1. Paper Review:"New Insight into Smile, Mispricing, and Value at Risk: The Hyperbolic Model" by E. Eberlein, U. Keller and K. Prause (1998). Anatoliy Swishchuk “Lunch at the Lab” Talk February 10, 2005

2. The Hyperbolic Density

3. Fitted Densities

4. Modelling Financial Assets (The most general Form )

5. The Hyperbolic Levy Motion is a Pure Jump Process

6. Drawback of the Model

7. Reformulation of the Model

8. Solution of the Basic Model

9. The Hyperbolic Model Infinitely Divisible A Levy Process (stationary and independent increments) Moment generating function is

10. Incomplete Market

11. Martingale Approach

12. Option Pricing

13. Comparison of Option Prices

14. Three-Dimensional Comparison

15. Black-Scholes Implicit Volatilities

16. Implicit Hyperbolic Volatility

17. Characteristic Function for the Levy Process

18. Martingale Measure

19. The Price Measure (density)

20. Choosing Parameter Theta

21. Calculating Theta to Define Martingale Measure I.

22. Calculating Theta to Define Martingale Measure II.

23. Calculating Theta to Define Martingale Measure III.

24. References I

25. References II Eberlein E, Keller U. (1995) Hyperbolic Distributions in Finance, Bernoulli, 1, 281- 99.

26. Thank You for Your Attention!