In a nutshell the martingale property of the beahviour of asset prices states that an accurate estimate of the future price of an asset may be obtained from a current price information, and the relevant information used to calculate asset price is the latest price information all these efficient market scenarios are well described by Fama.

Martingale approach is a way to express zero-sum game in diffusion terms.

It' s famous and wellknown by all here the simplest definition of martingale: A given stochastic process X with the property Et[ X(s) ]=X(t) for every s >= t any dX = 0dt + sigma(X,t)dz is called a martingale.Any zero drift process is a martingale -> d(X)=sigma dZand its expected value in the future is coincident with its value today.

Isn't it remarkable that our inability to predict, our inability to see beyond our noses, should result in the elegant Martingale Approach?

"Isn't it remarkable that our inability to predict, our inability to see beyond our noses, should result in the elegant Martingale Approach?"Instead of thinking in terms of "inability to predict", think in terms of "freedom to be".

I have the "vague" understanding of Martingales being essentially stochastic processes free from a forecastable drift element. Hence the no free lunch principle. I never was a big fan of measure theory but I get the vague idea. The type of question I am interested in has less to do with dynamic hedging and more to do with "detecting tradeable inefficiency".This poses questions in my mind of "near" martingales, perhaps along the lines of newtons' semi-martingale example. I have an information theory background so I always want to figure out just how unpredictable something may be (the quant's curse - I think I can forecast it, really...)Since the mathematical literature on martingales is vast (not my field) I imagine this is done to death somewhere :-)Very crudely, I imagine that if I added a sinusoidal predictable wiggle to a Martingale there would be some kind of "forecastable element" I might pick out of the noise, and that the ratio of the amplitude of this sinsusoid to something like the volatility cone breadth on one period of the wiggle would give me a crude measure of how "non-martingale" my process was??? Anybody have a clue on this one, like some engineering literature...The intuitive idea seems natural so I expect it has been fully analysed somewhere.

If one thinks of diffusion processes as a mix of signal (drift) and noise (local variability), then isn't it remarkable that all this driftless martingale does is make noise?-----------------------------------------------------------------------------------------------Mark Twain - "Cauliflower is cabbage with a college education."

I know I am posting well into the thread, but I felt that Neftci's Introduction to the Math of Financial Derivatives (2000) does an OK job of describing Martingales and gives a couple of examples.

One of the best ways to eplain martingale approach to pricing is to start from representation theory.if S is a martingale under measure Q, and Et[X] is a martingale no matter what measure it is, then according to representation theory, Et[X] can be expressed in a way of S. Baxter 's book give a very clear discription of the whole thing.

To price an option you need to find a portfolio of the underlying and a numeraire asset which will equal the option payoff. The portfolio has to be self-financing so that its current value is the only money that needs to be invested to maintain the portfolilo. It can be shown that all self-financing portfolios are just stochastic (ito) integrals with respect to the underlying (discounted (divided) by the numeraire asset) which by the properties of stochastic integrals will be martingales if the (discounted) underlying asset is a martingale. For certain underlying processes the converse is true - all martingales are equivalent to a stochastic integral and hence a self-financing portfolio. This is known as the martingale representation theorem. So if the underlying is a martingale then to find a self-financing portfolio matching the payoff you just need to find a martingale which will be equal to the payoff, and this is just the conditional expectation of the payoff. If the underlying is not a martingale then it is necessary to find a probability measure under which it is a martingale and then perform all the calculations using the new measure. QuoteOriginally posted by: OmarThere has to be a better, more intuitive way to explain the role of martingales in finance. The typical textbook explanation of the type quoted by Alan doesn't explain much......It's only a scenario, and I'm still unhappy: both the underlying process and the value of the call turn out to be martingales in one go. Simon, what do you think?The answer is that we are changing measure to help us identify the set of self-financing portfolios. Replicating the opttion payoff is a very minor issue because there are (almost?) always many ways to hedge an option payoff, even in incomplete markets, but they won't necessarily be s.f.

