Step 1: You are right -- The (constant) drift of the stock is a risk premium. The price of an (unhedged) option should be discounted by that. That brings the price of the zero-strike call option to the present price of the stock. That also takes care of your steps 3 and 5.Step 2: I can only handle constant drift -- Just as in BS. P/C parity should hold. Step 3: See step 1 above.Step 4: I am not familiar with the situations in which there are many possible risk neutral probabilities, and I don't see how all of them can lead to the same effective drift in the pricing formula, so I cannot respond to your step 4. Can you give me a reference to a textbook that discusses this situation?Step 5: See step 1 above.

A good reference is for instance a book called "Asset Pricing" by Cochrane, Princeton. Step 2: My point here is not about changing drift in time but comparing two situations with different constrant drifts. Try to think about it this way in terms of static comparison.

"But then, what is this vision of the market, where exchange and price communication are completely disabled?" It's the ultimate discrete hedging, where you do not hedge at all. Actually, that's what lots and lots of small time options traders do. They buy/sell an option, and sit there waiting for it to expire. My question is: How much should the sellers charge in order to break exactly even in the long run?

QuoteStep 1: You are right -- The (constant) drift of the stock is a risk premium. The price of an (unhedged) option should be discounted by that. That brings the price of the zero-strike call option to the present price of the stock. That also takes care of your steps 3 and 5.Omar,Of course, the stock price discounted by the drift is a martingale under the historic probability, and you can choose another porbability (log-normal as well) under which the stock price is a martingale when discounted with any other constant rate.Change of measure (risk neutral prob.) is completely equivalent to discounting with the drift for the stock itself. But, how would you price a call option with non zero strike without knowing its own drift ?Axel

"A good reference is for instance a book called "Asset Pricing" by Cochrane, Princeton. " That's one book I don't have. I'll look for it. "Step 2: My point here is not about changing drift in time but comparing two situations with different constrant drifts. Try to think about it this way in terms of static comparison." Then there should be no problem: My proposal is that in the absence of any hedging (for whatever reason that should be imposed on us), the fair price is given by the BS equation where the risk free rate (r) is replaced everywhere by the drift (\mu) of the underlying, including the discounting factor (which resolve the zero strike option problem that you raised in step 1). As \mu moves up and down, the puts and calls will move as they do in BS theory when the r moves up and down. I'll be delighted if you could point out to me where I go completely wrong.

I see. You would then discount constant future payoff by the rate mu? Again, this lead to an abvious arbitrage.

"But, how would you price a call option with non zero strike without knowing its own drift?" I think I miss your point -- My proposal is to price unhedged options using the BS equation but replacing the risk free rate everywhere by the drift of the underlying. That goes for all strikes, not just the zero strike option, which Philippe wished to test "my story" on.Incidentally, how do prices under discrete hedging go? Do they go up as the hedging time step (between rebalancings) gets larger and larger? My gut feeling is that dynamic hedging (with r showing up in the BS equation) gives the lowest possible price for a European call option, and that as the hedging becomes discrete, that price increases (all else being equal). My hunch is that in the limit that the time step becomes equal to the life time of the option (no hedging at all), the price becomes exactly equal to what one obtains from BS by using the drift of the underlying in place of r.

"You would then discount constant future payoff by the rate mu? Again, this lead to an abvious arbitrage.You mean arbitrage against another option that is hedged dynamically? You are definitely right: My 'fair price' will be higher than what dynamic hedging BS theory tells us that it should be. But I think that that's a problem with all discrete hedging prices: Fair pricing under discrete hedging schemes is different (must be higher -- more risk in holding a position) from that under dynamic hedging, and someone who can hedge better than me will arbitrage me (assuming I'm willing to buy an option at my 'fair price').But I still think it's meaningful to ask the question that I proposed earlier: How much should I charge for an option that I wish to sell but cannot hedge at all, if I wish to break even on the long run? I think we need to be very clear about this limiting case before we proceed to discuss the more interesting case of "partial hedging".

