1. When did people first start to begin their emails with "Good morning" or "Good afternoon"? It suggests that you're going to pay attention to the missive as soon as it arrives, another step in the morphing of email from letter to phone call.

2. I was at the Global Derivatives conference in Paris, and thinking about stochastic volatility, motivated in part by an old talk by Peter Jaeckel that someone emailed me. There's no doubt that volatility IS stochastic, and that the models are elegant, but … does it explain the smile, in particular the equity index smile?

Jaeckel points out how many of the economic causes of the smile in FX, equity and interest rates in particular, are related to what I think of as violations of scale invariance rather than the correlation between an asset price and its volatility. There is a more or less absolute meaning to 'low' and' high' for interest rates, as intuited by both crowds and the Fed. And for FX and equities, though they may be scale invariant in the long run, during each short-term period 'low' and 'high' have meaning too, a meaning imparted by investors, central banks and governments. There is something vaguely local about all of this.

There's an article in the Sunday NY Times Business Section and one in the Economist. I have to say I've never seen anything really comforting from a theoretical point of view on this topic, and I once wrote a paper on this too. The Economist does refer to a sort of probabilistic model from which one might perhaps back out some implied probabilities of bubbles.

It seems to me:

1. No one knows what fair value of anything is, though one can begin to tell when it gets to be unfair. Fischer Black wrote somewhere that a price anywhere between half and twice is fair value is fair value. If you run this recursively, you get a range from zero to infinity.

2. People are strongly influenced by other people's opinions.

3. People try to figure out what other people will do.

4. People have a tendency to get on moving trains.

Is there a way to turn all this into a convincing model?

This is an unconsidered piece.

I'm a little tired of reading about what a travesty Black-Scholes is. First of all, the real trouble isn't Black-Scholes, it's geometric Brownian motion. That's the underlying error.

Black-Scholes is an engineering construction that would work if stocks really did evolve under GBM. They don't. So, using Black-Scholes has plenty of problems. Stocks can jump, volatility isn't constant, you can't always short, there are transactions costs, and so on.

So, what can you do? Black-Scholes is a zeroth order approximation with (perhaps) a series of first- and second- and higher-order corrections. I say "perhaps" because claiming there are higher order corrections implies that someone knows the correct answer, and that's not true. You have to think of Black-Scholes as being the right answer is a Platonic world that doesn't match the one we live in.

It's true that many devotees of Black-Scholes are naive. They assume that if you correct it to accommodate the things it neglects you can get there. Instead, if you're a trader or a quant, you ought to think of Black-Scholes as a way of thinking about things, an ideal formula that doesn't hold in the real world, and now it's up to you to decide how to correct for its omissions. Live with it -- you can't do much better, at least for options. Even static hedging of the weak form (when there is no exact payoff matching) requires a model to construct the static hedge.

As someone I know once said: You can't give someone a Black-Scholes calculator and turn him into a trader.

People put too much faith in the model in the past. Now there's an over-reaction to its difficulties. What you have to do is look at the problems and then decide how to work your way around them, with a few calculations and a big dose of common sense.

The enthusiastic use of replication a la Black-Scholes is no doubt responsible for many disasters and market bubbles. But so is the naive reliance on anything: low future default rates, low P/E, The Nifty Fifty, you name it. Part of the trouble is the model itself -- P/E is a model of sorts. But another part of the trouble, perhaps an equal or greater part, is human enthusiasm, in particular desperate enthusiasm for some metric. Metrics in the social sciences (someone I know told me his father said that any field that has the word 'science' in its name isn't science) are always approximations.

If you get rid of Black-Scholes there will still be bubbles. Nevertheless, it's compulsory to understand its limiations. Wild horses couldn't drag me away??

I went to see "I'm Not There", a movie about Bob Dylan's double. It's a bit long, but what's impressive in the movie, besides his music and the covers of it (and perhaps in real life according to "Don't Look Back" as well as various biographies) is his refusal to be typecast, to be or do what his fans expect him to be. He accentuates that in his Chronicles - Volume One, which has an interesting first part but then gets too much (for me) into describing all his recording sessions.

There's a review of the movie in The New Republic which I skimmed though and it makes the point that the movie doesn't show the consequent sacrifices of this kind of position. It's a risky sacrificial position to take in real life, talented or not. I know few people capable of it. He's not a fat tail or an outlier; he's simply not part of the distribution, and hence his present is not the expected value of his future. He's not a martingale.

I suppose there are people like that in all fields, but I'm trying to think who. In finance Fischer Black was a little that way. In the fiction world the review mentions Pynchon, who refuses to enter the fan game at all, and wants no persona, just his books, to exist. In theater or movies, I don't know -- maybe Greta Garbo. There must be other and better examples.

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On the martingale/numeraire front, I've been reading bits and pieces and I still feel uncomfortable with any of the expositions I've seen. The binomially based ones are like scratching your left ear with your right hand by by putting your arm behind your neck. The binomial proof of option valuation is economically based, with clear intuition and a straightforward proof of risk neutrality. The binomially based proofs of the martingale theorems are irritating; they try to rephrase the simple results of the binomial risk-neutral model in terms of the theorems of martingale theory, and they're awkward and provide a very unnatural extension. But the continuous time proofs get buried in technicalities and intuition is very hard to extract.

I notice that Wilmott's book, or at least the previous one that I have with me right now, doesn't even attempt to cover this approach. That says something.

