Was Fermat\'s Theorem Really Proven?
 

Unlike scientific theories that are sometimes eventually shown to be wrong, generally accepted math proofs have never, to my knowledge, been overturned. Some like the proof that there is no largest prime, or that the harmonic series converges are so obvious that the odds it isn't true are greater than one in a googol (but less than one in a googolplex). More complex accepted proofs might be a quintillion to one favorites to be true.

But Wiles proof is different. It is lenghty, complex, has no obvious connection to the original question, and most importantly has only been double checked by a large handful of people. Furthermore I believe there is a probability argument that would allow Fermat's Theorem to have no counterexamples because of "chance". Also its proof had escaped the best minds for 500 years.

Personally I would take 100,000 to one odds that Wiles proof will eventually be shown to have a flaw. That's an exceedingly low number regarding a math proof, I think. But I am very unknowledgeable in this field. I wonder therefore what professional mathmeticans would make the odds.

Re: Was Fermat\'s Theorem Really Proven!
 

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But I am very unknowledgeable in this field.

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i wish i was as unknowledgable as you. Being overly modest is both stupid and disgusting. kinda like announcing "answered" prayers.

Re: Was Fermat\'s Theorem Really Proven!
 

Just an example of a mathematical theory that has been considered to be refuted is set theory. I don't claim to be a mathematical expert but I just thought I'd throw that out there.

Re: Was Fermat\'s Theorem Really Proven!
 

"Being overly modest is both stupid and disgusting."

I wasn't being modest regarding this subject. Actually I don't think I'm overly modest about any subject.

Re: Was Fermat\'s Theorem Really Proven!
 

I'm talking about a proof, not a theory.

Re: Was Fermat\'s Theorem Really Proven!
 

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"Being overly modest is both stupid and disgusting."

I wasn't being modest regarding this subject. Actually I don't think I'm overly modest about any subject.

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ya, i was joking. maybe i should've used a smiley face. don't forget that you should still value the fact that you know enough to truly know you're unknowledgable.

Re: Was Fermat\'s Theorem Really Proven!
 

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But I am very unknowledgeable in this field.

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i wish i was as unknowledgable as you. Being overly modest is both stupid and disgusting. kinda like announcing "answered" prayers.

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From what I've read here, sklansky is many things. Overly modest is not one of them.

Re: Was Fermat\'s Theorem Really Proven?
 

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proof that there is no largest prime, or that the harmonic series converges are so obvious that the odds it isn't true are greater than one in a googol (but less than one in a googolplex).

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There is no way you can come up with these numbers with any degree of accuracy.

You might as well say 'small enough to be considered negligible'

Re: Was Fermat\'s Theorem Really Proven?
 

I assumed that the only way these proofs are wrong is if every relevant molecule in everyone's brains was randomly twisted. Thats's between a googol and a googolplex to one.

Re: Was Fermat\'s Theorem Really Proven?
 

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I assumed that the only way these proofs are wrong is if every relevant molecule in everyone's brains was randomly twisted. Thats's between a googol and a googolplex to one.

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From my experiences with humanity, I'd say that everyone's brain is randomly twisted in some way. [img]/images/graemlins/wink.gif[/img]

Re: Was Fermat\'s Theorem Really Proven?
 

what about his assertion that the search for odd perfect numbers is pointless? what odds can i get that that turns out to be wrong?

Re: Was Fermat\'s Theorem Really Proven?
 

[ QUOTE ]
Unlike scientific theories that are sometimes eventually shown to be wrong, generally accepted math proofs have never, to my knowledge, been overturned. Some like the proof that there is no largest prime, or that the harmonic series converges are so obvious that the odds it isn't true are greater than one in a googol (but less than one in a googolplex). More complex accepted proofs might be a quintillion to one favorites to be true.

But Wiles proof is different. It is lenghty, complex, has no obvious connection to the original question, and most importantly has only been double checked by a large handful of people. Furthermore I believe there is a probability argument that would allow Fermat's Theorem to have no counterexamples because of "chance". Also its proof had escaped the best minds for 500 years.

Personally I would take 100,000 to one odds that Wiles proof will eventually be shown to have a flaw. That's an exceedingly low number regarding a math proof, I think. But I am very unknowledgeable in this field. I wonder therefore what professional mathmeticans would make the odds.

