Just reading a comment on martingale approach in Paul Wilmott's book: Quote The major drawback with this [martingale] approach is that it requires a probabilistic description of the financial world. Modern models are moving away from this simplistic idea, and martingale method may no longer be useful.Can somebody in the know enlighten me in what direction are modern models really moving? George

This is very interesting. Exactly what page and which Wilmott's book are you talking about ?

That is from PW Introduces Quant. Finance, chapter 8.12.1

I wouldn't agree with Paul's statement, but there are several alternatives to martingale pricing. Most of them amount to some kind of simulation of replicating portfolios. The key is that simulations do not assume drift at the risk-free rate of interest. Another alternative is to have a specific liquidity, risk-preference, supply/demand or behavioral models. Equilibrium models are older than martingale models. I think they are still useful, but I doubt they're what Paul had in mind.

Martingale methods will still be there in the long run, don't worry.

Aaron and madmax,thanks for your replies!The statement in the book really confused me.George

madmax: I have a question about your opinion on martingale methods... why do you think they will be there in the long run? surely it's ingenious idea... risk-neutral pricing... it simplifies the life a lot, and has not sooooo many restrictive assumptionsThe biggest problem with martingale approach (IMHO) is that it requires no arbitrage as justification. Do you think it's realistic? There are limits of arbitrage due to liquidity problems, etc.. even in "complete markets" due to different filtration of the agents we may have kind of arbitrage that will exist in equilibrium... transaction costs kill the martingale principle pretty well - it's still not quite solved for GE with fixed transaction costs.. SDF may not be unique even in the payoff space => martingale principle does not work... we may talk about some approximate martingale principle, giving us the transformed probabilistic description of the world, but not the exact, but rather an approximate one, or different views on the world for different classes of investors as in Information Economics papers, or in papers with limited participation economies.. IMHO, martingale principle in the form used now should not survive for long... but who knows - BS formulae are there for 30 years, and not only traders use them with some small modifications to get option pricing benchmarks, but rather these formulae determine the market levels themselves (to some extend) as our mind does not accept too large a deviation from the "common knowledge", from what was considered THE MODEL for so many years... I would appreciate your opinions on the topic.. Best, DROPlet

By the way, in the paper "Statistical Arbitrage and Securities Prices" Oleg Bondarenko @ University of Illinois at Chicago - published RFS Fall 2003 - talk about some of these issues... do you believe the is no better (as simple, but at least as efficient) alternative to the martingale pricing? Regards, DROPletPS I really hope for a discussion.. this is an interesting topic, though it's quite theoretical

QuoteOriginally posted by: phenomenologistCan somebody in the know enlighten me in what direction are modern models really moving?In my humble opinion The "hot" topic now is fractional Brownian motion models in finance, which introduce the non-Markov behavior of financial instruments.However these models have "bad" intrinsic features.As Sottinen put it in his presentation Fractional Brownian Motion as a Model in Finance , slide 7, QuoteWith generalized solutions to (2) the fractional pricing model is arbitrage free and complete, i.e. problems (b) and (c) are solved [HO]. However, it seems to be difficult to give an economical interpretation to the formulas. With path-wise solutions to (2) and with continuous trading one can do arbitrage in the fractional pricing model [C, Sh]. Surprisingly, the arbitrage arises in a modified binomial approximation [So]. As we see, both integral solutions are not quite satisfactory while using fractional Brownian motion to model processes in Finance.

You might also want to have a look at what Alexander Cherny termed the "Possibility Approach"; I heard his talk at the BFS Congress and found the idea very interesting.Here is the paper.The main difference from the traditional method is that he <i>does not</i> specify a probability measure on his space.A.