*Tid:***12 november 2001 kl 1515-1600 **

*Plats :***Seminarierummet 3733**, Institutionen för
matematik, KTH, Lindstedts väg 25, plan 7. Karta!

*Föredragshållare:***
Jonas Lundberg.
**

**Titel:** **
Option-pricing with a "smile" - derivation of probability
distribution via relative entropy minimization. (Examensarbete)
**

**Sammanfattning:**

The accuracy of the Black-Scholes model of option pricing has been questioned in recent years. The reason is that stock distributions have lost their lognormal shapes; the skewness has increased and the tails of the probability distribution functions have become fatter. This phenomenon is seen in option prices by the fact that different options with the same underlying stock are being traded on different volatilities, which is a clear violation of the Black-Scholes framework.

In this thesis I derive from option prices in the market, via relative-entropy minimization, alternative Q-measures, which describe the market opinion of the expected payoff function of the underlying security. While the probability distribution is expected to be lognormal in the Black-Scholes model, my calculations show different distributions. But the entropy-minimization method turned out to have some severe limitations. When the implied volatilities (given by the Black-Scholes formula) were changing a lot, and specifically when the implied volatility graph was very jerky, the entropy-minimization method was unable to find a distribution corresponding to the benchmark prices.

Because of the limitation of the method and problems of finding accurate data, no large investigation of the probability distributions for different stocks and for different time series was done. Only three stocks were investigated at a couple of different times.

With knowledge of the real expected payoff distribution (at least the market's opinion of it), you might be able to derive better option deltas. In order to compare the delta from this method with Black-Scholes deltas and get some statistical significance, you need to calculate a very large number of deltas though, which I was not able to do. A thorough investigation of the deltas will thus be left for future research.