Abstracts
Wednesday June 4, 16:30-17:00 | session 6.1 | Computational Finance | room AB
A new class of explicit Euler schemes, which approximate stochastic differential equations (SDEs) with superlinearly growing drift and diffusion coefficients, will be presented in this talk. It will be shown that, under very mild conditions, these explicit schemes converge in probability and in $\mathcal L^p$ to the solution of the corresponding SDEs. Key ideas from [1] and [4] which are used for the derivation of the aforementioned results will be highlighted. Moreover, rate of convergence estimates will be provided for strong and almost sure convergence. In particular, it will be shown that the strong order 1/2 is recovered in the case of uniform $\mathcal L^p$-convergence. Finally, a comparison will be made with the most recent developments in the field, namely tamed Euler (see [2] and [3]) and balanced (see [5]) methods. One of the examples will be the popular $3/2$ stochastic volatility model in Finance. Jump processes are also considered.
[1] I. Gyöngy, S. Sabanis, A note on Euler approximations for stochastic differential equations with delay, Appl. Math. Optim., 68 (2013), no. 3, pp. 391--412.
[2] Hutzenthaler, M. and Jentzen, A.: Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, \emph{ArXiv} \url{arXiv:1203.5809 [math.PR]}.
[3] M. Hutzenthaler, A. Jentzen, P.E. Kloeden, Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients. Ann. Appl. Probab. 22 (2012) 1611--1641.
[4] S. Sabanis, A note on tamed Euler approximations, Electron. Commun. Probab. 18 (2013), no. 47, 1–-10.
[5] M.V. Tretyakov and Z. Zhang, A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM J. Numer. Anal. 51(2013), no. 6, 3135–-3162.
Wednesday June 4, 17:00-17:30 | session 6.1 | Computational Finance | room AB
We study the shapes of the implied volatility when the underlying distribution has an atom at zero. We show that the behaviour at small strikes is uniquely determined by the mass of the atom at least up to the third asymptotic order, regardless of the properties of the remaining (absolutely continuous, or singular) distribution on the positive real line. We investigate the structural difference with the no-mass-at-zero case, showing how one can-a priori-distinguish between mass at the origin and a heavy-left-tailed distribution. An atom at zero is found in stochastic models with absorption at the boundary, such as the CEV process, and can be used to model default events, as in the class of jump-to-default structural models of credit risk. We numerically test our model-free result in such examples. Note that while Lee's moment formula tells that implied variance is \emph{at most} asymptotically linear in log-strike, other celebrated results for exact smile asymptotics such as Benaim and Friz (09) or Gulisashvili (10) do not apply in this setting--essentially due to the breakdown of Put-Call symmetry--and we rely here on an alternative treatment of the problem.
Wednesday June 4, 17:30-18:00 | session 6.1 | Computational Finance | room AB
We use an example for motivation: Consider the pricing of a simple European option. The underlying SDE is easily solved by Monte Carlo. Alternatively, the corresponding PDE can be discretized using finite differences in time and space followed, for example, by a Crank-Nicholson iteration. Our reference implementation prices the option on a 360 x 1000 mesh for free parameters including the constant interest rate, strike, price of the underlying at maturity, and 138 parameters of the local volatility surface (e.g. implied volatilities) in 2 seconds on the reference computer. Bumping delivers central finite difference approximations of the 141 gradient entries after 70 seconds with full compiler optimization enabled. Our adjoint solution based on dco/c++ computes the same gradient in only 4 seconds – a speedup of almost 20.
This talk introduces algorithmic differentiation and dco/c++ including challenges and pitfalls of the general methodology. The material is based on one-day workshops presented at the ICBI Global Derivatives meetings in Chicago (11/13) and Amsterdam (04/14). Additionally, we discuss recent progress in adjoint methods on GPUs.