# How to improve the Black-Scholes framework?

Since the distribution of daily returns are obviously not lognormal, my bottom line question is has BS been reworked for a better fitting distribution?

The best dist I've ever made fit is a double-sided exponential, but I'd easily settle for a regular exponential distribution for simplicity's sake.

If there aren't any papers showing what the net result could be, can the cumulative distribution function of the standard normal distribution simply be replaced with the cdf of the the exponential distribution? If so, do $d_1$ and $d_2$ have to be reworked?

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Prices, not returns, are assumed to be lognormal. –  Jase Dec 2 '12 at 12:42
I agree that $\frac{S(T)}{S(t)} = e^{(r-\sigma^2)(T-t)+\sigma(W(T)-W(t))}$ is lognormal, but I thought that the most common meaning of "stock returns" in finance is $ln \frac{S(T)}{S(t)}$. –  Jase Dec 2 '12 at 16:33
Just to clarify, BS assumes normal returns and hence lognormal prices. –  SRKX Dec 2 '12 at 21:20
Do you have a source for this definition of returns? (where they actually use to word "returns") –  Jase Dec 3 '12 at 2:43
@SRKX: Thank you. This is one of the most annoying (and surprisingly common) misnomer you see in a lot of papers/ discussion boards/ blogs but is never pointed out. –  emsfeld Dec 3 '12 at 2:57

You're not gonna find much off google, since nobody's gonna go public with anything they develop to make money. Power Law distributions are a much better fit for financial returns than normal, also if you apply variance instead, it'd explain the OTM option values in a more practical manner.

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yes that's why Goldman Sachs didn't care about their high speed trading code taken by an ex-employer intending to distribute for profit. wired.com/threatlevel/2011/03/aleynikov-sentencing Mind you by now I'm sure everyone in town and their grandmas have developed counter strats against this, haha. –  Rock Dec 16 '12 at 8:32
I have to partly concur with Rock here. I am not commenting on his suggestions about bs improvements but I agree that you can toss most of the improvement in academic literature into the garbage without fearing you missed out. Improvements to volatility modeling and pricing algorithms for options is a well guarded secret by most vol traders. There is (almost) no way an enlightenment makes it suddenly into the public domain. I worked as junior with a guy who really beat the crap out of the competition in long dated options and he brought the code inside a dll ... –  Matt Wolf Feb 14 '13 at 18:32
... And a tight IP agreement and the dll even called an external server for permissioning. That was his conditions upon joining. He never shared his approach in detail with anyone. He was an index vol trader but later on traded a CB book as well where he tossed out everything but the optionalities. I learned a lot from him but not how he priced those >1 year expiries –  Matt Wolf Feb 14 '13 at 18:35
Yeah, but do you think he was actually owned an awesome alternative to BSM or simply had a good model for relative dynamics of the implied vols? There seem to be plenty of high science alternatives to BS out there, you just can't really apply them to real world. –  Strange Feb 16 '13 at 3:05

Check out these resources:

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Right, it's a really nice method. Beside enabling you to consider any distribution for underlying price, I think you can use a transform to analyize historical data and fit an appropriate characteristic function to the data. However, I am still confused how can you change from an actual measure to a risk neutral one using this method. –  Amir Yousefi Dec 2 '12 at 3:20

Stochastic vol models with jumps are an updated version of Black-Scholes model. Because of volatility clustering and jumps in equity prices, stochastic vol models with jumps make sense (however, indicies do seem to follow a diffusion process with just stochastic vol as they do not have jumps, especially if you look at it from a point of view of trade time).

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Non-Gaussian Merton-Black-Scholes Theory would be a possible source of information on this type of model.

Note: I have glanced through this book, but have not read it thoroughly. However I can say that if you want to read this book you should be very comfortable with partial differential equations (especially the theory of pseudodifferential operators).

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I would like to provide an answer with a bit more embedded details.

The weaknesses of the Black-Scholes framework you refer come from the fact that it assumes that stock prices are following a Geometric Brownian Motion (GBM). This model assumes that stock prices evolve as follows:

$$dS_t = \mu S_t dt + \sigma S_t dW_t$$

You can solve this differential equation and get that, given $S_t$:

$$S_T = S_t e^{(\mu - \frac{\sigma^2}{2})(T-t) + \sigma (W_T-W_t)}$$

This means that stock prices are log-normally distributed, and that returns are normally distributed.

First, if you simply look at historical data, you can clearly see that returns do not seem to be normal. So it seems like GBM is an over-simplistic model for stock prices. Indeed, it fails to model (and this list is not exhaustive):

• Skewness
• Excess kurtosis (i.e. it underestimates the probability of rare events)
• Heteroskedasticity (the fact that, unlike in the GBM framework, it seems like $\sigma$ is not constant)

If you want to find improvements to the BS model, you could google for derivative pricing methods which assume models including the features listed above. For example, you could look at Monte-Carlo approach using the GARCH model.

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