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

The Hyperbolic Density

Fitted Densities

Modelling Financial Assets

(The most general Form)

The Hyperbolic Levy Motion is a Pure Jump Process

Drawback of the Model

Reformulation of the Model

Solution of the Basic Model

The Hyperbolic Model

• Infinitely Divisible• A Levy Process (stationary and independent

increments)• Moment generating function is

Incomplete Market

Martingale Approach

Option Pricing

Comparison of Option Prices

Three-Dimensional Comparison

Black-Scholes Implicit Volatilities

Implicit Hyperbolic Volatility

Characteristic Function for the Levy Process

Martingale Measure

The Price Measure (density)

Choosing Parameter Theta

Calculating Theta to Define Martingale Measure I.

Calculating Theta to Define Martingale Measure II.

Calculating Theta to Define Martingale Measure III.

References I

References II

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

Thank You for Your Attention!