Omar,Yes, by assets I mean non-derivatives, an equity portfolio for instance.I cannot agree with the way you propose to price a European call in absence of a hedge. You propose to compute the price as the discounted expected payoff of the option in the historic (or objective or true) probability, which amounts to say that you are risk neutral. Most agents are risk averse and should therefore modifiy the historic probability to take into account risk aversion. This is done through the marginal rate of substitution, another word for it is a pricing kernel. Only if you are risk neutral will this kernel be equal to unity. The absence of arbitrage theory tells us only that this kernel exists and that it should be positive. When markets are complete and a perfect hedge is available, then it is also unique. In the BS case, this unique kernel explains why the price of the option is computed with a probability where the drift is set to r. I believe that the interesting situation in finance is the case in between the two extremes: on the one hand the complete market assumption of BS is not realistic, but on the other hand saying that no hedging is possible is not right either. The reality seems to call for an imperfect hedge, and the pricing should take into account this imperfect hedge, as well as the quality of an optimal hedge.

Philippe, I cannot agree with the way you propose to price a European call in absence of a hedge. You propose to compute the price as the discounted expected payoff of the option in the historic (or objective or true) probability, which amounts to say that you are risk neutral. I am proposing an extremal situation, the very other extreme of perfect hedging, just to make things very simple and clear. So let me ask you again: Suppose I cannot hedge at all, for whatever reason. How would you price a European call option? Would you still modify the historic probability? What would this modification correspond to in terms of trading?

"Suppose I cannot hedge at all, for whatever reason. How would you price a European call option?"If you can't hedge at all, what does "price a European call option" mean? With perfect hedging one eliminates the stochastic component of the portfolio and a point estimate of the terminal value can be obtained, but without perfect hedging the stochastics remain and one is left with some sort of distribution of terminal values. What does "price" mean in such a situation? the mean of the distribution? a utility-weighted mean? a confidence interval about the mean?

In the absence of any hedging, I propose that the fair price of an option would be such that you to break even if you trade that option (under the same conditions) N times, in the limit N --> infinity. You end up being risk neutral, but only asymptotically. In the presence of perfect hedging, you break even on a trade-by-trade basis.

Well, in order to break even as N -> infinity, you have to still be around and trading at infinity. That means you have to eliminate the risk of ruin and adopt a risk averse pricing strategy. This would make your pricing indistinguishable from someone using a discounted, utility-weighted mean of the terminal distribution as the price.

Correction to my last post: If trading were sequential, you would initially use a risk adverse pricing strategy and adopt a more risk neutral asympotically as your trading capital grows (or a more risk averse strategy should your capital diminish).

I'm using the usual idealisations that one makes in situations like that: an infinitely deep pocket, etc. Just as in BS you assume that you are allowed unlimited borrowing, at a fixed rate, etc. None of these assumptions is realistic, that's very clear, but they are idealisations, and serve as starting points. Next time I derive BS in a class, I would like to do that in 3 steps:1. Price in the absence of any hedging, defining a fair price as above.2. Hedge in the presence of an interest rate = drift of stock.3. Take interest rate to be different (had better be lower) than drift.The trouble with the usual derivation one goes to (3) above immediately, and quite a bit of what's going on is lost and things become very mysterious. Of course, the real thing is 4. Hedge as much as you can and optimaly manage the residual risk. But that's tough, and we are trying to see what's going on step-by-step. So we start from (1).

Omar, your point is really interesting. But I'm not sure to understand well your 4th point.when you say 'to optimally hedge the residual risk' does it mean to minimize the variance (with a quadratic utility function) or to compute an efficient hedge frontier (with another utility function CRRA or log for the risk neutral case) ? Is the choice of the utility function important ?Does it mean that hedge depends on the anticipated return on the stock (ie. integral of the drift function) : in this case, the smile could be explained with a stochastic drift model... right ?Axel

