Surely the point of B&S is that, by making a certain set of assumptions, it is possible to ignore the preferences of investors when pricing an option. These assumptions are not really concerned with the distribution of returns (log normal, blah blah) so much as with the degree of market completeness relative to the products being priced and relative to the size of my balance sheet. For example, assuming that volatility and interest rates are deterministic and that transaction costs are zero is a "good enough" assumption when I'm pricing a small number of six-month Euro style ATM calls on the Dax. I can use other options to hedge myself. I can delta hedge the residual risk, which itself will always be small compared to my balance sheet.On the other hand, if I want to price a down-and-in barrier on the Thai Baht in huge size, then the assumption of market completeness is not good enough in terms of the product, the market or my balance sheet. So then I need to do something else.But the key assumption is of market completeness relative to the product and the size of the balance sheet.

QuoteOriginally posted by: ragsThe hedge may be possible in highly liquid markets, but aren't there general problem with liquidity risk? I know even that the most liquid markets, there is a problem with incomplete markets? But I'm coming straight out of my books. I have no trading floor experience.I agree with Johnny, but I'd add another point.BS shows you how to replicate vanilla options assuming perfect liquidity and constant, known volatility and risk-free interest rates. With slight tweaking it can relax those assumptions a little. The same basic argument can be used for exotic options, but then you sometimes need even stronger conditions, such as more complete markets.BS then argues that the cost of replication must equal the option premium or arbitrage exists. In the presence of illiquidity that may not be precisely true. But BS is still a reasonable price for the option. It's hard to think of a common situation in which the price would deviate too much from it. Market makers can still run a large option book with BS prices and get rid of most of their liquidity risk. And there is no reason to assume there will be a lot of excess demand or excess supply; liquidity risk is as likely to decrease the market price of the option as to increase it.

well i'm currently working on some problems using geometric brownian motion with poisson jump diffusions. I'll post some general results when I finish them. I'm pretty interested to see how it comes out.