# Applications of short-time asymptotic methods to option pricing and change-point detection for Lévy processes.

#### Abstract

The past decade has seen a tremendous growth in the literature on asymptotic analysis of financial models with jumps. In this thesis we characterize properties of option prices and implied volatility in an asymptotic regime where time-to-maturity and log-moneyness become small, which for liquidity reasons is of particular importance in practice. To begin with, we build on a recent result of Figueroa-López et al. [Mathematical Finance, doi:10.1111/mafi.12064], to obtain short-term asymptotic expansions for at-the-money (ATM) option prices, under a large class of exponential Lévy models with stable-like jumps of infinite variation. We do so under the weakest possible conditions on the Lévy measure for such an expansion to be well defined, and show that the continuous Brownian component can be extended to an independent stochastic volatility process with leverage. We also determine the existence of a short-term log-moneyness regime to which the formulas can be extended to include "near-the-money" options, which has important practical implications in that the most liquid short-term options have strike prices that are concentrated around the ATM strike. In the same model setting we also provide high-order asymptotic expansions for the ATM implied volatility skew (i.e. the strike derivative), which has received relatively little attention in the literature, but is actively monitored in practice by traders and analysts. We utilize a well known relation between the skew and the transition probability of the underlying process, so as auxiliary results we also obtain high-order expansions for ATM digital call prices and the delta of European call options. These results are markedly different from those obtained for near-the-money option prices, and shed further light on the relationship between important model parameters and the implied volatility smile near expiry. In particular, we are able to quantify the effect that the correlation (leverage) parameter of the volatility component has on the short-term skew in the presence of jumps, which provides an important model selection and calibration tool. The accuracy of the asymptotic expansions is assessed using Monte Carlo simulation. Our results indicate that for parameter values of relevance in finance, the approximations give a good fit for maturities up to one month, underpinning their relevance in e.g. FX markets where there are actively traded options with short maturities, and where the implied volatility skew plays a critical role in option pricing. The last part of this work is independent and considers change-point detection for Lévy processes. We show that the widely used CUSUM procedure is optimal, in a well defined sense, for detecting a change in the statistical properties of processes with independent and stationary increments, i.e. Lévy processes. This encompasses existing results on a change in the drift of a Brownian motion and a change in the jump intensity of a homogeneous Poisson process, and is a natural continuous-time extension of a widely known discrete-time result of Moustakides [Annals of Statistics, doi:10.1214/aos/1176350164], where the optimality of CUSUM is proved for a change in the distribution of a sequence of independent and identically distributed random variables.

#### Degree

Ph.D.

#### Advisors

Figueroa-Lopez, Purdue University.

#### Subject Area

Applied Mathematics|Statistics|Finance

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