Ju-Yi Yen

Advances in mathematical finance

ANHA Series Preface Preface Career Highlights and List of Publications / Dilip B. Madan Part I. Variance-Gamma and Related Stochastic Processes The Early Years of the Variance-Gamma Process / Eugene Seneta Variance-Gamma and Monte Carlo / Michael C. Fu Some Remarkable Properties of Gamma Processes / Marc Yor A Note About Selbergs Integrals in Relation with the Beta-Gamma Algebra / Marc Yor Ito Formulas for Fractional Brownian Motion / Robert J. Elliott and John van der Hoek Part II. Asset and Option Pricing A Tutorial on Zero Volatility and Option Adjusted Spreads / Robert A. Jarrow Asset Price Bubbles in Complete Markets / Robert A. Jarrow, Philip Protter, and Kazuhiro Shimbo Taxation and Transaction Costs in a General Equilibrium Asset Economy / Xing Jin and Frank Milne Calibration of Levy Term Structure Models / Ernst Eberlein and Wolfgang Kluge Pricing of Swaptions in Affine Term Structures with Stochastic Volatility / Massoud Heidari, Ali Hirsa, and Dilip B. Madan Forward Evolution Equations for Knock-Out Options / Peter Carr and Ali Hirsa Mean Reversion Versus Random Walk in Oil and Natural Gas Prices / Helyette Geman Part III. Credit Risk and Investments Beyond Hazard Rates: A New Framework for Credit-Risk Modelling / Dorje C. Brody, Lane P. Hughston, and Andrea Macrina A Generic One-Factor Levy Model for Pricing Synthetic CDOs / Hansjorg Albrecher, Sophie A. Ladoucette, and Wim Schoutens Utility Valuation of Credit Derivatives: Single and Two-Name Cases / Ronnie Sircar and Thaleia Zariphopoulou Investment and Valuation Under Backward and Forward Dynamic Exponential Utilities in a Stochastic Factor Model / Marek Musiela and Thaleia Zariphopoulou

Local Times and Excursion Theory for Brownian Motion

Prerequisites.- Local times of continuous semimartingales.- Excursion theory for Brownian paths.- Some applications of Excursion Theory.- Index.

Stochastic resonance and the trade arrival rate of stocks

We studied non-dynamical stochastic resonance for the number of trades in the stock market. The trade arrival rate presents a deterministic pattern that can be modeled by a cosine function perturbed by noise. Due to the nonlinear relationship between the rate and the observed number of trades, the noise can either enhance or suppress the detection of the deterministic pattern. By finding the parameters of our model with intra-day data, we describe the trading environment and illustrate the presence of SR in the trade arrival rate of stocks in the U.S. market.

Integral Representations of Certain Measures in the One-Dimensional Diffusions Excursion Theory

In this note we present integral representations of the Ito excursion measure associated with a general one-dimensional diffusion X. These representations and identities are natural extensions of the classical ones for reflected Brownian motion, RBM. As is well known, the three-dimensional Bessel process, BES(3), plays a crucial role in the analysis of the Brownian excursions. Our main interest is in showing explicitly how certain excursion theoretical formulae associated with the pair (RBM, BES(3)) generalize to pair (X, X ↑ ), where X ↑ denotes the diffusion obtained from X by conditioning X not to hit 0. We illustrate the results for the pair \((R_{-},R_{+})\) consisting of a recurrent Bessel process with dimension \(d_{-} = 2(1-\alpha ),\) α ∈ (0, 1), and a transient Bessel process with dimension \(d_{+} = 2(1+\alpha )\). Pair (RBM, BES(3)) is, clearly, obtained by choosing \(\alpha = 1/2.\)

Detailed study of a moving average trading rule

We present a detailed study of the performance of a trading rule that uses moving averages of past returns to predict future returns on stock indexes. Our main goal is to link performance and the stochastic process of the traded asset. Our study reports short-, medium- and long-term effects by looking at the Sharpe ratio (SR). We calculate the Sharpe ratio of our trading rule as a function of the probability distribution function of the underlying traded asset and compare it with data. We show that if the performance is mainly due to presence of autocorrelation in the returns of the traded assets, the SR as a function of the portfolio formation period (look-back) is very different from performance due to the drift (average return). The SR shows that for look-back periods of a few months the investor is more likely to tap into autocorrelation. However, for look-back larger than few months, the drift of the asset becomes progressively more important. Finally, our empirical work reports a new long-term effect, namely oscillation of the SR and proposes a non-stationary model to account for such oscillations.

Some Topics in Probability Theory

We present a succinct discussion of a number of topics in Probability Theory which have been of interest in recent years.

On Two Results of P. Deheuvels

We highlight some works in the probabilistic literature which are closely related to two results by P. Deheuvels.

The Feynman–Kac Formula and Excursion Theory

We provide a proof of the Feynman–Kac formula for Brownian motion, using excursion theory up to an independent exponential time θ. Call g(θ) the last zero before θ. The independence of the pre-g(θ) process and the post-g(θ) process and the representation of their laws in terms of the integrals of Wiener measure up to inverse local time, or first hitting times allow to recover a formulation of the Feynman–Kac formula via excursion theory.

Lévy’s Representation of Reflecting BM and Pitman’s Representation of BES(3)

This chapter is devoted to the discussion of two famous theorems: the first one, due to Levy, asserts that, if one subtracts Brownian motion from its one-sided supremum, the obtained process is distributed as the absolute value of Brownian motion; the second one, due to Pitman, asserts that if one subtracts Brownian motion from twice its one sided supremum, the obtained process is distributed as a BES(3) process. Extensions of these theorems to Brownian motion with drift are shown. The Azema–Yor explicit solution to Skorokhod’s embedding problem is shown; it involves a first hitting time by Brownian motion and its one-sided supremum.

A Simple Path Decomposition of Brownian Motion Around Time t = 1

The operation of random Brownian scaling is introduced. Applied to the random intervals (0, g), (g, d), (g, 1), where g is the last Brownian zero before time 1, respectively, d is the first Brownian zero after time 1, it is shown that the corresponding Brownian scaled processes are respectively the Brownian bridge, the BES(3) bridge, and the Brownian meander. Independence properties of the Brownian meander allow to study Azema’s remarkable martingale, which enjoys the chaos representation property, as shown by Emery.