Smile risk is most commonly modeled using a stochastic volatility model. We show that matching these models to market data yields vol-of-vols that decay in a power law with time to expiry. This motivates our development of a new model, based on Levy processes, which agrees with the available market data. Asymptotic methods are used to analyze the new model, obtaining effective Black volatilities for options under the new model.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

In this talk, we illustrate the use of sophisticated mathematical tools ranging from differential geometry to Hopf algebra to solve concrete problems in financial engineering: -Heat kernel on Riemannian manifold and calibration of Libor market models -Gaussian estimates of Schr"{o}dinger equation and Implied volatility asymptotics. -Hopf algebra and Monte-Carlo pricing.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

The first part of the talk will be concerned with moment explosions in stochastic volatility models. A moment explosion is said to occur, if a moment of the price process of some given order exists up to some finite time (the explosion time) and becomes infinite from there on. Through the results of Lee and Benaim & Friz, the moment explosion times are intimately connected to the asymptotic shape of the implied volatility surface. I will present the results of Andersen & Piterbarg and Lions & Musiela on moment explosions in SABR-type stochastic volatility models, and my own results on moment explosions in affine stochastic volatility models. In the second part of the talk I will present results on the long-term behavior of affine stochastic volatility models, which also have applications to the asymptotics of the volatility surface.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

This talk considers some properties of the implied volatility surface for large times to maturity. In particular, the implied volatility smile flattens at long maturities in a rather precise manner. The long implied volatility is almost surely non-decreasing as a function of calendar time, in analogy with the Dybvig-Ingersoll-Ross theorem on long interest rates. An asymptotic formula for the level and slope of the long implied volatility smile is given and is illustrated by examples.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

In the presentation, we give an asymptotic expansion of probability density for a component of general diffusion models. Our approach is based on infinite dimensional analysis on the Malliavin calculus and Kusuoka-Stroock's asymptotic expansion theory for general Wiener functionals. The initial term of the expansion is given by the `energy of path' and we calculate the energy by solving Hamilton equation. We apply our approach to asymptotic expansion of implied volatilities and obtain generalized SABR formula.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

Quadratic Wiener functionals , Van Vleck formula, and the KdV equation In this talk,we will give a survey based on a joint work with S. Taniguchi. First of all ,we recal known formulae obtained by R.H.Cameron-M.T.Martin and P.Levy. These are the analogues to the Van Vleck formula for fundamental solutions of Schrodinger equations in cases of potentials associated with harmonic oscillator and uniform magnetic field respectively. Next we will give a framework by which we can systematically discuss these analogues. Finally within the above framework ,we discuss topics related to soliton solutions of the KdV equation in terms of stochastic analysis.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

Implied volatility skew for models with fast mean-reverting stochastic volatility is well understood using singular perturbation methods (Fouque-Papanicolaou-Sircar, CUP 2000). Here, we study the Heston stochastic volatility model in the regime where the maturity is small but large compared to the mean-reversion time of the stochastic volatility factor. We derive a large deviation principle and compute the rate function by a precise study of the moment generating function and its asymptotic. We then obtain asymptotic prices for Out-of-The-Money call and put options, and their corresponding implied volatilities. Joint work with Jin Feng and Martin Forde.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

We study the asymptotic behavior of various distribution densities arising in analytically tractable stochastic volatility models. The main emphasis in this work is on the uncorrelated Hull-White, Stein-Stein, and Heston models. We obtain sharp asymptotic formulas with error estimates for the distribution density of the stock price and the distribution density of a time average of the volatility process. Applications are given to the problem of characterizing the asymptotic behavior of the implied volatility in stochastic volatility models. This is a joint work with E. M. Stein (Princeton University).