Help !Omar I think I really miss something :Suppose than the best delta hedge you can do imply a residual risk \epsilon, then you must have an expected return of f(\epsilon) + r, f > 0.1. I agree that f(\epsilon) and \epsilon should depend on \mu2. why is it equivalent, on average, to use \mu as a discount rate of the payoff function to price the option ?Axel

QuoteOriginally posted by: Omar"You would then discount constant future payoff by the rate mu? Again, this lead to an abvious arbitrage.You mean arbitrage against another option that is hedged dynamically? You are definitely right: My 'fair price' will be higher than what dynamic hedging BS theory tells us that it should be.No, I think NN2 means arbitrage against borrowing and lending money.The "constant future payoff" is just the payoff of a zero coupon.Let me try to summarize NN2's reasoning:1. No arbitrage between the family of option prices I produce (of strike zero or greater than zero) imposes a linear operator formally identical to an expectation.2. Pricing the "payoff of a zero coupon" right means my expectation should be discounted by the borrowing rate r.3. Pricing the "payoff of the underlying" right - and this means pricing it at the spot underlying price - means the drift implied by my expectation operator has to be the rate r.So Omar, in your world, you have to forgo at least one of the three propositions above, i.e. you have to accept either of the following:1. Arbitrage opportunities.2. Not pricing the bond right.3. Not pricing the underlying right.These three points all imply that you accept arbitrage opportunities.But probably this is OK with you because you forbid that someone buys from you or sells to you either of the underlying, the bond, or indeed any option, because you forbid any form of hedging.So the prices that you display are not prices meant to be "taken" by other market participants.What are they, then?Your answer: "the fee that the sellers should charge in order to break even"But, if I may ask, why is it allowed in your world that somebody buys an option from your seller, in the first place?In other words, is not the notion of a "price" simply breaking down in your world?

"But I still think it's meaningful to ask the question that I proposed earlier: How much should I charge for an option that I wish to sell but cannot hedge at all, if I wish to break even on the long run?"Isn't this the same as pricing a derivative on a non-traded underlying state variable? Purely as an example to make a point, assume that the non-traded state variable X follows process:dX = Drift(X,t) dt + Sigma(X,t) dWand you want to price a derivative whose value is a function V(X,t). In this case you end up with the pricing pde:dV/dt + 0.5*Sigma(X,t)*d2V/dX2 + (Drift(X,t) - Sigma(X,t)*Lambda(X,t))*dV/dX - r.V = 0Where Lambda(X,t) is the market price of risk of the non-traded state variable X, as a function of X and t:Lambda(X,t) = (Drift(X,t) - r) / Sigma(X,t)In the case of a non-traded underlying, the best you can do is to estimate Drift(X,t) and Sigma(X,t) on day one, use these estimates to form an estimate of Lambda(X,t) and hope that you've got it about right. Contrast this with the case of a traded underlying, where you can think of a replicating portfolio of shares and riskless bonds as allowing you to re-estimate Drift(X,t) continuously. (In the case of a market where you can form a replicating portfolio of shares, volatility swap and riskless bonds, you can think of the replicating portfolio as allowing you to re-estimate both Drift(X,t) and Sigma(X,t) continuously).So the value of an option on a non-traded underlying variable will differ from the value of an option on an identical, but traded, variable by the amount of mis-estimation of the market price of risk. This is not the same thing as substituting "drift" for "riskless rate" in the BS pde. However, the question is a good question as it focuses attention on the precise nature of the famous BS cancellation. I have tried to argue here that this cancellation can be thought of as allowing the replicating portfolio to continuously re-estimate the parameters used in the market price of risk. This mis-estimation error will be zero in the case of a traded underlying variable, but not zero for a non-traded underlying variable.

Johnny, what you say applies when the derivative is underlain by a non tradable variable, for instance the interest rate.In that case, this variable is not what NN2 calls a price. When a variable is not a price, it may indeed not have a drift equal to r. However, Omar is in a different situation. The underlying in his case is a price (i.e. is tradable), yet Omar wants to refrain from trading it (for hedging purposes, among other things).NN2's point stands. Not trading the underlying does not make it non tradable (i.e. does not stop it from being a price).