Kerry Back's excellent Springer book entitled "A Course in Derivative Securities" makes a remark about the numeraire/martingale way of looking at options pricing:

"It seems worthwhile here to step back a bit from the calculations and try to offer some perspectives on the methods developed in this chapter. The change of numeraire technique probably seems mysterious. Even though one may agree that it works after following the steps in the chapter, there is probably a lingering question about why it works. The author's opinion is that it may be best simply to regard it as a 'computational trick'. Fundamentally it works because valuation is linear. … The linearity is manifested in the statement that the value of a cash flow is the sum across states of the world of the state prices multiplied by the size of the cash flow in each state. The change of numeraire technique exploits the linearity to further simplify the valuation exercise … After enough practice with it, it will seem as natural as other computational tricks on might have learned."

Thinking of it as a computational trick indicates how unintuitive the result is. Anyone have a better introductory proof?

GAIM recently had an all-day conference in New York City on hedge fund replication. There are three approaches, and I'm a little sceptical/skeptical about them all.

Hedge fund returns display all sorts of nonlinearities -- some because the underlying instruments are noninear, others because they modify their exposure to linear securities as markets move, inducing a nonlinearity.

The first method is linear factor replication -- a sort of least squares regression of observed hf returns to market factors, an APT model.

The second is nonlinear factor replication -- a similar statistical analysis using nonlinear factors or nonlinear trading strategies.

The third is distribution replication -- an attempt to build a payoff distribution, out of any underlying security (copper, electricity, the S&P, the price of sugar -- pick one) that will have the same shape as the hedge fund returns your trying to replicate. Value the distribution as an option, compare its value to the value of the actual hedge fund you're interested in, and then, if it looks cheaper, replicate it by dynamic delta hedging. The idea is that since you have the same distribution of returns, you should earn the same expected return. Same (expected) risk, same (expected) return.

There's a funny cartoon in Grant's Interest Rate Observer that has a picture of a road sign saying "Entering Greenwich CT: 2 and 20." Getting hedge fund returns on the cheap is what this activity of replication is all about.

Does it work well enough to use reliably? I'm a little scepticalskeptical.

The main problem is that you don't really know the payoff function for hedge funds, whereas you do for options. Probably this stuff is more useful for creating synthetic beta-driven hedge funds, with betas to anything you like, that can run algorithmically for cheap and may expose you to synthetic merger arb funds or vol trading funds.

The distribution approach is in the true spirit of finance, driven by the idea of equal risk equal return even when lognormality doesn't hold. But it requires so much statistics on such poor data that it's hard to swallow. And furthermore, since you replicate the eventual return distribution (if there is a stable one and if the method works) but not the month by month exposure, you don't know how long it'll take to generate the same return you expect from the hedge fund.

Necessity is the mother of invention. Invention often requires desperate measures.

An amalgam of the answers I liked best came from Umut Gokcen and Mario G. It went something like this, modified a little by me.

In physics a law of motion tell you exactly where a particle whose position you know right now will be at a very small instant of time later. Used again, the same law of motion then tells you where it will move to another instant later.

Calculus is the branch of mathematics that tells you how to add up all the small future exact position changes over instants of time to determine exactly where the particle will be in the future.

But not everything obeys such exact laws of motion. A stock price which you know today will not take some definite value an instant later, but rather will have a probability of being in some range of values. This price motion is called random or 'stochastic.'

Stochastic calculus is an extension of calculus that tells you how to add up all the small future ranges of movements over each instant of time to determine the final range of values and probabilities the stock price can take at some future time.

A student today asked me how one would explain stochastic calculus to someone who knew no math.

I think it's a tough question and a clever one. I'd be curious to hear short answers –  paragraphs, not essays. You don't have to spell out everything, just a line of approach. If you have a good idea, send it to me at eman.derman.blog@mac.com.

is the title of an article today in the Science section of the NY Times. It's a report of a b ook called "Why Environmental Scientists Can't Predict the Future."

Here are a few excerpts:

"They also discuss concepts like model sensitivity — the analysis of parameters included in a model to see which ones, if changed, are most likely to change model results.

But, the authors say it is important to remember that model sensitivity assesses the parameter’s importance in the model, not necessarily in nature. If a model itself is “a poor representation of reality,” they write, “determining the sensitivity of an individual parameter in the model is a meaningless pursuit.” …

Besides, they acknowledge, people seem to have such a powerful desire to defend policies with formulas (or “fig leaves,” as the authors call them), that managers keep applying them, long after their utility has been called into question …

Models should be regarded as producing “ballpark figures,” they write, not accurate impact forecasts.

The key result of options pricing is that if you hedge at implied volatility, then over the next instant dt a long options position produces

dP&L = (1/2)Gamma*S^2(realized vol^2 - implied vol^2) dt

Of course this is in theory, assuming Wiener process for the innovations, etc.

I learned this years ago, but not right when I learned options theory. It was several years later. I think the equation is almost more general than Black-Scholes -- it tells you how to benefit from curvature. You can understand the equation even if you don't understand PDEs.

I've been doing a lot of teaching lately, both to practitioners and students, and many students and many many professors know all about Black-Scholes and how to derive it and how to solve it, but many many don't recognize this equation or the information embodied in it.

Paul and Reza Ahmed have a beautiful paper on this in Wilmott.