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

Using the short-time expansion of the heat kernel associated with the logarithm of the stock price in a local volatility model, we will calculate the neary-expiry first order deviation of the implied volatility from the harmonic mean of the local volatility. The general method can be used to obtain more precise asymptotic behavior of the implied volatility. One prominent feature of the first order deviation is that it is relatively stable with respect to the local volatility in the sense that it depends only on the certain integrated deviations of the local volatility from its harmonic average, a property not shared by higher order deviations. A similar theory can be developed for certain stochastic volatility models.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

We translate in semi-group theory a lot of tools of stochastic analysis. If the formula come from the stochastic analysis, they are simpler to check by the theory of P.D.E. as well for instance the Girsanov formula or the Wong-Zakai approximation of a diffusion, as well as Malliavin's theorem on hypoellipticity and various applications to heat-kernels.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

We introduce a new analytical approach to price American options. Using an explicit and intuitive proxy for the exercise rule, we derive tractable pricing formulas using a short-maturity asymptotic expansion. Depending on model parameters,this method can accurately price options with time- to-maturity up to several years.The main advantage"of our approach over existing methods lies in its straightforward application to models with stochastic volatility and stochastic interest"rates. We exploit this advantage by providing an analysis of the impact of volatility mean-reversion, volatility of volatility,and correlations on the American put price.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

This presentation reviews an asymptotic expansion approach to numerical problems for pricing and hedging derivatives. As examples, plain-vanilla and average options under stochastic volatility models are presented.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

We discuss simple and universal formulae that give quantitative links between tail behaviour and moment explosions of the underlying on one hand, and growth of the volatility smile on the other hand. Practical relevance comes from model calibration and smile extrapolation from market data. Sometimes heat-kernel estimates, or formal applications thereof, can be used although it appears that the existing theory does not fully provide what is needed.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

We discuss market completeness for diffusion-driven factor models beyond the classic requirement that the volatility matrix of traded instruments is invertible. We show that the market generated by a finite-dimensional diffusion model is complete as soon as the coefficients of the SDE are d(x) dP almost surely C1 with locally Lipschitz derivatives. As a consequence, when factor models are considered as diffusions in Hilbert spaces, then any such factor model which admists a finite dimensional representation creates a (locally) complete market. (the limit of locality is given by the existence of the FDR). This is illustrated on the example of Variance Swap Curve Market Models.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

The fast Fourier transform (FFT) technique is now a standard tool for the numerical calculation of prices of derivative securities. Unfortunately, in many important situations, such as the pricing of contingent claims of European type near expiry, and the pricing of barrier options close to the barrier, the standard implementation of this technique leads to serious systematic errors. We propose a new, fast and efficient, variant of the FFT technique, which is free of these problems, and is as easy to implement as the most common version of FFT. We apply this techniques to computing the prices and sensitivities of barrier options and first-touch digital options on stocks whose log-price follows a L\'evy process. The numerical results obtained via our approach are demonstrated to be in good agreement with the results obtained using other (sometimes fundamentally different) approaches that exist in the literature. However, our method is computationally much faster (often, dozens of times faster). Moreover, our technique has the advantage that its application does not entail a detailed analysis of the underlying L\'evy process: one only needs an explicit analytic formula for the characteristic exponent of the process. Thus our algorithm is very easy to implement in practice. Finally, our method yields accurate results for a wide range of values of the spot price, including those that are very close to the barrier, regardless of whether the maturity period of the option is long or short. Joint work with Mitya Boyarchenko. Ahikiko Takahashi "An Asymptotic Expansion Approach in Finance" Abstract: This presentation reviews an asymptotic expansion approach to numerical problems for pricing and hedging derivatives. As examples, plain-vanilla and average options under stochastic volatility models are presented. Olivier Scaillet "Pricing american options under stochastic volatility and stochastic interest rates" Abstract: We introduce a new analytical approach to price American options. Using an explicit and intuitive proxy for the exercise rule, we derive tractable pricing formulas using a short-maturity asymptotic expansion. Depending on model parameters,this method can accurately price options with time- to-maturity up to several years.The main advantage"of our approach over existing methods lies in its straightforward application to models with stochastic volatility and stochastic interest"rates. We exploit this advantage by providing an analysis of the impact of volatility mean-reversion, volatility of volatility,and correlations on the American put price.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)

The authors report on a new higher-order weak approxmation scheme for SDEs and concrete algorithms based on the scheme. The ODE-valued random variables whose averages approximate the given SDE are constructed by using the notion of free Lie algebra. It is proved that the classical Runge--Kutta method for ODEs is directly applicable to the drawn ODE from the random variable.

• Event: Workshop "Small time asymptotics, perturbation theory and heat kernel methods in mathematical finance" (2009)