Was Fermat's Theorem Really Proven?

[jason_t]

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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.

[buck_thunder]

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!!

[chiachu]

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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]

[mosta]

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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.

[PairTheBoard]

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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.

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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

[mosta]

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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.

[PairTheBoard]

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

[mosta]

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.

[PairTheBoard]

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

mosta --
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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.

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Sure smells fishy to me.


PairTheBoard

[mosta]

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.