Is log-normal a good model for stock dynamics

Khoa Tran · 11-10-2007, 07:40 AM

So, the standard model that most quants start with is

$dS_t=\mu S_tdt + \sigma S_tdW_t$

(note that the dynamics is interpreted in Ito sense).

Why is this a good model of the stock price?
How do you test a stochastic process model in general? I know very little about Statistics.

YADD · 11-12-2007, 11:30 AM

Quote:

Originally Posted by Khoa Tran

So, the standard model that most quants use is

$dS_t=\mu S_tdt + \sigma S_tdW_t$

(note that the dynamics is interpreted in Ito sense).

[*]Why is this a good model of the stock price?

No, it is not.

* It has no fat tails.
* It has no skew - developed markedly post 1987-crash.
* It has no jumps.
* It has only one (BM) driver. Its volatility surface should be driven by one or more factors; in other words, sigma should be made stochastic.

Historically, it was chosen probably because:

* of the central limit theorem (and the efficient market hypothesis prevailing at the time).
* it prevents stock from going negative.
* it is soluble.
* it has the fewest parameters possible.
* it is in line with the conventional notion of risk/reward (sigma/mu).
* it is easy to work with.
* it is not too dumb an assumption.

Quote:

[*]How do you test a stochastic process model in general? I know very little about Statistics.

Awaiting answers from statisticians/econometricians....

neomikeo · 11-13-2007, 04:32 AM

Quote:

Originally Posted by YADD

No, it is not.

* It has no fat tails.
* It has no skew - developed markedly post 1987-crash.

Log normal variable x=exp(z) if z is N(mu, sigma ^2).

1. Lognormal variable has positive skew and positive excess kurtosis.

2. You could estimate the log normal distribution using matching moments or methods of moments.
Expectation (x) = exp(mu+1/2 sigma^2).

+ mu could be estimated by looking at the first moment of z or log(x).
+ signma could be obtained by solving mu and the first moment of x.

shinichi9htv · 11-13-2007, 06:15 AM

I think anh Hoang meant the smiles are flat (no skew)

YADD · 11-13-2007, 08:58 AM

Quote:

Originally Posted by neomikeo

Log normal variable x=exp(z) if z is N(mu, sigma ^2).

1. Lognormal variable has positive skew and positive excess kurtosis.

In derivative pricing, (convex) smile and skew are defined in terms of z (your notation). Lognormal variables do not have smile nor skew.

Khoa Tran · 11-14-2007, 07:33 AM

I actually didn't focus on log-normality (sorry about not making it clear at first) but rather on the philosophy modeling in general. So, I'm also waiting for answers from statisticians/econometricians and some kind of behavioral finance answer: how the behavior of investors/traders leads to this or that dynamics.

Btw, Julian Shaw in How I became a quant actually criticized many quants heavily for not having a business-mind (something like they're proud of their incredibly complicated mathematics but too lazy to learn any econometrics). I'll write a review on his memoir soon. Gosh, I just realize that I talked too much today!

ngtridung · 11-14-2007, 11:39 AM

I am neither a statistician nor econometrician; I just want to make some comment based on my understanding:

1. Estimation: From the lognormal process $S_t$ , it follows that $d(ln(S_t)) = (\mu - \sigma^2/2) dt + \sigma dW$ . With a discretization $S_1, S_2,... ,S_n$ (e.g. the observed daily prices of $n$ consecutive trading day (assuming the stock is traded everyday)). Defines $Y_t = ln(S_{t+1}) - ln(S_t) = ln(S_{t+1}/S_t)$ , then $Y_t$ should follow a normal distribution $N((\mu - \sigma^2/2) , \sigma)$ if the lognormal assumption on $S_t$ is correct. From here, the parameters $\mu, \sigma$ can be estimated by many different approaches. The most popular one is the maximum likelihood estimations. (by forming the likelihood function $L(\mu, \sigma | Y_t) = \prod_{t=1}^{n-1} f(Y_t | \mu, \sigma)$ . The MLE satisfies: $\mu - \sigma^2/2 = \bar{Y}; \sigma^2 = Var(Y_t)$ .

2. Testing the lognormal model: The lognormal model only holds if $Y_t = ln(1+Return_t)$ follows a normal distribution. This test is among the tests of the random walk hypothesis (or the predictability of stocks returns) that researchers have been trying to support/reject since 1960s. In 1988, Lo, A. and Mackinlay A.C. wrote a paper “Stock Market Prices do not follow random walk: evidence from a simple test” rejected the random walk hypothesis. Their work is based on the variance ratios test. (The literature review of their paper should tell the results from ealier tests) Further reading on statistical tests on stochastic process can be found in another paper by Lo, A. “Maximum likelihood estimation of generalized Ito processes with discretely sampled data”. Econometric theory, 4, 231-247, 1986.

3. As many of you have pointed out the problems of constant interest rate and volatility, interest rate skew is model through stochastic volatility model (SVM), the most complex one (I think) is the SABR (stochastic, alpha, beta, rho) model (you can search and see wiki for more details).

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