# Abstracts

**Mathematical models for the formation of financial bubbles**

*Francesca Biagini (Ludwig-Maximilians Universität München, Germany)*

Tuesday June 3, 11:00-11:30 | session 1.1 | Computational Finance | room AB

The notion of an asset price bubble has two ingredients. One is the observed market price of a given financial asset, the other is the asset's intrinsic value, and the bubble is defined as the difference between the two. The intrinsic value, also called the fundamental value of the asset, is usually defined as the expected sum of future discounted dividends. Here we study a flow in the space of equivalent martingale measures and focus on the corresponding shifting perception of the fundamental value of a given asset in an incomplete financial market model. This allows us to capture the birth of a perceived bubble and to describe it as an initial submartingale which then turns into a supermartingale before it falls back to its initial value zero. We illustrate our results in two examples, one due to Delbaen and Schachermayer [2] and the other given by a a variant of the stochastic volatility model discussed by Sin in [3]. This talk is based on the paper [1].

[1] Biagini F., Föllmer H., Nedelcu S. Shifting Martingale Measures and the Birth of a Bubble as a Submartingale, Finance and Stochastics, accepted, 2013.

[2] F. Delbaen and W. Schachermayer. A simple counter-example to several problems in the theory of asset pricing. Mathematical Finance, 8:1-12, 1998.

[3] C.A. Sin. Complications with stochastic volatility models. Advances in Applied Probability, 30(1):256-268, 1998.

**Efficient numerical methods for local Levy models**

*Oleg Reichmann (ETH, Zurich, Switzerland)*

Tuesday June 3, 11:30-12:00 | session 1.1 | Computational Finance | room AB

We consider a certain type of multidimensional normal Markov processes, which are Feller processes. This class of processes includes as special cases Lévy processes, many local volatility and local Lévy models. Due to their nonstationarity the considered processes can exhibit qualitative behaviour that is substantially different from that of Lévy processes such as state-space dependent jump activity. The nonstationarity also has substantial repercussions on their computational and analytical treatment: whereas for Lévy and the closely related affine models, Fast Fourier Transformation (FFT) algorithms form the basis for fast and powerful option pricing algorithms, the nonstationarity implies that FFT based numerical methods are, in general, not applicable in the numerical solution of their Kolmogorov equations (with the notable exception of, for example, affine processes).

From an analytical point of view, Feller processes are rather well understood. This is due to the fact that generators of Feller processes are pseudodifferential operators with symbols that admit a Lévy-Khintchine representation. Contrary to Lévy processes or diffusions with local volatility, domains of the infinitesimal generators for semigroups induced by Feller processes are, generally, variable order Sobolev spaces. Accordingly, the use of standard discretization schemes (based on Finite Differences or Finite Elements) for numerical solution of the Kolmogorov equations associated to such models is not straightforward; the same applies to the numerical analysis of these discretization schemes, i.e. the mathematical analysis of stability, consistency and convergence of these schemes.

One central theme of this talk is therefore to describe recent progress in the design and the numerical analysis of discretization schemes which allow a unified numerical treatment of the Kolmogorov equations for a large class of normal Markov processes. These schemes are based on variational, multiresolution schemes which use spline-wavelet bases of the domains of the processes' infinitesimal generators.

**Pricing barrier and Bermudan style options under time-changed Levy processes: fast Hilbert transform approach**

*Pingping Zeng (Hong Kong University of Science and Technology, Hong Kong)*

Tuesday June 3, 12:00-12:30 | session 1.1 | Computational Finance | room AB

We construct efficient and accurate numerical algorithms for pricing discretely monitored barrier and Bermudan style options under time-changed Levy processes by applying the fast Hilbert transform method to the log-asset return dimension and quadrature rule to the dimension of log-activity rate of stochastic time change. Some popular stochastic volatility models, like the Heston model, can be nested in the class of time-changed Levy processes. The computational advantages of the fast Hilbert transform approach over the usual fast Fourier transform method, like exponential decay of errors in terms of the step size in the transform and avoidance of recovering option prices at the monitoring time instants, can be extended to pricing path dependent options under time-changed Levy processes. We manage to compute the fair value of a dividend-ruin model with both embedded reflecting (dividend) barrier and absorbing (ruined) barrier. We also consider pricing of Bermudan options in conjunction with the determination of the associated critical asset prices. Finally, we successfully construct accurate Hilbert transform algorithms for pricing finite-maturity discrete timer options under stochastic volatility processes. A finite-maturity timer option expires either when the accumulated realized variance of the underlying asset has reached a pre-specified level or on the mandated expiration date, whichever comes earlier. The challenge in the pricing procedure is the incorporation of the barrier feature in terms of the accumulated realized variance to deal with the expiration condition. Our numerical tests demonstrate high level of accuracy, efficiency and reliability of the fast Hilbert transform approach when compared to other numerical schemes in the literature.

**Multi-Level Monte Carlo Simulations with Importance Sampling**

*Przemyslaw Stilger (Manchester Business School, UK)*

Tuesday June 3, 12:30-13:00 | session 1.1 | Computational Finance | room AB

We present an application of importance sampling in a Monte Carlo simulation for multi-asset options and in a Multi-Level Monte Carlo simulation. We demonstrate that applying importance sampling only on the first level of the Multi-Level Monte Carlo significantly improves its effective performance. We extend the Likelihood Ratio Method Based on Characteristic Function to estimate the Greeks of multi-asset options and in a Multi-Level Monte Carlo in a computationally efficient manner. Moreover, we combine it with the importance sampling to reduce the variance of the Greeks. Finally, we study the impact of the skew on the effective performance of importance sampling.