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David,

I am a PhD student in mathematics. I sat through a semester long course on the proof of Fermat's theorem a couple of years ago. I set the odds that a flaw is shown to exist in the Taylor/Wiles proof at 10^100:1.

Historically, there are two examples of accepted math proofs being overturned that come to my mind; I am sure that there are others. In the last 1800s, a proof of the four-colour conjecture stood for approximately ten years and in the 1960s a proof of the so-called invariant subspace problem stood for a couple of years before it was overturned. A proof of the former was given by Appel/Haken in the late 1970s and the latter has still never been proven.

Also, I am sure it was just a typo, but the correct statement is that the harmonic series diverges, not converges as you wrote.

Jason.

Re: Was Fermat\'s Theorem Really Proven?
 

I'm a mathematician, but I'm not a number theorist, so some of what I'm about to say may be wrong. But here's my understanding. Wiles proved the Shimura-Taniyama-Weil conjecture. This was a deep conjecture in number theory, much bigger than Fermat's Last Theorem. The fact that STW implies FLT had been known for a long time. The proof that STW implies FLT is not incredibly long and complex. Wiles's original proof of STW did have flaws, which were found and corrected. Since then, a stronger version of STW has been proven by other people, which means there is yet another proof out there. I don't have the expertise to verify these proofs, but I personally know people who do and have. Given all of that, I think the proof of FLT is on very solid ground.

OMG!! SKLANSKY!
 

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Personally I would take 100,000 to one odds that Wiles proof will eventually be shown to have a flaw.

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David, I'd take .00001:1 .... Do you see why??????????????

Re: Was Fermat\'s Theorem Really Proven?
 

</font><blockquote><font class="small">En réponse à:</font><hr />
Historically, there are two examples of accepted math proofs being overturned that come to my mind; I am sure that there are others. In the last 1800s, a proof of the four-colour conjecture stood for approximately ten years

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Ok, If I remember correctly the four-color conjecture has been shown to be true through extensive computer modelling, but there isn't a proof per se to show it to be so.

Re: Was Fermat\'s Theorem Really Proven?
 

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Historically, there are two examples of accepted math proofs being overturned that come to my mind; I am sure that there are others. In the last 1800s, a proof of the four-colour conjecture stood for approximately ten years

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Ok, If I remember correctly the four-color conjecture has been shown to be true through extensive computer modelling, but there isn't a proof per se to show it to be so.

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The Appel/Haken proof reduced the problem to verifying the conjecture for roughly 1800 different cases which they programmed a computer to do. The computer program output that all 1800 cases were 4-colourable and thus the theorem was proven. A few years ago a similar proof was given that required roughly 500 cases to be checked.

Re: Was Fermat\'s Theorem Really Proven?
 

Ok, here's a question I've never gotten a straight easy answer too. So, we know that if we have a graph with node in the shape of a wheel, it can't take more than 4 colors to make it so that no arc begins and ends on the same color.

http://www.f2f2s.com/images/colorgraph.jpg

So, why can't we look at a map as a collection of these wheel graphs, where the states/countries/whatever are the nodes, and the borders are the arcs?

Re: Was Fermat\'s Theorem Really Proven?
 

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Ok, here's a question I've never gotten a straight easy answer too. So, we know that if we have a graph with node in the shape of a wheel, it can't take more than 4 colors to make it so that no arc begins and ends on the same color.

http://www.f2f2s.com/images/colorgraph.jpg

So, why can't we look at a map as a collection of these wheel graphs, where the states/countries/whatever are the nodes, and the borders are the arcs?

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That is how the maps are viewed when thinking about this problem. Each country is represented by a node and adjacent countries have an arc drawn between them. The maps that satisfy the hypotheses of the four-colour problem turn out to be planar (none of the arcs cross) and the four-colour theorem is more commonly stated as any planar graph (that's what the object with the nodes and arcs is called) is four-colourable (you can colour the nodes with four colours so that no arc has the same colour at its two endpoints). Not all maps will necessarily reduce to such a collection of wheel graphs. Just draw an incredibly complicated planar graph and then it's easy to construct a map corresponding to that graph.

Re: Was Fermat\'s Theorem Really Proven!
 

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Just an example of a mathematical theory that has been considered to be refuted is set theory.

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As a mathematician, I have no idea what you mean when you say set theory has been refuted. I don't know of any problem with set theory. I suggest that you are remembering something that got garbled.

