Asymptotics of Volatility Surfaces and Option Prices

Asymptotic Approximations are widely used in mathematical finance. The present proposal is concerned with the asymptotic behavior of option prices, implied volatility, and local volatility, as some crucial parameter (typically, strike or maturity) becomes very large or small. Such results have immediate practical applications: fast approximate pricing (in risk management, e.g.), fast model calibration via good approximation formulas, and the choice of good parametrizations of the volatility smile. Asymptotics can also answer qualitative questions of the kind: "To what extent are the wings of the smile influenced by the mean reversion level of the underlying`s variance process? Or by the initial value of the variance process?" The main goals of the project are as follows. 1. The local volatility associated to a call price surface is the diffusion coefficient of a process reproducing the given prices. The short maturity behavior of local vol has been of interest to several authors, but almost nothing is known about its wings (when the state variable becomes very large or small). We have a candidate for an approximate wing formula (preprint with P. Friz, 2011), and presume that it is essentially model-free (just as Lee`s 2004 formula for the wings of implied vol). This remains to be proven, though. Moreover, we want to prove a rigorous statement of the kind "the local vol of a jump process explodes as maturity tends to zero". This would confirm intuition; quantitative refinements would allow to inspect a given local vol surface for jump behavior of the process that generated it. 2. While there are many papers about implied vol asymptotics, it is a mystery so far which kind of asymptotic regime gives the best results in concrete situations. We want to determine effective bounds for expansions w.r.t. large strike and/or large maturity, which allow to compare different regimes. For some expansions, one can use known effective analyses; for others (large strikes, in particular), new bounds have to be developed. 3. Out-of-the-money options with short maturity can be handled in diffusion models by large deviation methods. We want to complement these results (going back to Varadhan in the 60ies) by a central limit theorem. As applications, we envisage new approximations for binary options and the implied vol slope. 4. We want to find a good and rigorous short-maturity approximation of the price of an arithmetic Asian option. Geometric Asian options are well studied, since their pricing is simpler than that of arithmetic ones in many models, and since they can serve as control variates for pricing arithmetic Asian options by Monte Carlo. Note, however, that Monte Carlo simulation, as well as most other proposed methods, run into severe numerical difficulties when pricing short-maturity arithmetic Asian options. We therefore want to estimate their prices asymptotically, by a (technically demanding) multivariate saddle point approach. There are partial results on this problem by Dufresne (2004), Barrieu, Rouault & Yor (2004) and the proposer (2011). We apply for one predoc and one postdoc position, each over a period of three years.

Banks and other financial firms use mathematical models to price financial instruments and manage their risks. These models have to be calibrated to current market data, which requires involved computations. One of the main goals of the project was to support these computations by providing fast approximate formulas. Such approximations not only allow for faster computations, but also give insights into the qualitative behavior of financial market models. In particular, different models can be compared, in order to choose the most suitable one. The approximations developed in this project are particularly well suited to be used for practical applications.Another problem treated in this project concerns the consistency of option prices. Option prices are inconsistent if they admit the construction of a portfolio with risk-free gain. We developed new criteria to test given option price data for consistency. Our results allow for a bid-ask spread, which has not been considered for this problem before. As theoretical underpinning, we generalize a classical theorem from probability theory (Strassen 1965).