This is most marvellous, indeed... If you claim that you have replicated the payoff of a 55% delta vanilla call, congratulations, you just answered your own original question (below). Otherwise, I really don't understand what it is that you're asking/arguing/disputing. I think it would help if you could phrase your question/statement explicitly, rather than running circles arnd it.
Do you agree or disagree with the hedged bets comparison to the vanilla option? jcapital gave an interesting insight, by noting that hedged positions may lower premium compared to what they might request under no hedge. If we take the idea a bit further, could an implicitly hedged position (like vanilla call) be affected because of its internal hedge in addition to the hedging external to it? Could the hedging process by its nature of moving a bit of risk away to the other guy leads to systematic relative underpricing and overpricing of options? If you were someone interested in playing the delta effect, would you use bet1, bet 2, vanilla call, a combination of the three/etc? Explain how and why.
For the sake of your argument you produced two prices for a single, model-dependent assumption. That is, that the two digital bets need to be priced at .55 fairval, not your 45 & 55 valuation. The dissection results in two identical payoffs, but one pays AON (asset or nothing) in stock and one pays AON (all or nothing) in cash. I know what you're trying to convey, but it's wrong. You cannot replicate or make fungible a digital and a vanilla option. You cannot make a moneyness "valuation" argument for a vanilla.
Is that right? It seems that in the case of cash-or-nothing, you get the exact same payoff regardless of how far from the strike the underlying ends up, ie, a fixed dollar amount. But in the case of asset-or-nothing, you get participation in however far above the strike it ends up, because you're being given the asset itself, which you can (presumably) immediately liquidate for cash. This could be good or bad, relative to the cash-option, depending on how far it's moved.
I was only looking in terms of probability. Never traded an AON (asset) option, but plenty of cash digitals. I haven't given it much thought. Of course you're correct. Disregard. The AON in stock is a synthetic (Euro-exercise) vanilla call. The only correct comment I made in that post was that the "cash" digital is not replicable with a vanilla. :eek:
It looked correct, at first glance, although I confess to not really spending too much time on it. I don't quite understand what particular insight you might be referring to here. From what I could tell, Ran.Cap's point was simply an illustration of a well-known and well-understood fact that one of the "fudgey" assumptions that are indispensable to option pricing is the so-called completeness of mkts. It's also a well-known and well-understood fact that option sellers are short implicit liquidity risk, i.e. they're the ones that get screwed if the completeness assumption gets violated. Option prices do take this into account, although it's understandably difficult, if not impossible, to decompose option prices into "volatility premium" and "liquidity premium". If this is the insight you're struggling towards, I wish you the best of luck. If/when you manage to quantify liquidity risk, pls let me know and I will personally pay for your ticket to Stockholm, so that you can claim your Nobel Prize. If the insight you're referring to is something else, I don't know what it is, I am sorry. Firstly, relative to what? Secondly, see above. I have absolutely no idea what makes a vanilla call "an implicitly hedged position". I also don't really understand what an "internal hedge" is. A vanilla option, to me, is the basic irreducible contract that I can price from first principles, if necessary (with some assumptions about the mkt and using a replicating portfolio that consists of basic non-contingent instruments). If I am not mistaken, it seems to me that your notion of "internal hedge" depends on the ability to replicate a vanilla option payoff using digital options. That, to me, is invalid circular reasoning, as I don't know how to price digitals unless I know how to price vanilla options. Again, I apologize if I have misunderstood your point. In practice, the answer is obvious. In theory, I believe you will find the passage above relevant to this question as well. Would it be too much to ask you to just clearly say what you want to say? I am happy to play along, but it would be mighty nice to know where we're all headed with this.
Since this is going exactly nowhere... how about that Germany, eh? Made my fav Argentina look foolish. Blanked 'em 4-zip!
What a team! I confess I was not a fan and I was really backing Argentina (after Ghana's tragic exit, of course). However, the Germans do play beautiful football. Like clockwork, in the best possible sense. I didn't think they could crush Argentina as easily as they dispatched the sh1tty England, but they're really somethin' else!
There is no substitute... Yeah, it seems as if they had been saving it for the quarters. They look unbeatable.
Assume for the moment that we know how to price an arbitrary binary option. Can't we view a vanilla option as a collection of (hypothetical) binary options at different strikes, with different payoffs? Payoff would then be an integration along the strike price axis, from 0 to Price@Expiry. Thinking out loud here...a perfect hedge for a binary with strike $X and a payout of $1.00 could be made with a call spread at $X and $X + $0.01. The price of that spread should equal the price of the binary divided by 100 ($1 = 100*$0.01). Anyway, I'm meandering now...and looking forward to a possible SuperClassico Ultra-Retro Germany-Holland rematch in the final.