Was Fermat's Theorem Really Proven?

mosta · #1 07-30-2005, 01:18 PM

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saying you do equally well with the Call-Hedge combo as with a Put-Hedge combo regardless of how the stock moves. Let's say your Call-Hedge Combo has you all-in with 100 total shares sold at say $80/sh and your Put-Hedge combo has you all-in with 100 total shares bought at $40/sh. Due to put-call parity you paid the same price for them. Now suppose due to the extreme underlying upward "Trend" in the stock - assuming there could be such a thing - the stock triples in price straight to $180/sh. Your Call-Hedge strategy has made you something like $10/sh on 50 shares. Your Put-Hedge strategy has made you what? As the stock moves up you gradually sell your 50 shares until it hits $80/sh and you make something like $10/sh on 50 shares.

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A. long the 60 call for $1, short 50 shares at $60.
let's do this as discrete one time period to expiration model.

1. stock goes to $80. sell 50 shares. net sold 100 shares at average price of $70. exercise call to buy 100 shares for $60. all positions closed. profit = $20 on 50 shares and $0 on 50 shares, minus $1 on call contract.

2. stock goes to $40. buy 50 shares. net zero stock position (sold 50, bought 50). call expires worthles. so no position. profit = $20 on 50 shares, minus $1 on call contract.

B. long the 60 put for $1, long 50 shares at $60.

1. stock goes to $80. sell 50 shares. net zero stock position (bought 50, sold 50). put expires worthless. so no position. profit = $20 on 50 shares, minus $1 on put contract.

2. stock goes to $40. buy 50 shares. net bought 100 shares at average price of $50. exercise put option to sell 100 shares at $60. all positions closed. profit = $20 on 50 shares and $0 on 50 shares, minus $1 on put contract.

now you said stock goes to $180. we add one time period. it matters whether it went to $180 from $40 or from $80. consider each of the four scenarios above as a starting point.

A.1. stock goes from $80 to $180. you have sold 100 shares (but you haven't exercised your call yet and it hasn't expired because there's a time period left). your position is neutral. profit/loss is zero.

B.1. stock goes from $80 to $180. you sold the 50 shares you bought already. your stock position is zero. your put is way out of the money. your position is neutral. profit/loss is zero.

A.2. stock goes from $40 to $180. you had zero stock at $40 because you bought back your short 50 shares. now on this gap move to $180 you will sell 100 shares at $180. your call expires and is exercised so you cover your short 100 shares buying for $60. you make $120 on 100 shares in addition to the scalp you made in time period one.

B.2. stock $40 to $180. you were long 100 shares total at $40 against your puts. on this gap move to $180 you will sell those 100 shares at $180. you have zero stock position and your put expires worthless, out of the money. you make $120 on 100 shares in addition to the scalp you made in the first time period.

Here's how it works from the trader's perspective, analytically. you buy a call or a put with a given strike and expiration. let's say the 60 strike again, but this time stock is at $70 when you do the trade. you look at your sheets to get the delta of the option. the call delta is 65. the put delta is 35.

A. buy ITM call for $11 ($10 parity plus $1 intrinsic/time value). sell 65 shares. now you are delta neutral. you have 35 shares to sell as stock goes up and 65 to buy as stock goes down.

B. buy OTM put for $1 ($0 parity, plus same $1 intrinsic/time value). buy 35 shares. now you are delta neutral. you have 35 shares to sell as stock goes up and 65 to buy as stock goes down.

the next morning stock is up $5, to $75. you look at your sheets to see what your new delta is. the call and the put had the same gamma, let's say 2 (deltas per $1 of stock move). so in both cases your delta on your sheets is now long 10. so you say, I better sell 10 shares.

A. you now have sold a total of 75 shares. you have 25 more to sell as stock goes up and 75 to buy as it goes down. you are delta neutral again. and your gamma is now a less (stock farther from strike). your gamma is like 1.5 now.

