Black-Scholes Extensions for Non-Gaussian Returns and Changing Volatility

# How to improve the Black-Scholes framework?


# 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?

Google searches give me nada.

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?

## Answer by Rock (score 2, accepted)

https://quant.stackexchange.com/a/4706

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.

## Answer by Amir Yousefi (score 6)

https://quant.stackexchange.com/a/4662

Check out these resources:

- The book Levy Processes in finance.

- This paper basically enabling you to use any distribution for asset prices: Option Valuation Using the Fast Fourier Transform

## Answer by Andrew (score 2)

https://quant.stackexchange.com/a/4765

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).

## Answer by GAM (score 2)

https://quant.stackexchange.com/a/7037

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).

## Answer by SRKX (score 1)

https://quant.stackexchange.com/a/4668

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.