Re: Was Fermat\'s Theorem Really Proven?
 

Something that you might find interesting to read that's kind of related to this topic is Proofs and Refutations by Lakatos. The meat of the book consists of a dialogue between an imaginary teacher and students regarding "proofs" involving the Euler characteristic. I thought it was quite good.

Re: Was Fermat\'s Theorem Really Proven?
 

As a previous poster mentioned, Wiles's proof has been around for over 10 years now and been presented to a lot more than a handful of mathematicians. When he first released it there were flaws to which attention was called almost immediately. Since the revised proof of 1994, no additional flaws or corrections have come to light in another 10 years of close scrutiny by thousands of individauls.

If you had asked this question in 1995, you could probably get odds of 100:1 for finding another flaw. Sometime in 1996 or early 1997 you could probably have gotten 100000:1 odds. But now, enough work has been done, enough alternative proofs given and extensions of Wiles's work made, that it's pretty much a done deal.

Frankly, I would rate the chance of a flaw in the proof four-colour theorem as higher than the chance of a flaw in Fermat. It was much less accessible than Fermat ever was. (But I won't give 100000:1 on it either.)

Something you might be interested in is the Great Internet Mersenne Prime Search's "proofs" of what the 31st to 38th Mersenne primes are, by either factoring or doing two Lucas-Lehmer tests with identical results for every candidate. The chance of a hardware failure causing a single test to come out wrong have been watched for several years and are on the order of 1 in 100; but the chances of two out of three tests (if the first two tests don't match, a third test is done, and whatever answer comes up twice is accepted as correct) on different machines both falling victim to hardware errors and giving the *same* wrong residue for the L-L test is down around 1 in 10^20. Millions of possible exponents have been tested now. So there is something like a 1 in 10^12 chance that there is one number with a wrong L-L residue in the GIMPS database, and something around a 1 in 10^7 chance that that number, if retested, would turn out to really be a Mersenne Prime. There is a mathematical proof you can bet on: how does 10^19:1 on "there are exactly 37 Mersenne primes smaller than 2^6972593-1" sound?

I am on a budget of course so I can't take toooo much action on this; but if you'll transfer me one attodollar (plus $3.95 shipping and handling) I will set aside ten bucks for you in case a smaller Mersenne prime ever turns up.

Re: Was Fermat\'s Theorem Really Proven?
 

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Personally I would take 100,000 to one odds that Wiles proof will eventually be shown to have a flaw. That's an exceedingly low number regarding a math proof, I think. But I am very unknowledgeable in this field. I wonder therefore what professional mathmeticans would make the odds.

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Personally, I believe the proof. I watched the Nova special about Andrew Wiles three or four times--I still have it. I didn't understand most of it, but it was fascinating none-the-less. Early on he thought he solved it, but there was a problem. That problem was solved. You should view this tape. You might be able to find it at your local library. Nova repeats their programs from time to time as well.

But here is a more interesting question. Fermat himself said he had simple proof of this theorem which the margins of his book was too small to contain.

Will anyone ever be able to come up with such a "simple" proof, and perhaps more importantly--was Fermat telling the truth?

Re: Was Fermat\'s Theorem Really Proven?
 

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Will anyone ever be able to come up with such a "simple" proof, and perhaps more importantly--was Fermat telling the truth?

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Proofs will get simpler over time. How much simpler is hard to say.

I think "almost everyone" agrees that Fermat thought he had a proof, but that it was, in fact, flawed just like the thousands of alleged proofs that came in the years after his book was published.

For that matter, a great many 17th and 18th century proofs, judged correct at the time, would now be considered flawed because of the increased rigor that came into fashion in the 19th century. For instance, if a student were to follow exactly the same steps that Euler used to "prove" that 1+1/4+1/9+..+1/n^2 = Pi^2/6, his paper would be marked wrong (for playing too fast and loose with factoring a polynomial of infinite degree) - Euler just didn't write down a justification for why he could get away with certain manipulations of infinite series, and now we find it easier to prove it by a different method that to fill in all of the background for why Euler's method worked.

Re: Was Fermat\'s Theorem Really Proven!
 

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Just an example of a mathematical theory that has been considered to be refuted is set theory. I don't claim to be a mathematical expert but I just thought I'd throw that out there.

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I'm pretty sure set theory is logically consistent. Are you referring to something like Russell's Paradox or the Continuum Hypothesis?