"In other words, is not the notion of a "price" simply breaking down in your world?" You are right. The way that I phrased my question gives rise to the ambiguities that you mention. Let me phrase things differently:Suppose I live in a BS world. I sell a certain option O at BS price x. I do not hedge my short position in any way. Why not? Let's say it's because I believe that "hedging is for pussycats", and that I have an infinitely deep pocket. Suppose I repeat the same exercise (always selling O, always under exactly the same initial conditions) N times. My question now is: How much money would I lose per trade, in the limit N --> infinity?The above way of phrasing things is precisely equivalent to my earlier statement, as far as requiring "a number" is concerned, but I hope it removes the conceptual difficulties that you and Philippe have raised.

At last the forum thread I have been waiting for. I more or less agree with Omar;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.Now, just a brief Bayesian/anti-sigma-algebra-gobbledegook diatribe:Many of the obstacles to a more powerful, coherent and intuitive theory of pricing, in my opinion, arise from over formalization and a receding understanding of what probabilities are. This is compounded by the martingale approach, which gives the impression that there is something magical about the risk neutral process - it is just the process which, by luck as it were, corresponds to a perfect hedging strategy. Thus, why not call this an expected value under your (real) process/distribution for the stock price:Suppose we wish to price a European call, in the absence of any hedging. I would like to claim that the "fair price" (in the absence of any hedging) would be such that a trader breaks even after trading that very option N times, in the limit N -> very large number.My question now is: How much money would I lose per trade, in the limit N --> infinity?One also should be careful to distinguish between questions such as 'How much should I charge for an option that I wish to sell but cannot hedge at all, if I wish to break even on the long run?', and 'How much do I expect to lose?'. The former requires a utility function because it requests the formalism to make a judgement call, the latter merely asks the weighted average (although of course more broadly speaking selection of the averaging operator corresponds to a previous choice of utility).Nevertheless in my opinion Omar's question, 'How much do I expect to lose?', is valid and pedagogically & conceptually useful. It can and must be answered simply. Actually my view is that it is more interesting to say what is the probability distribution of my losses, with no hedging. Then the next step is to say what is the probability distribution with hedging a la X. Finally when X = perfect delta, we have BS.Anyway my boss is looming over the cubicle, so; I think NewNumberTwo is mistaken in his argument on put-call parity (Omar's #2). Indeed PC parity relates the price of put, call and stock (or forward). Also as mu increases so must call price, and put price must fall. However this does not threaten PC parity because the forward price, which must be defined as the expected price of the stock, also increases. What is affected is that the forward price is not equal to the arbitrage forward price, but this is OK because we are talking about what we expect to happen, not what we know will happen. Got to go!

QuoteOriginally posted by: SimplicioMany of the obstacles to a more powerful, coherent and intuitive theory of pricing, in my opinion, arise from over formalization and a receding understanding of what probabilities are. This is compounded by the martingale approach, which gives the impression that there is something magical about the risk neutral process - it is just the process which, by luck as it were, corresponds to a perfect hedging strategy.Wrong. "Risk-neutral probability" is just another (fancy, I agree) word for "pricing system." It is formally identical to a probability measure, and the pricing algorithm is then identical to discounted expectation under that formal probability.There are as many such risk-neutral probabilites as there are expectation operators (probability is the expectation of the indicator function), and as many such operators as there are positive, linear operators.They all imply a drift or r for the underlying, if the underlying is tradable and the money account is tradable.Perfect hedging in a BS world (i.e. Brownian) simply picks a unique risk-neutral probability (or pricing system), the one consistent with the initial cost of the dynamic hedging strategy.It is because of dynamic hedging that the volatility number in the BS formula had better be the real volatility of the underlying (i.e. the one driving the rebalancing of the delta).When hedging is not allowed (following Omar), the number of pricing systems is again infinity, even in a BS world. Just use the BS formula with any volatility number (yet with drift r), however different that number may be from the real volatility of the underlying. (Of course you have to use the same number across all strikes and maturity dates).The discounted expected value under the real probability distribution of the underlying (not the formal "risk-neutral" probability) is what Omar is contemplating.Only this will not price correctly the underlying!