## Section: New Results

### Numerical methods for option pricing

#### Call-Put Duality

Participants : A. Alfonsi, B. Jourdain.

It is well-known that in models with local volatility functions (t, x) and constant interest and dividend rates, the European Put prices are transformed into European Call prices by time-reversal of the volatility function and simultaneous exchanges of the interest and dividend rates and of the strike and the spot price of the underlying. All the existing proofs of this Call-Put duality rely on some partial differential equation argument. For instance, it is a consequence of Dupire's formula. By a purely probabilistic approach based on stochastic flows of diffeomorphisms, B. Jourdain has generalized the Call-Put duality equality to models including exponential Lévy jumps in addition to local volatility. He has also recovered various generalizations of Dupire's formula to complex options recently obtained by Pironneau.

Aurélien Alfonsi and Benjamin Jourdain have obtained a similar Call Put duality for perpetual American options when the local volatility function does not depend on the time variable. The perpetual American Put price is equal to the perpetual American Call price in a model where, in addition to the previous exchanges between the spot price and the strike and between the interest and dividend rates, the local volatility function is modified. This duality result leads to a theoretical calibration procedure of the local volatility function from the perpetual Call and Put prices. Then they have investigated generalizations to payoff functions of the form (x, y) when is the positive part of a function concave in each variable and non-increasing (resp. non-decreasing) in x (resp. y). The duality result remains valid for specific choices of more general than the Call-Put case (y-x) + . This means that the nature of the Perpertual American duality is different from the one of the European duality. Indeed, in the European case, the fact that the second order derivative of (y-x) + with respect to y is the Dirac mass at x plays a crucial role.

#### Parisian options

J. Lelong has studied double barrier Parisian options. He has established explicit formula for the Laplace transforms of their prices with respect to the maturity time and have established accuracy results for the procedure we use to numerically invert the Laplace transforms. Moreover he has implemented an algorithm for pricing Parisian options with simple barrier based on a Laplace transform method as described in [35] in the software Premia.

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