Re: Was Fermat\'s Theorem Really Proven!
 

I think they're talking about the fact that originally it was thought that set theory could encompass all of mathematics and make it 'complete'.

Re: Was Fermat\'s Theorem Really Proven?
 

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Will anyone ever be able to come up with such a "simple" proof, and perhaps more importantly--was Fermat telling the truth?

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Fermat later claimed less than Fermat's Last Theorem. If he really had a simple proof, he wouldn't have done that.

I don't know how simple a proof there will be at some point. I doubt there will be a something you can show a bright high school student in an hour. Something to keep in mind is that the statement of FLT fails in some extensions of the integers, so you need to use something that is true about the integers, and false for the extensions. Simple algebraic manipulations won't do that.

Re: Was Fermat\'s Theorem Really Proven?
 

I will lay you 1 quintillion:1 that there is no largest prime.

Edit: in all seriousness, Andrew Wiles' proof is taught in many universities as a graduate level seminar. So there are a lot more people that understand the proof as compared to when it was first published. If there was a flaw in it, we would have found it by now.

Re: Was Fermat\'s Theorem Really Proven?
 

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Personally I would take 100,000 to one odds that Wiles proof will eventually be shown to have a flaw.

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if you specify a time period, i will take your bet for 1$.

Re: Was Fermat\'s Theorem Really Proven?
 

[ QUOTE ]
Unlike scientific theories that are sometimes eventually shown to be wrong, generally accepted math proofs have never, to my knowledge, been overturned. Some like the proof that there is no largest prime, or that the harmonic series converges are so obvious that the odds it isn't true are greater than one in a googol (but less than one in a googolplex). More complex accepted proofs might be a quintillion to one favorites to be true.

But Wiles proof is different. It is lenghty, complex, has no obvious connection to the original question, and most importantly has only been double checked by a large handful of people. Furthermore I believe there is a probability argument that would allow Fermat's Theorem to have no counterexamples because of "chance". Also its proof had escaped the best minds for 500 years.

Personally I would take 100,000 to one odds that Wiles proof will eventually be shown to have a flaw. That's an exceedingly low number regarding a math proof, I think. But I am very unknowledgeable in this field. I wonder therefore what professional mathmeticans would make the odds.

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I'm suspicious of Black-Scholes. It concludes that the value of an option is independent of any Trend in the stock, and as I see it does so by assuming an infinite bankroll that allows infinite hedging if the Trend goes against you. Since the Planet's entire options market is based on this Theorum you can actually get bets down if you want to gamble on a flaw.

PairTheBoard

Re: Was Fermat\'s Theorem Really Proven?
 

Many times mathematicians go about writing proofs to problems they feel is true. x^n + y^n = z^n seems true for every case you can think of.. the problem is proving the generality. If Wiles' proof is flawed.. it will be in how it's justified, and not because someone discovered a counter-example.

What's more interesting is proving something that mathematicians believed true for centuries isn't right... like when they tried to prove Euclidean geometry - then found out they couldn't.

Re: Was Fermat\'s Theorem Really Proven?
 

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Many times mathematicians go about writing proofs to problems they feel is true. x^n + y^n = z^n seems true for every case you can think of.. the problem is proving the generality. If Wiles' proof is flawed.. it will be in how it's justified, and not because someone discovered a counter-example.

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If someone discovers a counterexample, the Taylor/Wiles proof is flawed. However, the Taylor/Wiles proof is correct and no one will ever discover a counterexample.

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What's more interesting is proving something that mathematicians believed true for centuries isn't right... like when they tried to prove Euclidean geometry - then found out they couldn't.

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Mathematicians tried to prove that the parallel postulate follows from the other axioms of Euclidean geometry. It was proven that this is impossible.

Re: Was Fermat\'s Theorem Really Proven?
 

You guys baffle me. If this is the level of intelligence that I am up against on the felt - I QUIT !! Thanks for ruining my dreams. Seriously, I'm a smart guy (and very good with numbers, although I never studied mathematics, my finance degree came rather easily). What you guys study and know is just ridiculous. I guess I will have to rely on being deceptive rather than outsmarting any of you.

David - I am very happy to see you pushing intelligent discussion and conversation in this forum. The religion thing was getting old.