B. you bought 35 shares before and now you sold back 10. you have 25 more to sell as stock goes up and 75 to buy as it goes down. delta neutral. gamma 1.5.

both are the same. the gamma (convexity) ratchets your delta up or down as stock moves. that signals you to trade stock and improve your average price or scalp. as stock moves away from strike you run out of gamma, so run out of stock to trade and end up fully hedged--and then you hope that stock reverses and goes back to strike so you get gamma again and can scalp again.

graphically you know how a call or a put is a curve inside a hockey stick (laid flat on it's shaft). when you trade stock you're applying a linear transformation to that graph which means that you rotate it. so whichever way you started--the call was pointing to the right, the put was pointing to the left--after the stock hedge the graphs have rotated to be exactly the same bowl. delta neutral means the level point on the bowl is where stock is. and the graph curves up in either direction. that curve is your gamma. the curve up is also your profit as stock moves. as stock moves up the bowl, that means you are getting a delta. so you trade stock to get neutral and rotate the bowl flat again, so it's symmetric. the stock trades that rotate the bowl flat lock in little bits of profit by either improving your average price as you compelete your hedge or scalping as you take off your hedge.

mosta · #2 07-30-2005, 01:40 PM

jason's probabilistic argument and my trading/hedging strategy are two different ways to see put-call parity. here's a third one: consider the combination of long call and short put on the same strike. what is that position? if stock is above strike at expiration you exercise the call. if stock is below strike at expiration you get assigned on your put. so regardless, the combo amounts to a contract to buy the stock at the strike price at expiration, whether stock is above or below. what's that? a forward contract. what's the value of a forward? parity plus basis. there is no time value/intrinsic value to a forward. thus all the time value/optionality of the long call and short put must cancel out--must be the same.

suppose stock is at strike. the call is trading $1 and the put is trading $2. (suppose the specialist is bearish and doesn't know how to price options and thinks the put should be worth more because he "knows" stock is going down.) what do you do? buy the call, sell the put, sell 100 shares, and never look at the position again. you pay $1, collect $2, sell stock at strike, and it back at strike at expiration either by exercising your call or being assigned on your put depending on whether stock has gone up or down. riskless $1 profit/arbitrage. (okay there are interest and dividend factors/risks, and that is why call and put are different in that regard.) you have a forward hedged/arb'd against the underlying. you have no risks, not for stock price or time (yes for interest and dividend...).

you can also do it where stock is not at strike. then there will be parity in one optoin. but the value above parity must be the same or you get an arb.

PairTheBoard · #3 07-30-2005, 07:31 PM

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jason's probabilistic argument and my trading/hedging strategy are two different ways to see put-call parity. here's a third one: consider the combination of long call and short put on the same strike. what is that position? if stock is above strike at expiration you exercise the call. if stock is below strike at expiration you get assigned on your put. so regardless, the combo amounts to a contract to buy the stock at the strike price at expiration, whether stock is above or below. what's that? a forward contract. what's the value of a forward? parity plus basis. there is no time value/intrinsic value to a forward. thus all the time value/optionality of the long call and short put must cancel out--must be the same.

suppose stock is at strike. the call is trading $1 and the put is trading $2. (suppose the specialist is bearish and doesn't know how to price options and thinks the put should be worth more because he "knows" stock is going down.) what do you do? buy the call, sell the put, sell 100 shares, and never look at the position again. you pay $1, collect $2, sell stock at strike, and it back at strike at expiration either by exercising your call or being assigned on your put depending on whether stock has gone up or down. riskless $1 profit/arbitrage. (okay there are interest and dividend factors/risks, and that is why call and put are different in that regard.) you have a forward hedged/arb'd against the underlying. you have no risks, not for stock price or time (yes for interest and dividend...).

you can also do it where stock is not at strike. then there will be parity in one optoin. but the value above parity must be the same or you get an arb.

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That looks right to me. Your Long-Call Short-Put Arbitrage really seems to nail it down. After thinking about it some more I'm not so sure your original Call-Hedge Put-Hedge combos nail it down so well after all. I'm thinking the Hedge associated with the Put amounts to basically buying a Call, like how Jason1990 explained that the Call is a kind of convenience instrument that gives you the same result as if you employed an equivalent strategy with the actual stock. So it's no suprise that essentially buying a put and call gives the same result as essentially buying a call and put. Ah ha. That is, unless there's no put-call parity in which case the better method would be the one that actually buys the cheaper option and uses its corresponding hedge.

