Below is a compact summary of the Black-Scholes result for option pricing, emphasizing the importance of perfect hedging. With perfect hedging you can price the option as long as you know the future probability distribution for the underlying -- it doesn't have to be log-normal or have fixed variance.
The solution, in hindsight, is wonderfully simple. The proper price to put on an option should equal the expected value of exercising the option. If you have the option, right now, to sell one share of a stock for $10, and the current price is $8, the option is worth exactly $2, and the option price tracks the current stock price one-for-one. If you knew for certain that the share price would be $8 a year from now, the present value of the option would be $2, discounted by the cost of holding money, risklessly, for a year --- say $1. Every $2 change in the stock price a year hence changes the option price now by $1. If you knew the probability of different share prices in the future, you could calculate the expected present value of the option, assuming you were indifferent to risk, which few of us are. Here is the crucial trick: a portfolio of one share and two such options is actually risk-free, and so, assuming no arbitrage, must earn the same return as any other riskless asset. Since we're assuming you already know the probability distribution of the future stock price, you know its risk and returns, and so have everything you need to know to calculate the present value of the option! Of course, the longer the time horizon, the more we discount the future value of exercising the option, and so the more options we need to balance the risk out of our portfolio. This fact suffices to give the Black-Scholes formula, provided one is willing to assume that stock price changes will follow a random walk with some fixed variance, an assumption which "did not seem onerous" to him at the time, but would now be more inclined to qualify.Scholes on models, mathematics and computers:
Starting from an economic issue and looking for a parsimonious models, rather than building mathematical tools and looking for a problem to solve with them, has been a hall-mark of Scholes's career. "The world is our laboratory", he says, and the key thing is that it confirm a model's predictive power. There is a delicate trade-off between realism and simplicity; "tact" is needed to know what is a first-order effect and what is a second-order correction, though that is ultimately an empirical point.
The evaluation of such empirical points has itself become a delicate issue, he says, especially since the rise of computerized data-mining. While by no means objecting to computer-intensive data analysis --- he has been hooked on programming since encountering it in his first year of graduate school --- it raises very subtle problems of selection bias. The world may be our laboratory, but it is an "evolutionary" rather than an "experimental" lab; "we have only one run of history", and it is all too easy to devise models which have no real predictive power. In this connection, he tells the story of a time he acted as a statistical consultant for a law firm. An expert for the other side presented a convincing-looking regression analysis to back up their claims; Scholes, however, noticed that the print-out said "run 89", and an examination of the other 88 runs quickly undermined the credibility of the favorable regression. Computerization makes it cheap to do more runs, to create more models and evaluate them, but it "burns degrees of freedom". The former cost and tedium of evaluating models actually imposed a useful discipline, since it encouraged the construction of careful, theoretically-grounded, models, and discouraged hunting for something which gave results you liked --- it actually enhanced the meaning and predictive power of the models people did use!