Some intuition based on standard economic arguments. Note that these are not based on arbitrage, hedging or anything like that. They do not assume continuous time, continuous rebalancing, delta neutral positions etc.Economic theory tells us that any random future payoff will have a time-0 price that is given by the so called Euler equation:X_0 = E{ exp(-b T) \frac{U'(W_T)}{U'(W_0)} X_T }The quantity N_T = exp(-b T)\frac{U'(W_T)}{U'(W_0)} is the Marginal Rate of Substitution (MRS), and the utility here measures wealth (but could be consumption, the bonus or any other quantity that might make us happy). b is a subjective factor that discounts future preferences. The utility function has to satisfy the conditions: U'>0 and U''<=0 (more is better, but at a diminishing rate). The second derivative will reveal the risk aversion: if the inequality is strict we are risk averse, since a $ drop of wealth has a higher 'utility cost' than a $ gain. If instead the utility function is linear, $ changes map into utility changes and we are risk neutral. In this case, the MRS is always equal to a constant: N_T = exp(-b T) and payoffs are just equal to their (discounted) statistical expectations.We can infer some valuable information, by considering the payoffs of assets, such as the bond, the stock or the option. * In the case of the bond, we have that X_T=B_T=1, which implies B_0 = E{ N_T } = exp(-b T) E{ \frac{U'(W_T)}{U'(W_0)} }The interest rate is of course r = -log(B_0)/T. If we are risk neutral, we would have r = b, or that the market discount factor is equal to the subjective discount factor. We can normalize N_T with its expectation, defining M_T = N_T/E{N_T}, the pricing kernel. Then, we can write the value of any payoff X_T asX_0 = E{ N_T X_T } = E{N_T} E{ M_T X_T } = B_0 E{ M_T X_T }* In the case of the stock, we have that X_T=S_T, which will make the current priceS_0 = B_0 E{M_T S_T}* The option price will be given byC_0 = B_0 E{ M_T max{S_T-K,0} }If we were risk neutral, M_T=1 and all expectations would be simplified as straight discounted payoff expectations: S_0 = B_0 E{S_T} and C_0 = B_0 E{ max{S_T-K,0} }. The same would happen if our utility (wealth) is not affected by changes of X_T (the difference between a 'punter' and a 'manager'). If we are risk averse, the payoffs are weighted with their impact on our utility via M_T.We saw why future payoffs have to be weighted with the MRS, in order for expectations to be taken. We can view this weighting in two ways:* We can define a new random variable Y_T = M_T X_T and compute its expectation, based on the bivariate density of (W_T,X_T). This expectation will depend of course on the shape of the utility function, the discount factor, and the correlation between the wealth and the payoff.* We can think of M_T as a Radon-Nikodym derivative, that implies a set of risk-adjusted probabilities. (That's one of the reasons we actually normalized N_T, in order to make M_T a martingale, so it can serve as a R-N derivative). These probabilities, Q, will reflect our risk aversion, namely they will attach higher weights to events we consider to be 'bad'. * Then, Girsanov's theorem tells us that how we can compute expectations under the new probabilities:E{ M_T X_T } = E_Q{ X_T }* It is easy to verify that under these new probabilities the discounted prices of all random payoffs form martingales, by constructionB_0 E_Q{ X_T } = B_0 E{ M_T X_T } = X_0* This applies of course to the stock price, S_0 = B_0 E_Q{ S_T }Now since we have our math hat on, and since we don't want to have anything to do with utility functions, we move backwards:* We are looking for a set of probabilities that make the stock price a martingale* If everything is Brownian and time is continuous, then Girsanov's theorem will indicate the R-N derivative M_T, which will be unique. Otherwise, we will need further conditions to select the R-N out of all possible ones.* We also know that the price of any other claim (e.g. option) will also be a martingale under Q, therefore* We compute its price as C_0 = B_0 E_Q{ C_T } = B_0 E{ M_T C_T }Apparently, this is an heuristic description. There are plenty of technical conditions and other bells and whistles attached. Just wanted to indicate that whenever we use a purely mathematical approach for option pricing purposes, we are actually making a number of serious financial assumptions that are routinely overlooked.Kyriakos