Hi,Thank you very much for your replies. Numbersix, I have thought about your explanation on conditionnal probabilities (which you also explained in your article in wilmott) and came to the following questions :I have tried to solve the problem for a small number of spot and time step. So I take a simple binomial scheme (the recombining one), starting from So and going through time t1, t2 and t3. On time t1, I can go to Su and Sd, on time t2, Suu, Sud and Sdd, and on time t3, Suuu, Suud, Sudd, Sddd. From the vanilla option prices, I can access the following probabilities : So -> S...To price my barrier options, I need the other 9 conditionnal probabilities (from one point, the stock can only go up or down one step). Using the fact that from one point, the departure probabilities must add to one (after all, the stock has to go somewhere !), I have 5 equations. Moreover, using the conditionnal probabilities rule, I have 7 more equations (one for each point in time t2 and t3). So, I have 12 equations, 9 unknowns and no repetitive equation. I tried to solve it and could find a solution.Of course, if I suppose that from one point, the spot can go to every other point in the next time step (not only going up or down), then I haven't got enough equations to solve my problem. This is basically what you suppose in the article : the stock can jump from one state on time t to another one on time t+dt. But for a simple binomial tree, I have enough information. Don't you think that if we split the asset and time scale enough, we could go back to the binomial scheme ? It seems to me that you are basically including jumps in your underlying process (which is of course true because the underlying moves at least one tick by one) and that is the reason why the diffusion process do not yield the same solution.

QuoteOriginally posted by: decombh4Don't you think that if we split the asset and time scale enough, we could go back to the binomial scheme ? It seems to me that you are basically including jumps in your underlying process (which is of course true because the underlying moves at least one tick by one) and that is the reason why the diffusion process do not yield the same solution.NO!!We cannot go back to the binomial scheme.The binomial scheme is the equivalent of diffusion in the discrete set-up.You cannot discretize jump-diffusion in a binomial tree, no matter how fine the time step!Also, note that the prices of the vanillas struck at nodes of your binomial tree are always enough to determine all the transition probabilities. Hence determine everything. This is known as "implied tree."But like I say, this works only in the binomial, or equivalently in the continuous set-up, in diffusion.The whole point is that when considering alternative processes (that you can no longer discretize but on a whole grid, and with transitions going from each node to every other node), then vanillas are no longer sufficient to determine the transition probabilities.

NumberSix,I am thinking again about the way you want model to the effect of non-stationary variable by introducing a markovian process of switching regimes. I've got the impression the non-stationarity is hidden in the transitory regime of the markov process to its stationary state. In some sense if you have infinitely many states of regimes-process or simply a diffusion on it (and jumps occur often), it will be obliged to move from one state to another over the time step. But as long as if jumps are rare you can just say that it didn't happen and nothing has changed.now imagine you use your model for a sufficiently long period and the market parameters doesn't change (so as to keep the calibration results stable). In that case your calibration will say that you are stuck in one of your regimes and that will start to contradict your model after some time. Wouldn't this give bad hedges over the period?Alexei

Alexei,I think you nailed it. The time effect in these Markov time homogeneous models (call it non stationary if you want) come indeed from the transitory path to some stable steady state. That is precisely the beauty of it. This steady state of course need not be stuck in one regime!The transitory path may be very long and assume very curious shapes. That is all we need as a modeling device. Now what does Nature (or Market) actually do? Who knows. There are things that you cannot prove right or wrong:- is the dynamic of prices stationary or not,- is is time homogeneous or not.What really matters is that we can always represent it (or so I conjecture until proven wrong) as a path of some time homogeneous process. The big word here is representation. The goal is to represent the market vision about the future in a concise and robust way. We all know that tomorrow will be another day and that the market need not be consistent tomorrow with its view of yesterday. That is why it is so difficult for us to talk to Econometricians and Phycisists, always on the hunt for the definitive Data Generating Process. I view personnaly the world as highly non stationary! That does not prevent me from using a stationary representation to model the view of the future as expresed by the smile and the term structure of CDS. The potential sources of non stationarity being so extreme and so wild, I think that it is fruitless to try to grasp them in a meaningful model. But now I have to live with the fact that tomorrow we shall certainly have to recalibrate, in full contradiction with the model itself. Calibrtion, Co-calibration, Recalibration...

Thank you numbersix for your comments. You are right about the diffusion process.A couple of questions on the paragraph entitled "Risk-Neutral vs Real Probability" (which is I think close to another thing I asked in another thread) :- In BS, I have zero variance only if I hedge continuously. When I hedge discretly my gamma, my variance is not anymore zero. And this is without accounting for jumps, change in volatility, etc. I just suppose that the asset trade continuously but that I hedge discretly (due to transaction costs; otherwise, I would hedge all the time). I suppose that this is a case of which you were talking about when you said that real probability (and for example, the real drift of the stock) matters for the hedge, no ? Can we say that in this case the market is incomplete ? - In more advanced models (for example, stoch vol or jumps), everything becomes more complicated to me. From what I have understood, even if I trade continuously, my variance is not zero. Stoch vol : I suppose that I only hedge with the underlying (no other options for the moment). If I am right on my definition of the market price of risk, is the market complete ? By this I mean : if I input (by pure luck) the right number for the market price of risk, can I hedge only with the stock and have zero variance ? If I can use other options for my hedge, I can possibly hedge this price of risk by hedging the lamba in Heston, no ? What happens if I am not right about the form of the market price of risk (I think Heston takes it proportionnal to variance) ? Jump : I think that there are two possibilities here : include again a market price of risk (just like in Heston) or make the assumption that the risk does not exist (using the risk-free for the expectation).Is it because of this use of hidden utility functions or diversification assumption (nearly the same to me) that you prefer to use your HERO ?

In the pure BS setting, just hedging at discrete interval does a very good job and the drift is a second order problem. As soon as you enter the incomplete market realm with say stoch vol or jumps, things are much more complex. The big idea here is that pricing and hedging are no longer uniquely defined as in BS. Severql non arbitrage prices are possible, each corresponding to a different choice of prices of risk (which could possibly be extremely complex processes).Since exact hedging is not possible, you need some sort of objective function to decide what is a good hedge. And finally, there is no guarantee that the market price of a derivative is close to the value of some decent replication strategy, whatever this means.Not much is left from the BS world, which explain why people will do everything they can (local model without jump where spot is the only state variable) to maintain the nice BS results: unique exact hedge and therefore unique pricing by absence of arbitrage.I believe that Finance is slowly growing out of this fantasy land. What will be left from BS will be the huge idea that dynamic strategies increase tremendously the opportunity set available to investors.