Good Luck Jason_T with your PHD. I am always truly impressed with someone who can put that much dedication into there education. After my B.S. (and that is exactly what it turned out to be [img]/images/graemlins/wink.gif[/img]) I was ready to blow my brains out!!

Re: Was Fermat\'s Theorem Really Proven?
 

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You guys baffle me. If this is the level of intelligence that I am up against on the felt - I QUIT !!

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My thoughts exactly...
and those colored maps are giving me bad flash backs of my combinatorics class last quarter [img]/images/graemlins/frown.gif[/img]

Re: Was Fermat\'s Theorem Really Proven?
 

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I'm suspicious of Black-Scholes. It concludes that the value of an option is independent of any Trend in the stock, and as I see it does so by assuming an infinite bankroll that allows infinite hedging if the Trend goes against you. Since the Planet's entire options market is based on this Theorum you can actually get bets down if you want to gamble on a flaw.

PairTheBoard

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BS is not an empirical theory, even though it sounds like it means to be. Essentially BS says that a stock's "volatility" determines the value of its options. But in practice, there is no meaningful empirical definition of "volatility". Traders use "implied volatilities". What this means is that instead of trying to observe some kind of empirical volatility (eg actual standard deviation of underlying's returns over some period), they pick their price and work backwards to _derive_ the (implied) volatility. It's like saying I have a theory that the "M" characteristic determines how fast a person runs. But the only way for me to determine people's "M"s is to have them run and see how fast they are. You'll find this kind of analysis is very typical in economics (eg revealed preference theory), and yet they're still allowed in universities. BS actually is very useful (unlike most econ), because it allows you to compare related options and control complex hedging strategies with a single parameter. But if you go look up some options prices in the market, you'll see that every option on a single stock trades at a different implied volatility, and they bounce around constantly. So there is no sense in which you might "beat" BS--BS in practice takes market price as in input, it doesn't determine it.

Also, BS does not assume an infinite bank roll.

Re: Was Fermat\'s Theorem Really Proven?
 

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I'm suspicious of Black-Scholes. It concludes that the value of an option is independent of any Trend in the stock, and as I see it does so by assuming an infinite bankroll that allows infinite hedging if the Trend goes against you. Since the Planet's entire options market is based on this Theorum you can actually get bets down if you want to gamble on a flaw.

PairTheBoard

[/ QUOTE ]

BS is not an empirical theory, even though it sounds like it means to be. Essentially BS says that a stock's "volatility" determines the value of its options. But in practice, there is no meaningful empirical definition of "volatility". Traders use "implied volatilities". What this means is that instead of trying to observe some kind of empirical volatility (eg actual standard deviation of underlying's returns over some period), they pick their price and work backwards to _derive_ the (implied) volatility. It's like saying I have a theory that the "M" characteristic determines how fast a person runs. But the only way for me to determine people's "M"s is to have them run and see how fast they are. You'll find this kind of analysis is very typical in economics (eg revealed preference theory), and yet they're still allowed in universities. BS actually is very useful (unlike most econ), because it allows you to compare related options and control complex hedging strategies with a single parameter. But if you go look up some options prices in the market, you'll see that every option on a single stock trades at a different implied volatility, and they bounce around constantly. So there is no sense in which you might "beat" BS--BS in practice takes market price as in input, it doesn't determine it.

Also, BS does not assume an infinite bank roll.

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Here's the thing that I don't get. If there is a trend in the stock - just like volatility, something you couldn't know until after the fact - it would theoretically not affect the value of the option, other than by how it affects the volatility. For example, it the trend has the stock doubling in a month then the option is worth more because a doubling stock has more volatilty than a flat one. But the high volatility due to the doubling trend is equally high volatilty for both puts and calls. The theory seems to imply that a put in such a stock is worth just as much as a call. Makes no sense.

PairTheBoard

Re: Was Fermat\'s Theorem Really Proven?
 

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The theory seems to imply that a put in such a stock is worth just as much as a call. Makes no sense.

