Duy Nguyen

A general framework for discretely sampled realized variance derivatives in stochastic volatility models with jumps

After the recent financial crisis, the market for volatility derivatives has expanded rapidly to meet the demand from investors, risk managers and speculators seeking diversification of the volatility risk. In this paper, we develop a novel and efficient transform-based method to price swaps and options related to discretely-sampled realized variance under a general class of stochastic volatility models with jumps. We utilize frame duality and density projection method combined with a novel continuous-time Markov chain (CTMC) weak approximation scheme of the underlying variance process. Contracts considered include discrete variance swaps, discrete variance options, and discrete volatility options. Models considered include several popular stochastic volatility models with a general jump size distribution: Heston, Scott, Hull–White, Stein–Stein, α-Hypergeometric, 3/2 and 4/2 models. Our framework encompasses and extends the current literature on discretely sampled volatility derivatives, and provides highly efficient and accurate valuation methods. Numerical experiments confirm our findings.

A General Valuation Framework for SABR and Stochastic Local Volatility Models

In this paper, we propose a general framework for the valuation of options in stochastic local volatility (SLV) models with a general correlation structure, which includes the stochastic alpha beta...

First Hitting Time of Integral Diffusions and Applications

ABSTRACT We study the first hitting time of integral functionals of time-homogeneous diffusions, and characterize their Laplace transforms through a stochastic time change. We obtain explicit expressions of the Laplace transforms for the geometric Brownian motion (GBM) and the mean-reverting GBM process. We also introduce a novel probability identity based on an independent exponential randomization and obtain explicit Laplace transforms of the price of arithmetic Asian options and other derivative prices that non-linearly depend on the integral diffusions. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method.

Integral Representation Of Probability Density Of Stochastic Volatility Models And Timer Options

This paper contributes a generic probabilistic method to derive explicit exact probability densities for stochastic volatility models. Our method is based on a novel application of the exponential measure change in [Z. Palmowski & T. Rolski (2002) A technique for exponential change of measure for Markov processes, Bernoulli 8(6), 767–785]. With this generic approach, we first derive explicit probability densities in terms of model parameters for several stochastic volatility models with nonzero correlations, namely the Heston 1993, 3/2, and a special case of the α-Hypergeometric stochastic volatility models recently proposed by [J. Da Fonseca & C. Martini (2016) The α-Hypergeometric stochastic volatility model, Stochastic Processes and their Applications 126(5), 1472–1502]. Then, we combine our method with a stochastic time change technique to develop explicit formulae for prices of timer options in the Heston model, the 3/2 model and a special case of the α-Hypergeometric model.

Magnitude and Speed of Consecutive Market Crashes in a Diffusion Model

In this paper, in a time-homogeneous diffusion setting, we study a sequence of last passage times of generalized drawdown processes before the first passage time of another monitoring generalized drawdown process. These quantities are closely related to consecutive small market downward movements before a final big market crash modeled by the first passage time of the monitoring generalized drawdown process. Our theoretical framework simultaneously incorporates the first passage time, the absolute and relative drawdown times, and generalizes the scope of Zhang and Hadjiliadis (2012). We explicitly determine the path decompositions bridging these random times, the stochastic differential equations (SDE) governing each of the path fragments, and derive the Laplace transform of durations between any two of these random times. Applications include designing various risk measures summarizing several market information (e.g. historical maximum/minimum, absolute drawdown, relative drawdown) altogether. They allow us to study, in a microscopic way, a sequence of smaller drawdowns (signals) before a big drawdown (market crash) actually realizes. They can have applications to the detection of a potential market crash from signals of market drawdowns. We illustrate our results using a drifted Brownian motion.

A General Framework for Time-changed Markov Processes and Applications

Abstract In this paper, we propose a general approximation framework for the valuation of (path-dependent) options under time-changed Markov processes. The underlying background process is assumed to be a general Markov process, and we consider the case when the stochastic time change is constructed from either discrete or continuous additive functionals of another independent Markov process. We first approximate the underlying Markov process by a continuous time Markov chain (CTMC), and derive the functional equation characterizing the double transforms of the transition matrix of the resulting time-changed CTMC. Then we develop a two-layer approximation scheme by further approximating the driving process in constructing the time change using an independent CTMC. We obtain a single Laplace transform expression. Our framework incorporates existing time-changed Markov models in the literature as special cases, such as the time-changed diffusion process and the time-changed Levy process. Numerical experiments illustrate the accuracy of our method.

An Integral Representation for Elasticity and Sensitivity for Stochastic Volatility Models

This paper presents a generic probabilistic approach to study elasticities and sensitivities of financial quantities under stochastic volatility models. We describe the shock elasticity, the quantile sensitivity and the vaga value of cash flows with respect to perturbation of the volatility function of the model. The main contribution is to establish explicit formulae for these elasticities and sensitivities based on a novel application of the exponential measure change technique in Palmowski and Rolski (2002). We carry out explicit calculations for the Heston model and the 3/2 stochastic volatility model, and derive explicit expressions in terms of model parameters.

Integral Representation of Probability Density of Stochastic Volatility Models and Timer Options

This paper contributes a generic probabilistic method to derive explicit exact probability densities for stochastic volatility models. Our method is based on a novel application of the exponential measure change in Palmowski & Rolski (2002). With this generic approach, we first derive explicit probability densities in terms of model parameters for several stochastic volatility models with non-zero correlations, namely the Heston (1993), 3/2, and a special case of the α-Hypergeometric stochastic volatility models recently proposed by Da Fonseca & Martini (2016). Then, we combine our method with a stochastic time change technique to develop explicit formulae for prices of timer options in the Heston model, the 3/2 model and a special case of the α-Hypergeometric model.