Hello,I followed a class in capital market theory. This class was very difficult for me since my math level is poor. I have an oral exam and the professor told us that we can make a small project on a specific chapter. My idea is to start from the B&S formula and talk about a recent improvement in the calculation. I know that the orginal formula is very old and is no more used in the option pricing. Could you recommend me a paper or a project that talk about an improvement of the formula and that is still used in the option pricing nowadays?Here is how we learned the B&S formula:After reviewing the probabilities and the complete markets theory, we began with stochastic calculus: Brownian motion and stochastic integral, Ito processes and Girsanov theorem, Ito's lemma, multidimensional processes and multidimensional ito's lemma. After, we saw the arbitrage and pricing: self financing strategies, arbitrage and martingale measure, pricing. At the end we saw the model itself of B&S and the B&S by changes of numeraires.I'm a beginner in this class and I'm just beginning to try to understand the course. If you could help me, it would be great. I don't know if the description is clear enough, don't hesitate to ask me if I haven't been clear with my request.

QuoteOriginally posted by: MattcmtHello,I. I know that the orginal formula is very old and is no more used in the option pricing.The formula is still used it will never be old. Most option pricing models have some black-scholes favor in them. It true most practitioners do not use it exactly as you learned it in your course, but they use it as a starting point and modify it. If you remember the first time you were taugth the formula, your teacher probably told you that interest rate and volatility were assumed to be constant. In subsequent models you will learn about models in which interest rates and volatility are stochastic....

hi mattcmtin general, option prices just are what they are, similarly to the way that a stock price should be the sum of all its discounted future earnings, but nobody really uses this to value a stock. working slightly back to front, given an option price, a trader can work out what the volatility should be in the Black & Scholes formula that will give the correct option price. This is called the *implied* volatility and is closer to what traders do in practice; looking at the volatility as an alternative measure of price. this allows them to compare the relative prices of options of different maturity, where the dollar prices would not be directly comparable.problem is that often, options with different maturities or strikes give different implied volatilities. this contradicts the Black/Scholes assumption of constant volatility; the volatility of the same stock cannot be constant but different for different options!the range of implied volatilities across different strikes & maturities is called the implied volatilty surface. there are different ways to modify the model to attempt to account for this, such as allowing the volatility to change depending on time and on the level of the underlying stock, or as quantwanabe mentioned, to make it stochastic. these ways tend not to give nice pricing formulas though; they have to be solved by numerical computations.

Try Black-Karasinski. Shouldn't be too much of a technical leap.You can find the details in Hull or google (bond and option pricing when short rates are lognormal).

there are a number of interest rate models based on the short-rate; vasicek is the simplest, there is also hull/white, black/karasinki, cox/ingersoll/ross. maybe black/karasinki is the most used; none is the cutting edge of interest rate modelling though, if you're particularly looking for something more current & widely used in practice.

Thanks a lot for the information... I'm going to try to understand Karasinski.