PhD Defense by **Allen Hoffmeyer**

Iam defending my thesis titled "Small-time asyptotics of call prices and implied volatilities for exponential Lévy models" on January 6th at 3pm. My committee members are:

Christian Houdré (Adviser, School of Mathematics)

Yuri Bakhtin (Courant Institute of Mathematical Sciences, NYU)

José Enrique Figueroa-López (Department of Statistics, Purdue)

**Small-time Asymptotics of Call Prices and Implied Volatilities for Exponential L´evy****Models**

Directed by Professor Christian Houdr´e

We derive call-price and implied volatility asymptotic expansions in time

to maturity for a selection of exponential L´evy models. We consider asset-price models

whose log returns structure is a L´evy process. In particular, we consider L´evy

processes of the form (Lt + σWt)

t≥0 where L = (Lt)

t≥0

is a pure-jump L´evy process

in the domain of attraction of a stable random variable, W = (Wt)

t≥0

is a standard

Brownian motion independent of L, and σ ≥ 0.

Call-price asymptotics for in-the-money (ITM) and out-of-the-money (OTM) options

are extensively covered in the literature; however, at-the-money (ATM) callprice

asymptotics under exponential L´evy models are relatively new.

In this thesis, we consider two main problems. First, we consider very general

L´evy models for L. More specifically, L that are in the domain of attraction of a

stable random variable. Under some relatively minor assumptions, we give first-order

call-price and implied volatility asymptotics.

Interestingly, in the case where σ = 0 new orders of convergence are discovered

which show a much richer structure than was previously considered. Concretely, we

show that the rate of convergence can be of the form t

1/α`(t) where ` is a slowly

varying function. We also give an example of a L´evy model which exhibits this new

type of behavior and has a new order of convergence where ` is not asymptotically

constant.

In the case where σ 6= 0, we show that the Brownian component is the dominant

term in the asymptotic expansion of the call-price. Under more general conditions on

L (even removing the requirement of L to be in the domain of attraction of a stablerandom variable), we show that the first-order call-price asymptotics are of the order

√

t.

Finally, we investigate the CGMY process. For this process, call-price asymptotics

are known to third order. Previously, measure transformation and technical

estimation methods were the only tools available for proving the order of convergence.

In the last chapter, we give a new method that relies on the Lipton-Lewis

(LL) formula. Using the LL formula guarantees that we can estimate the call-price

asymptotics using only the characteristic function of the L´evy process. While this

method does not provide a less technical approach, it is novel and is promising for

obtaining second-order call-price asymptotics for ATM options for a more general

class of L´evy processes.