You guys are definitely convincing me but I'm afraid I'm still not quite out of the woods on the thing.

ok. so if the put-call are not issued in parity the market would immediately arbitrage them to parity per your explanation of the risk free arbitrage opportunity.

But I still can't get over the notion that if there is a Tripling Trend in the stock, AND EVERYONE KNOWS IT, then immediately upon issue of the Call-Put in parity the market would eagerly buy up the Calls and move them to a higher price that's out of parity with the put. Yes, the arbitragers would profit from the riskless opportunity, but wouldn't it worth it to the naked buyers of the calls to let them do so?

I suspect that if EVERYONE KNOWS THE TREND it changes everything. The Trend is irrelevant as long as it's unknown to the market? I'm still surrounded by trees.

PairTheBoard

jason1990 · #4 07-31-2005, 12:45 AM

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I suspect that if EVERYONE KNOWS THE TREND it changes everything. The Trend is irrelevant as long as it's unknown to the market? I'm still surrounded by trees.

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Okay, so imagine there's people selling calls and people selling puts. If everyone suddenly gets some information that the stock is going to go through the roof, then everyone's going to buy the call and no one's going to buy the put. But that can't drive up the price of the call. If the price of the call increases, then everyone will just buy the stock itself. In a sense, the call has the same trend as the stock, because it is derived from it.

Say you're at a casino watching Joe play blackjack. A stranger comes up behind you and wants to offer you a deal. If Joe gets a blackjack on this next hand, the stranger will pay you $25. How much should you pay the stranger for this deal? We'll you wouldn't pay him any more than $10, since you could just go put $10 on top of Joe's bet.

But suppose the deck has only As and Ts in it and everyone knows it. Do you suddenly want to pay the stranger more money because his offer has a higher EV? Hell no. Will he get a lot more sales? Of course. He'll get more sales and there will also be more people playing blackjack. But his price can't increase.

I know this is kind of a vague explanation and the BJ analogy has some holes, but I'm just trying to demonstrate something qualitative about what's going on.

PairTheBoard · #5 07-31-2005, 01:34 AM

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I suspect that if EVERYONE KNOWS THE TREND it changes everything. The Trend is irrelevant as long as it's unknown to the market? I'm still surrounded by trees.

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Okay, so imagine there's people selling calls and people selling puts. If everyone suddenly gets some information that the stock is going to go through the roof, then everyone's going to buy the call and no one's going to buy the put. But that can't drive up the price of the call. If the price of the call increases, then everyone will just buy the stock itself. In a sense, the call has the same trend as the stock, because it is derived from it.

Say you're at a casino watching Joe play blackjack. A stranger comes up behind you and wants to offer you a deal. If Joe gets a blackjack on this next hand, the stranger will pay you $25. How much should you pay the stranger for this deal? We'll you wouldn't pay him any more than $10, since you could just go put $10 on top of Joe's bet.

But suppose the deck has only As and Ts in it and everyone knows it. Do you suddenly want to pay the stranger more money because his offer has a higher EV? Hell no. Will he get a lot more sales? Of course. He'll get more sales and there will also be more people playing blackjack. But his price can't increase.

I know this is kind of a vague explanation and the BJ analogy has some holes, but I'm just trying to demonstrate something qualitative about what's going on.

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I think I see what you're getting at. Just trying to clarify here. Suppose the game is roll a die, get paid 5-1 if a 6 comes up. If the stranger gives you $6 if a player wins you'd pay $1 for the op. Or you could bet a dollar with the player. Now the Casino offers a same odds Christmas game but uses a die with 5 sixes. Would you pay the stranger more for the op? Not if you could just bet with the player. ok. But what would really happen? Any player with a seat would probably charge you $4 to place a $1 bet for you and the stranger would probably up his op price to $5. This translates to what would happen at the opening if news came out overnight that would cause a jump in the stock price. It would cause a corresponding jump in the price of call options that had been trading the day before. For someone selling a new call at the opening he would price it according to the expected higher opening of the stock. But he would do the same thing with new puts he might sell that day and there's no reason why any of this should disrupt the principle of put-call parity.

Now if an individual knew this news was coming out in the middle of the day he would most certainly profit biggest by loading up on mispriced calls at the opening. But the calls wouldn't be mispriced due to a problem with put-call parity. The calls would be mispriced due to a mispriced stock.