QuoteOriginally posted by: Omar Next time I derive BS in a classI think this is the really interesting question.Basically, Omar, you are looking for the "simplest" way of understanding option pricing (or building up this understanding).I agree that BS is very mysterious when you look at it deeply.But then is it not because it is based on a very improbable coincidence?(And what does one see, when one looks deeply into a coincidence? Might not the coincidence be hiding the depth?)I think that the real innovation in BS is dynamic hedging and self-financing portfolios (by the way, I am not sure they have invented it).As a matter of fact, the concept of dynamic hedging and self-financing portfolios is very general and applies to all kinds of underlying processes. I claim that the BS theory is based on a coincidence because you can easily imagine a pedagogical situation where you wish to restrict dynamic hedging to the simplest and most natural underlying process you could think of (Brownian motion, of course), only you could not imagine that this very natural pedagogical act was going to bring about a whole new theory, BS theory and complete markets, even less so that this theory was going to become the textbook theory of option pricing!Likewise, you can conceive of dynamic hedging in discrete time and discrete underlying price. Only you could not imagine that when you proposed the binomial model for simplicity of exposition, there would also arise a whole specific theory: complete markets, and the ensuing option pricing in a discrete setting!Sadly, the binomial model has also become the textbook case of option pricing theory in discrete time.In other words, I think the simplest way of looking at BS (or its binomial homologue) is to see that they should not exist!I think the right way of teaching option pricing theory is to begin with incomplete markets, self-financing portfolios, dynamic hedging as a stochastic control problem, residual risk, etc., then take a few minutes to amuse the students and show them the degenerate case of Brownian motion (or its binomial counterpart), and the collapse of the residual risk! Complete markets should not exist either. They are a concidence too.Of course coincidences do exist, and they are very amusing. But the most important thing to note about their existence is that is noticed a posteriori. Complete markets should not a priori exist. Can you now see the scandal in teaching them as option pricing theory!To take a physical analogue, I believe the simplest way of teaching Newtonian mechanics is to teach Quantum mechanics first.And of the two theories, I think the first is the most mysterious. Its general statement is: "A physical object has a certain property, and will display a certain behaviour when subjected to a certain experiment." But then, what is an "object"? What is a "property"? And what is an "experiment"?

I fully agree with you -- starting from vanilla BS theory, too many things fall into place at the same time and fit in together so well and one has no clue what's going on. The trouble is that one thinks one knows what's going on while one doesn't. Proof: no two people agree on why the drift doesn't show up in the BS equation. Look up MJ's post the other day. He gives many reasons. As Feynman once said, "One gives many reasons, when one doesn't have a single good reason". To my mind, that's all a direct consequence of our collective lack of understanding of BS.The situation is even more dramtic when you start from the binomial model. I read frequently that "One should make things as simple as possible, but not simpler". I think the binomial model makes things so simple to the point that a beginner misses the point. If you can only teach continuous time BS and the binomial model, I highly recommend covering BS first, asking what the story on dynamic hedging is, then going down to the binomial model to try to see what's going on.

QuoteOriginally posted by: acabrolOmar, your point is really interesting. But I'm not sure to understand well your 4th point.Neither do I really. I know nothing about stochastic control, but the ITO33 people solve HJB before breakfast.

QuoteOriginally posted by: OmarThe situation is even more dramatic when you start from the binomial model. I read frequently that "One should make things as simple as possible, but not simpler". I think the binomial model makes things so simple to the point that a beginner misses the point. If you can only teach continuous time BS and the binomial model, I highly recommend covering BS first, asking what the story on dynamic hedging is, then going down to the binomial model to try to see what's going on.I agree that the binomial model makes things so simple to the point that a beginner misses the point, but I disagree with the general tone of your comment which makes it sound as if the binomial was a simplification of BS itself, hence an over-simplification!On the contrary, I think the binomial and BS are two independent theories of complete markets, the one in discrete time and the other in continous time. (It is just a mathematical coincidence that the binomial option prices happen to be the discretization of BS prices - I mean that the continuity argument from the binomial to BS is just a numerical argument, not a financial argument; and the real hypocrisy in the binomial derivation is that it says: "Well, for the sake of simplicity, let us discretize Brownian motion in the simplest way we can, matching first and second moment, and this is the binomial world!" while it hides the fact that binomial world is precisely what it needs in order that the discrete hedging be perfect). And the argument that I would like to draw from the caricatural simplicity - even simplism - of the binomial, is rather the following:- Much in the same way that it strikes us as ridiculously improbable and outré that one should postulate a binomial world just for the sake of establishing complete markets (in discrete time), it must strike us as equally outré that one should postulate a Brownian world for the sake of complete markets in continous time.

