R training now that I think of it

do you see yourself staying at this HF or you wanna move somewhere else longterm/short term ?

>> No.290200

>>290172

My master's thesis was on computation linear algebra, so I'm biased. Linalg is cool as shit, but not terribly applicable.

>>290173

>ballpark AUM

Somewhat more than $100k, and somewhat less than $100bn.

>when you say liquid and deep

Stocks, currencies, futures, etfs, commodities via futures, etc. Not really bonds directly.

>portofolio managers

Doesn't really exist, single fund. Two partners which I suppose generally map to the position of PM, though only one really interfaces with operations, the other tends to be client facing.

>> No.290205

>>290181

You're right. Trinomial is generally used for exotic options, not for pricing in fat tails. For that, you'd probably want o use something like monte-carlo simulations assuming some suitably fat-tailed distribution.

>do you see yourself staying at this HF

Yes. I may retire from finance at some point, but I doubt I'll leave this fund for another.

>> No.290204

>>290180

from what i've read about what a typical quant program looks like and personal stories, I highly agree with the statements about theoretical shit. You might need to know fuckall about groups, rings, and modulo arithmetic but the concepts (and most of all the fucking proofs) require and develop mathematical skills you won't usually see. Logic has helped me a great deal in diff eq and math modeling, and is great in stats too (mgf's, mostly)

>> No.290214

>>290200
oh man I want to read your thesis now. if i post an email will you email it/the link to me?

>> No.290220

>>290200

u are a cheeky cunt u m8 ill give you that, can you discuss recent returns atleast or anything that tells me how youre performing and not just gonna shut your doors soon or if youre the next bridgewater ?

also do you see yourself ever breaking into the 7 figures range or those insane fuck off money some of the big names pull in ? does anyone in your fund pull in that kinda money ?

>> No.290254

>>290214

>I want to read your thesis now

Sorry, nope. I'd say it's Q classified, but that would be a lie. It's pretty mediocre work, and it also would be easy to tell who I am from it.

>>290220

>recent returns

We target for a low and stable positive return (around 10% net of fees) with minimal volatility and mainly market to institutions, which should give you a general idea of our AUM. We had a great 2011, pulled around 20% after fees, and have averaged at around our target most other years. Of course, we looked shitty in 2013, but almost 30% total returns on the S&P make everybody look shitty. I doubt we'll be the next Bridgewater, and the partners will probably return capital if we get anywhere near that number, but we're generally pretty effective and low-risk, and I think we'll do fine in the long run.

>7 figures range

Perhaps. I would't be long for finance if I started making that kind of money, though. I already have over a million in the bank and the fund, and I'd probably retire once I hit ten million or so.

>> No.290267

>>290254

human question, what would you like to do in your retirement ? how old are you ? got any family ?

and those numbers still sound damn fucking good when comparing to the average hf returns over the time period

also how did you end up on 4chan ?

>> No.290274

>>290254
can't shoot a guy for trying. Would you know of anything else that can be tackled by a 4th year math major that has to do with options pricing? And please don't say linear regression, I already know that shit is terrible.

>> No.290297

>>290267
>human questions

Ugh, I hate these.

30's, unmarried with no kids. I'm thinking I might like to teach at a university. Not finance, though, or if finance some higher-level mathematical or computational finance.

>how did you end up on 4chan

What can I say, I'm a NEET at heart. I've been lurking here on and off for a while, the creation of /biz/ made me start posting.

>>290274
The best book I've found is

>http://www.amazon.com/Undergraduate-Introduction-Financial-Mathematics-Edition/dp/9814407445

It should cover just about everything at a reasonably basic level. Apparently there's a new edition, I have the 2nd one.

>> No.290298

>>290080
well, now that I REALLY read it, read this and make sure I'm following accurately:

We first assume u% = d% = 1/2 or 0.5

X[i] describes the process that involves the change in stock price on the millisecond timescale.

To forecast the price one minute into the future, use S[T] = Sum(X[i]) , 0 <= i <= 600

Then, stock price is a random variable with distribution:
S[0] + 0.01*(binpdf[x, 600, 0.5] -3.0)

I haven't yet re-read the binomial dist. chapter, but where is the - 3.0 coming from?

>> No.290306

>>290297

oh and thank you for the book.

>> No.290309

>>290298

Nonono. p = 1-p = 1/2 is the probability of an up or down move. u and d are the size of the move up or the move down- either in absolute or relative terms (depending on if you use prices, returns, or log-returns, which depends on the application).

X[i] is the process at 100 ms time intervals, and we forecast 1 minute into the future.

3.0 is the expected value of the process, since it's an uncentered bionmial ditribution. We're interested in (X-mu), not X itself.

>> No.290313

>>290297

seeing as youre unmarried and no kids how much of the 'models and bottles' lifestyle do you live ? do you have all the typical shit like a supercar and expensive toys ? do you have any desire to stay in the game for the really big toys like a private jet or a yacht or idk, fuck knows what ? do you fuck any 10/10 hookers or gold diggers ?

>> No.290327

>>290313

>'models and bottles'

Fuck this. I-bank douchebags and other assholes in the finance industry give us a bad name, though, like, the financial crisis probably didn't help.

