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#72
08-03-2005, 01:02 AM
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Re: Common Sense Black-Scholes
when you pose a scenario: stock is going down, what does that do to the value of the stock/call/put--you have to allow that stock and call and put are essentially equivalent, a priori. any one of them can be replicated by the others. these equivalency relationships supercede your outlook in the underlying's market, because they provide *riskless* trading opportunities if they get out of line. if you jack the puts by a dollar, arbitrageurs will take the free dollar all day, and you will be synthetically getting short the stock--"great", you think stock's going down--but doing it at one dollar below market. if you want to get short, why give away a dollar in the process?
I know this is just repeating. I do see how there is an intellectual dissonance between the asymmetry in stock distribution and the symmetry in options of opposite delta. I think the points are 1. option arbitrage relations are independent of underlying distribution (jason would be the authority on this, not me) and 2. the real value in the option is its convexity, its gamma-theta-vega. you can make the delta whatever you want. as far as convexity goes, call and put hvae the same. therefore they are the same option. I've read through all of jason's posts and I followed them completely until his proofs on the asymmetry of up and down strikes. as I studied hull on my own (crash course), when I looked back I realized I didn't fully comprehend all the steps separately, and distinguish which steps relied on geeomtric brownian motion, which depended on the continuous dynamic hedge, which depended on stock price distributions, etc. it didn't matter at work, but I want to go back to that on my own time. I like to refer to the dynamic hedging strategy and I know it doesn't all depend on that. but I find that very comprehensible. but I think I'm correct thta you don't need any math or distributions or dynamic hedges to see that long a call and short a put on the same strike is equivalent to a foward in the underlying (there is a qualification for risk of stock closing exactly at strike on expiration and there being ambiguity over whether either or both options will be exercised, but forget that...). a forward is just like the underlying (plus basis), so long call, short put is called synthetic stock. once you have that you have +stock = +call -put -call = -put -stock etc etc etc these relatoinship control, because arbitrage trading opportunities dominate. 
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#73
08-03-2005, 02:32 AM
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Re: Common Sense Black-Scholes
mosta --
"but I think I'm correct thta you don't need any math or distributions or dynamic hedges to see that long a call and short a put on the same strike is equivalent to a foward in the underlying " I agree that parity is easy to see this way. I asked jason1990 this, PTB -- "I'm now thinking that what Black-Scholes did was construct the theoretical Equivalent Trading Strategy for an option and calculate the theoretical cost of implemening the strategy - thus providing a rigorous pricing method for the option. Was Black-Scholes even needed then for the Put-Call parity you express above? Or does that follow automatically from the kind of arguments mosta has been making?" He responded with a proof of parity along your lines above, although he refered to the equivalent trading stragegies rather than the actual put and call when equating to the long stock postion, then brought the put and call back in. He made this comment in summary. jason1990 -- "For this to work, we must assume that options are priced according to the startup cost of an equivalent trading strategy. In particular, we must know that such strategies exist and are unique." ============== With C(K) and P(K) the value of the call and put at strike K, S the stock price, and ' the calculus derivitive, jason1990 pointed out that the relation P(K)=C(K)+K-S implies P'(S)=C'(S)+1 So the assymetry flows from the fact C'(S) is not -.5 . That the call becomes worth more than the put as their respective strikes move closely away from S follows from the fact -.5 < C'(S) < 0 so, P(S-2) < C(S+2) I guess we could also notice on the other side that when the Put and Call are equally in the money, the Put has a higher premium than the Call. ie. P(S+2) > C(S-2) Maybe it's not important but I'm still curious as to "why" this is going on. Does this really explain it? mosta -- "is it not just because stock prices are log-normally distributed, not normally, and therefore not symmetric?" PairTheBoard
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#74
08-08-2005, 07:16 PM
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Re: Was Fermat\'s Theorem Really Proven?
The harmonic series does not converge [img]/images/graemlins/wink.gif[/img]
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#76
08-10-2005, 05:18 PM
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Re: Was Fermat\'s Theorem Really Proven?
David,
I think your bet is a bad one. Pascal wouldn't even make such a bet. The proof is correct. It is perhaps the most scrutinized big theorem around. This only comparable modern mega-result is the classification of finite simple groups (which is much much longer...). Plus, you have to worry about bankroll concerns if you get much action. See my July magazine article on factoring in variance for longshot (high variance) bets.
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Linear Mode
