Financial Market and Derivatives
Winning and losing streaks
Winning streaks appear frequently in all financial markets including equity, commodity, foreign exchange, real estate, etc. Most stochastic process models for financial market data in the current literature focus on stylized facts such as fat-tailedness relative to normality, volatility clustering, mean reversion. However, none of existing financial models captures the pervasive feature of persistent extremes: financial indices frequently report record highs or lows in concentrated periods of time. The new model in this paper enables us to measure and assess the impact of persistent extremes on financial derivatives and to more accurately predict option values. This model in this paper describes the phenomenon of market overreaction at the macro level, which complements existing behavior finance literature on this subject that explain market reactions by psychological reasoning and evidence. The paper also explores the possibility of using the model for measuring the tendency of overbought stocks and indices.
R. Feng, P. Jiang, H. Volkmer. (2020) Modeling financial market movement with winning and losing streaks: sticky extrema processes. Download
Global Association of Risk Professionals (GARP) Best Paper Award for Quantitative Methods in Finance Conference 2019
Conditional Asian options
Conditional Asian options are recent market innovations, which offer cheaper and long-dated alternatives to regular Asian options. In contrast with payoffs from regular Asian options which are based on average asset prices, the payoffs from conditional Asian options are determined only by average prices above certain threshold. Due to the limited inclusion of prices, conditional Asian options further reduces the volatility in the payoffs than their regular counterparts and have been promoted in the market as viable hedging and risk management instruments for equity-linking life insurance products. There has been no previous academic literature on this subject and practitioners have been only known to price these products by simulations. We propose a first analytical approach to computing the prices and deltas of conditional Asian options in comparison with regular Asian options. In the numerical examples, we put to the test some cost-benefit claims by practitioners.
R. Feng, H.W. Volkmer. (2015) Conditional Asian options. International Journal of Theoretical and Applied Finance, 18 (6), 1550040. Download
Credit value adjustment
Downgrade-triggered termination clause is an innovation in credit risk management to control counterparty credit risk. It allows one party of an over-the-counter derivative to close off its position at marked-to-market price when the other party’s credit rating downgrades to an agreed alarming level. Although the default risk is significantly reduced, the non-defaulting party may still suffer losses in case that the other party defaults without triggering the termination clause prior to default. At the heart of the valuation of credit value adjustment (CVA) is the computation of the probability of default. We employ techniques from ruin theory and complex analysis to provide solutions for probabilities of default, which in turn lead to very efficient and accurate algorithms for computing CVA. The underlying risk model in question is an extension of the commercially available KMV-Merton model and hence can be easily implemented. We provide a hypothetical example of CVA computation for an interest-rate swap with downgrade triggered termination clause.
R. Feng, H.W. Volkmer. (2012) Modeling credit value adjustment with downgrade-triggered termination clause using a ruin theoretic approach, Insurance: Mathematics and Economics, 51 (2), 409–421. Download
Extrema of Diffusion Processes
The stochastic integral representations (martingale representations) of square integrable processes are well-studied problems in applied probability with broad applications in finance. Yet finding explicit expression is not easy and typically done through the Clack-Ocone formula with the advanced machinery of Malliavin calculus. To find an alternative, Shiryaev and Yor introduced a relatively simple method using Itô’s formula to develop representations for extrema of Brownian motion. In this paper, we extend their work to provide representations of functionals of time-homogeneous diffusion processes based on the Itô’s formula.
R. Feng. (2016) Stochastic integral representations of the extrema of time-homogeneous diffusion processes. Methodology and Computing in Applied Probability, 18(3), 691–715. Download
Geometric Brownian motion with affine drift and its time-integral
The joint distribution of a geometric Brownian motion and its time-integral was derived in a seminal paper by Yor (1992) using Lamperti’s transformation, leading to explicit solutions in terms of modified Bessel functions. In this paper, we revisit this classic result using the simple Laplace transform approach in connection to the Heun differential equation. We extend the methodology to the geometric Brownian motion with affine drift and show that the joint distribution of this process and its time-integral can be determined by a doubly-confluent Heun equation. Furthermore, the joint Laplace transform of the process and its time-integral is derived from the asymptotics of the solutions. In addition, we provide an application by using the results for the asymptotics of the double-confluent Heun equation in pricing Asian options. Numerical results show the accuracy and efficiency of this new method.
R. Feng, P. Jiang, H. Volkmer. (2021) Geometric Brownian motion with affine drift and its time-integral. Applied Mathematics and Computation, to appear. Download