You'd think I'm pretty much the stereotype of an introverted mathematician, if you met me. I put most of his money in the bank. I have a nice apartment, I drive a nice Lexus, but I still dress in cheap g-man suits and wear $50 watches, and I certainly don't have yacht or a private jet, nor would I want one. The closest thing I have to a hobby is collecting a library of math books and nifty original manuscripts, and I do like eating out all the time at nice restaurants.

>> No.290335

>>290309
gotcha. I guess my only real question left is how would I determine X[i]? This may be a stupid question and I could simply be overthinking it.

>> No.290345

>>290335
like basically, if X[i] is the process at 100 ms time intervals, obviously the stock price doesn't have a conscience and says to himself "oh well fuck, if my time interval is an even integer i'll go up and if its odd i'll go down but only if the odd integer is divisible by 3" unless I'm way off on what you're saying a process is

>> No.290368

>>290345

(On phone now)

Think about the process like this. Every 100ms, the stock gremlin flips a biased coin with probability p of heads. If it's heads, he moves the price up by u, and if it's tailed, he moves it down by d.

>> No.290414

>>290327

fair enough, I totally understand that and I respect you for not getting suckered in to the whole culture, tbf when you told me youd like to teach in your retirement I already had the feeling you werent that type, but was still curious to ask, do you know people around you (in your fund or elsewhere) that are like that ?
and that collection actually sounds pretty badass, atleast you didnt pick something typical rich people do like art or fucking stamps or something

My last question is I guess my hardest one, do you, idk, like what youre doing ? like when you wake up in the morning do you look forward to your day or is it just 'get work out of the way so I can go home and relax', is this what you wanted to do ? when you introspect do you feel like you picked the right thing or are you in it for the money and its bearable but then youre out ?

>> No.290418

>>290368
i'm not expecting replies, so if you have RL shit to do by all means, I really do appreciate your time and willingness to teach, so I'm trying to understand the best I can

so lets assign some probability values to the stock gremlin's coin. lets say heads has a .51 chance, and tails has a .49 chance.

so its has Bernoulli distribution, which is a special case of the binomial distribution, so X[i] = B(# of flips, probability of heads) = B(1, .51) = .51 chance of increasing price by u?
Just running with it now, if the above is incorrect you don't need to read:
S[T] = 600*.51

S[0] = .51, so

.51 + 0.01 *( Bin(x, 600, 0.51)-3.0) is our distribution

I think I just confused myself.

>> No.290423

>>290418
i can't remember how we evaluate a 0th trial, so my CS studies might be interefering (for example, 0 is counted as the first step in CS, etc). I really should retake this class.

>> No.290433

>>290418

yeah and as this anon said, really thanks for all your time man, you should trip, you would offer alot of valuable knowledge/insight, although I can imagine people getting on your balls all the time if you did

>> No.290446

>>290414

Our fund is pretty low-key, none of the quants are like that, though one of the partners and I guess some of the traders probably are something like a lower-key M&B person.

I like my job. Much of what I do is essentially math research, and only a relativity small portion can e applied to finance. I have a long string of interesting results that didn't ultimately result in a tradeable model that I've been given the go-ahead to publish when I leave, so my job is essentially a super-well compensated academic who doesn't have to teach, and I really like that. The days can be kind of long and the sword of damocles-that if you fuck up a bunch of pensions and endowments will lose a lot of money- is there and stressful, but overall I find my job as or more enjoyable than my prior one witha lot less bureaucratic bullshit, and it's five to ten times as lucrative, so yeah, I'd say I'm happy.

That said, I think I undersand, and the people at work understand, that all this could change quickly. If the fund started to take losses, or it grew too much and the culture changed, or if I lost my nest egg and HAD to work for the money to make my mortgage, I could see the job becoming a miserable chore pretty quickly.

>> No.290461

>>290152
>I've done that in high school
Not sure I have a reaction image for such a post. Seeing baby concepts of these distributions, particularly poisson is no where near the real deal.

>> No.290489

>>290418
disregard my posts I understand now

>> No.290515

>>290433
>you should trip

Ehhh, not a fan of tripcodes. Besides, I already have a pretty distinctive signature. If there's every somebody shilling for VWINX or VWIAX, then that's probably me. And if people do shill it to impersonate me, then as long as they aren't also endorsing daytrading or shitcoins then it's probably fine.

Tell you what, I'll trip up temporarily, and then tomorrow when I'm more sober I'll post a knowledge dump on optimal long-term investing and asset allocation. I doubt people will mind

>> No.290940

>>290200
> Linalg is cool as shit, but not terribly applicable.

What? Linear algebra is probably one of the most useful and directly applicable things in the math curriculum

>> No.291539

>>290940

Sorry, I should have made clear I was speaking more to my experience here. Obviously it's very useful in, eg, MPT and asset allocation, and there's all sorts of discrete-time models that use pretty hefty linear algebra, I just never really encounter them on a regular basis.

Incidentally, re that dump on long-term investing, I woke up this morning and found I had written five pretty detailed pages on it and still had a long way to go. I'll probably spend a while editing and expanding it. Give me some time, it looks like something people might be interested in.

>> No.291546

>>290515
>VWIAX
so you work for vanguard?

>> No.291596

>>291546

Lol nope, it's just my single largest holding and all my retirement account has in it.