PairTheBoard

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yes, put-call parity is one of the first things you learn as an options trader. all that put-call parity amounts to is that buying a (at the money) call on 100 shares of stock and hedging it by selling 50 shares of stock short against it amounts to entirely the same thing (except for a minor wrinkle with basis exposure) as buying the (same strike) put on 100 shares and hedging it by buying 50 shares of stock. same risk exposure for stock price movement, time decay, and implied volatility change (different for interest rate and dividend change but those are usually pretty minor).

the way to understand BS is to understand the hedging/trading strategy that they use to derive the option value. suppose a stock is trading at $60 and I buy 1 contract of the 60 call for $1. one strategy would be to hold the calls and hope stock goes up over $61, where my profit starts. here's the BS strategy. (remember a "contract" is for a call on 100 shares--it's like 100 calls you might say.) when I buy the call at the same time I can sell 50 shares $60. (remember stock is currently trading at the strike price, 60.) now the game changes. now I don't need stock to go up, rather I can profit on an up move or a down move. consider the two scenarios:

a. stock goes up. what's my strategy here? I have the right to buy 100 shares at 60 a share (my call contract). I paid $1 for this. so to break even I need to sell 100 shares at an average price of $61. I already sold 50 shares at $60. so now if I sell 50 more shares at $62, I break even. or to be more cautious, what I might do is sell some at $60.50, $61, $61.50, $62, $63, etc a little at a time. once I'm short 100 shares, I could call it a day. hopefully I've worked my price average up over $61. I cover those short shares with my call contract and close everything out. my profit is the differency between my average sale price starting with teh first 50 shares I sold at $60, and $61 (60 for the exercise price of the call plus $1 for what the call cost me).

b. but what if stock goes down? well now the call is not looking so useful anymore. but I can still make money. I can buy my 50 short shares back. I break even if I can make a $1 buying the shares back. I sold 50 shares at $60, so if I can buy 50 shares at an average price of $58, I break even. hopefully I can buy it better than that. now my call is not something I'm thinking about using (I wouldn't exercise the right to buy at $60 when stock is under $60), but I made money on my short stock hedge.

now take another step. suppose you bought the calls at the money and sold the 50 shares, and then the stock kept running up and you kept selling a little at a time until you had sold 50 more shares. in a sense you're done. but actually you're not: now suppose stock drops back down suddenly to $60. what do you do? start the whole process over! sell 50 shares again. if stock runs up again, sell more and more until you've sold 50 more. if stock drops in price, buy more and more until you've bought back the 50. you can do this indefinitely. you can scalp stock against your options, because they stop you out from having any loss on an extreme price move. in fact yo make money on an extreme price move in either direction. one more refinement: you don't have to wait till stock as at strike and sell exactly 50 shares. whenever stock moves up you sell some and whenever it moves down you buy some. your stock position will vary between 0 and short 100 shares. if you can scalp on a continuous basis, then you can use calculus and differential equations to model this strategy.

and that is how we value the option. it doesn't matter whether stock goes up or down. I will scalp stock either way. I don't care which _way_ it moves, but I do care how much it moves. I need to make the price of the option in by trading stock against my options as it moves up or down. the more it moves the more I can trade stock and make little profits putting my hedge on and off (or holding my hedge and trying to catch a single big move-same thing either way, it scales). this is why the standard deviation of stock price movement is all we need to konw to value an option--because we have a money making strategy to trade against the option that is indifferent to which way stock moves. but it does need stock to move. time passing is lost opportunity for the stock to move and for me to trade against it. so as time passes options drop in value. you're biggest risk when you buy the calls and sell the 50 shares is that the stock price might not move at all.

Re: Was Fermat\'s Theorem Really Proven?
 

mosta --
"and that is how we value the option. it doesn't matter whether stock goes up or down. I will scalp stock either way. I don't care which _way_ it moves, but I do care how much it moves. "

When you buy a call you don't care whether the stock goes up or down?

PairTheBoard

Re: Was Fermat\'s Theorem Really Proven?
 

not if you trade stock against it in the way that I explained. and for that strategy there is no difference between the call and put.

Re: Was Fermat\'s Theorem Really Proven?
 

PTB --
"When you buy a call you don't care whether the stock goes up or down?"

mosta --
[ QUOTE ]
not if you trade stock against it in the way that I explained. and for that strategy there is no difference between the call and put.

[/ QUOTE ]

Sure smells fishy to me.


PairTheBoard

Re: Was Fermat\'s Theorem Really Proven?
 

I'm not sure how serious you are, but: do you understand the strategy I outline? you don't need to have black scholes and differential equations to get it. if all you're getting out of it is a "smell" I would suggest, if I may be so bold, that you work harder on the analyticcs. get a pen and paper and work through several scenarios of stock price movement and trades.