#### Date of Award

8-2016

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Frederi Viens

#### Committee Chair

Frederi Viens

#### Committee Member 1

Kiseop Lee

#### Committee Member 2

Raghu Pasupathy

#### Committee Member 3

Jonathon Peterson

#### Abstract

Asymptotic behavior of implied volatility is of our interest in this dissertation. For extreme strike, we consider a stochastic volatility asset price model in which the volatility is the absolute value of a continuous Gaussian process with arbitrary prescribed mean and covariance. By exhibiting a Karhunen-Loève expansion for the integrated variance, and using sharp estimates of the density of a general second-chaos variable, we derive asymptotics for the asset price density for large or small values of the variable, and study the wing behavior of the implied volatility in these models. Our main result provides explicit expressions for the first five terms in the expansion of the implied volatility, based on three basic spectral-type statistics of the Gaussian process: the top eigenvalue of its covariance operator, the multiplicity of this eigenvalue, and the *L2* norm of the projection of the mean function on the top eigenspace. Strategies for using this expansion for calibration purposes are discussed.

For small time, we consider the class of self-similar Gaussian stochastic volatility models, and compute the small-time (near-maturity) asymptotics for the corresponding asset price density, the call and put pricing functions, and the implied volatilities. Unlike the well-known model-free behavior for extreme-strike asymptotics, small-time behaviors of the above depend heavily on the model, and require a control of the asset price density which is uniform with respect to the asset price variable, in order to translate into results for call prices and implied volatilities. Away from the money, we express the asymptotics explicitly using the volatility process' self-similarity parameter *H,* its first Karhunen-Loève eigenvalue at time 1, and the latter's multiplicity. Several model-free estimators for *H* result is discussed. At the money, a separate study is required: the asymptotics for small time depend instead on the integrated variance's moments of orders 1/2 and 3/2, and the estimator for *H* sees an affine adjustment.

#### Recommended Citation

Zhang, Xin, "Extreme-Strike and Small-time Asymptotics for Gaussian Stochastic Volatility Models" (2016). *Open Access Dissertations*. 894.

https://docs.lib.purdue.edu/open_access_dissertations/894