# Abstracts

**A direct method for calculating Greeks under some Levy processes**

*Hueiwen Teng (National central university, Taiwan)*

Wednesday June 4, 11:30-12:00 | session 4.1 | Computational Finance | room AB

Empirical evidence has shown that some Lévy processes provide a better model fit for market option prices compared with the traditional Black-Scholes models. Greeks are price sensitivities of financial derivatives and are essential for hedging and risk management. But to calculate the Greeks under Lévy process is a challenging and timely task. To overcome this difficulty, this paper proposes a direct method for calculating the Greeks. Briefly speaking, our proposed method provides a ``differentiation' of an indicator function, so that the product rule and chain rule remain valid once the order of the integration and differentiation is switched. Explicit examples for calculating deltas, vegas, and gammas of European and Asian options under Merton's model and the variance-gamma process are given. Numerical results confirm that the proposed method outperforms existing methods, in terms of unbiasedness, efficiency, and computational time.

**Fast Convergence of Regress-Later Estimates in Least Squares Monte Carlo**

*Janina Schweizer (Maastricht University, The Netherlands)*

Wednesday June 4, 12:00-12:30 | session 4.1 | Computational Finance | room AB

Many problems in financial engineering involve the estimation of unknown conditional expectations across time. In the econometrics literature a well-known solution is estimation through sieve. This approach has been exploited in Least Squares Monte Carlo (LSMC) where a simulation-based regression approach is taken. Unlike conventional algorithms where the value function is regressed on a set of basis functions valued at an earlier time, the “Regress-Later” method regresses the value function on a set of basis functions valued at the same time. The conditional expectation across time is then computed exactly for each basis function. We provide sufficient conditions under which we derive the convergence rate of Regress-Later estimators. Importantly, our results hold on non-compact sets. We show that the Regress-Later method is capable of converging significantly faster than conventional algorithms and provide an explicit example. Achieving faster convergence speed provides a strong motivation for using Regress-Later methods in estimating conditional expectations across time.

**Regularity results for degenerate Kolmogorov equations of Affine type**

*Nicoletta Gabrielli (D-MATH ETHZ, Switzerland)*

Wednesday June 4, 12:30-13:00 | session 4.1 | Computational Finance | room AB

A common problem faced in mathematical finance is the efficient computation of expectations of functionals arising from the pricing of derivative contracts. A possible way to look at this quantity is by means of the Kolmogorov equation corresponding to the pricing problem. One of the main features of affine-type operators is its degeneracy and the lack of Lipschitz regularity. In this talk we analyze a new representation of affine processes as path-space valued Lévy processes. This new representation not only leads to a new perspective on numerics of affine processes but is also essential to prove regularity of degenerate Kolmogorov equations with unbounded initial condition.