Am I getting anywhere close to green pastures?

PairTheBoard

jason1990 · #6 07-31-2005, 02:41 AM

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Any player with a seat would probably charge you $4 to place a $1 bet for you and the stranger would probably up his op price to $5.

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Okay, fine. So we knew the EV of the bet had increased because of the As and Ts. But because of that knowledge, suddenly the EV decreased as people with seats charged us more. But in the end, everyone still knows the EV of the bet, so they know the trend, and yet the stranger's price must be in sync with the price we get from the players. Effectively, he's just placing the bet for us. He might as well take our $5 and hand it to someone with a seat. He's just the middle man. His job remains the same whether the deck is full of Ts or full of 2s.

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Now if an individual knew this news was coming out in the middle of the day he would most certainly profit biggest by loading up on mispriced calls at the opening. But the calls wouldn't be mispriced due to a problem with put-call parity. The calls would be mispriced due to a mispriced stock.

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Sounds right to me. He could also profit just as much by appropriately loading up on the (mispriced) stock itself. The stock and its derivatives are in sync, like the stranger's price and the price of the players. There's nothing special about put-call parity here. The same is true of any derivative option. Say I want to sell you a contract that will pay you $50 if the stock ends below $60/share at expiration, but only if the stock had exceeded $100/share sometime between now and then. The price of this contract must be "in sync" with the stock. If you want to place this bet (if you think the stock will do this), then you could work out some complicated method of directly trading in the stock which gives you this outcome. How about I do that for you and save you the time? I'll be the middle man and place the bet for you. I've worked out the procedure and here is the initial cost. Is it a good bet for you to make? I don't know. Maybe it's a great bet. Maybe it's horrible. Whatever. The price is still the same.

jason1990 · #7 07-31-2005, 02:59 AM

I just fully noticed that you changed the game from BJ to dice and that the $4 charged by the seated players makes it EV neutral for the observers. If that's the case, okay. But the point was that the stranger's price must be in sync with what you can get at the table, even if it's not EV neutral. Whether the EV is positive or negative, he still must be in sync.

And I don't think they'd charge us $4. I think they'd charge us a little less and "share the wealth." Taking money from the house is an experience worth sharing. [img]/images/graemlins/smile.gif[/img]

By the way, were you suggesting by way of analogy that, barring inside information, stock prices are naturally EV neutral? Because I don't think anyone in the financial world would agree with this.

mosta · #8 07-31-2005, 01:38 PM

if the call price were to get jacked up out of alignment with the puts, you could do one of two things. you could buy puts and stock one-to-one, if you believe in the imminent upward move, and get the equivalent position at a better price. or you could buy put, sell call, buy stock, one to one to one, deciding that riskless arbitrage is a better game than taking a big shot on stock price movements. and if you don't do number 2, someone else will. when the exchanges cross listed their issues in 1999, various anonymous trading groups made a killing in the arbitrage game using the auto-execution system (small orders sent in and automatically filled on screen prices instantly). a price gets out of line and in seconds the auto-ex screen fills with trades. reversals for credits, butterflies for credits, calendar spreads for credits. when you get butterflied it takes a second to add up all the trades and confirmed what happened, and then you just had to grin even though you'd given money away. it wsa awesome. the primary arbitrageur was represented by brokerage that appropriately enough had the acronym B29--"The Bomber".

when options were singly listed calls and puts were generally kept out of line. that's what you call, raping the customers.

calls and puts do still get out of line sometimes, but it has to do with tender deals. the THC-PALM spin-off was a classic one where the reversal was trading for an $8 credit, palm was hard to borrow, etc etc. but it's only in the context of deals.

mosta · #9 07-31-2005, 03:04 PM

let me just emphasize: put + stock = call. it's the same position. that's why they stay in line. if they didn't you could arbitrage the option versus its synthetic equivalent. both options have the same optionality--same gamma, theta, vega--they only differ in delta. and we can change delta arbitrarily by trading stock against the option--without changing the option characteristics, gamma, theta, vega. therefore we can turn either option into the other. +call -stock = +put. graph a long put, long stock, and the sum. you get a long call.