QuoteOriginally posted by: OmarIn the absence of any hedging, I propose that the fair price of an option would be such that you to break even if you trade that option (under the same conditions) N times, in the limit N --> infinity. You end up being risk neutral, but only asymptotically. In the presence of perfect hedging, you break even on a trade-by-trade basis.Dear OmarWe discussed your puzzle with NumberSix (or his double, I am not sure) this morning on Eurostar. I propose to proceed step by step.Step 1Do you agree that your method should price a very special call, the one with zero strike price, namely the underlying itself ?I assume you answer yes.So let us see what your theory implies for the underlying. Because a stock has a rather large drift (a fact of life) you end up on average always making money if you buy the stock today and sell it later at some given maturity. Following your story, this means that the underlying is not well priced today because it is too cheap. Would you call that an arbitrage? Surely not, you earn a return on the stock, also called a risk premium, because this investment is risky. This is really Investment 101. Of course nothing here hinges on the ability to hedge. Step 2So you want calls and puts to depend on the drift? Following again your story, for a given spot price you should see calls becoming dearer and puts becoming cheaper as the drift increases. Do you agree?But then you would violate one of the Golden Rules of option pricing: the Put-Call parity, which implies that Puts and Calls must go up and down together.Step 3Let us move one step higher in theory. In absence of arbitrage, a pricing system must be a positive and linear operator. Because the price of the bond is one (assunming r=0 here for simplicity), this positive operator can be understood as being an expectation operator applied to the payoff at maturity. If r is non zero, you end up with a discounted expectation. What can this expectation be? Under which probability? Nobody knows, and absence of arbitrage alone will not tell you in general. You know however one thing: this operator should price the underlying itself correctly (go back to step one if you have a problem here and start again, but stop if you loop more than three times). Pricing the underlying correctly means that the price today is the discounted (at the risk free rate) expectation of the price at maturity, for our unknown probability. This proves that for this mysterious probability, there is no risk premium for the underlying, or that its drift is r. That is precisely why any such probability is called a risk neutral probability. To repeat, Risk Neutral Probability is another word for "a linear and positive pricing system". All we need here is the knowledge of the current price of the underlying today, no hedging argument is needed so far. Step 4Surely you may ask, are there many possible risk neutral probabilities? In general the answer is yes. Before going any further, remark that they all yield a drift of r for the underlying (and by the way, for any other security). Enters the dynamic hedge. If markets are complete, every payoff may be perfectly replicated, and in absence of arbitrage, the price of a derivative must be equal to the initial value of the hedging strategy. You end up with a unique pricing system, a unique risk neutral probability which yields also the value of the perfect hedge. This is what happens in continuous time for the BS setting or in discrete time with the binomial tree. In a more general case with incomplete markets, if the security cannot be hedged perfectly, then many pricing systems may be consistent with absence of arbitrage. Bottom line: the drift is r in BS not because of perfect hedging but because all pricing systems (consistent with no arbitrage) imply a drift r and there happens to be only one such system in BS. Step 5And what, you may again ask, if we live in a BS world but are not allowed to hedge? What I said above applies and there is an infinity of possible pricing systems. For instance, consider the BS formula with various implied volatility numbers. The "pricing system" you propose is however not even included in the large set of possible pricing systems consistent with no arbitrage because you do not price the underlying! Go back again to step 1, but do not loop this time. In incomplete markets, it may be useful to derive a pricing system from an imperfect hedge. It is also quite interesting to learn how imperfect a hedge can be. Again this is a very different issue to r being the drift in all pricing systems. I'll be waiting for you in "Le Pas Au Delà" (the Step Not Beyond).

suggestion : when Omar says 'imperfect hedging' he may refer to arbitrage opportunity in the very short term ? in this case, the absence of arbitrage argument you mention doesn't hold any more, and there is no more expectation operator which works for the underlying itself.... hope it'll helpAxel

If Omar means by "absence of any hedging" - that he doesn't mind that the family of option prices he produces may generate arbitrage opportunities between them, - and that he doesn't mind that because anyone wishing to put in place such an arbitrage strategy would in effect have to buy some option and sell some other option against it, or in other words, perform some form of "hedging,"then, I agree, anything is possible.But then, what is this vision of the market, where exchange and price communication are completely disabled, and every prisoner has to wait indefinitely in his isolation cell, in order to break even?