Research Projects

Published/Accepted Papers

Outperformance and tracking: Dynamic asset allocation for active and passive portfolio management    [PDF]    [GitHub]
with S. Jaimungal; Applied Mathematical Finance, 2018, Volume 25, Issue 3.

Portfolio management problems are often divided into two types: active and passive, where the objective is to outperform and track a preselected benchmark, respectively. Here, we formulate and solve a dynamic asset allocation problem that combines these objectives in a unified framework. Using stochastic control techniques, we provide explicit closed-form expressions for the optimal allocation and show how the optimal strategy can be related to the growth optimal portfolio. The admissible benchmarks encompass the class of functionally generated portfolios (FGPs), which include the market portfolio values. Finally, some numerical experiments are presented to illustrate the risk-reward profile of the optimal allocation.


Technical uncertainty in real options with learning    [PDF]    [GitHub]
with Á. Cartea and S. Jaimungal; The Journal of Energy Markets, 2018, Volume 11, No. 4.

We introduce a new approach to incorporate uncertainty into the decision to invest in a commodity reserve. The investment is an irreversible one-off capital expenditure, after which the investor receives a cashflow stream from extracting the commodity and selling it on the spot market. The investor is exposed to price uncertainty and uncertainty in the amount of available resources in the reserves (i.e. technical uncertainty). She does, however, learn about the reserve levels through time, which is a key determinant in the decision to invest. To model the reserve level uncertainty and how she learns about the estimates of the commodity in the reserve, we adopt a continuous-time Markov chain model to value the option to invest in the reserve and investigate the value that learning has prior to investment.

Submitted Papers and Preprints

A Variational Analysis Approach to Solving the Merton Problem
with S. Jaimungal

We address the Merton problem of maximizing the expected utility of terminal wealth using techniques from variational analysis. We assume a general semimartingale market model with stochastic parameters driving both risky and risk-free asset values. We obtain a characterization of the optimal portfolio for general utility functions in terms of a forward-backward stochastic differential equation (FBSDE), as well as closed-form solutions for a number of well-known utility functions. Our results complement a previous study conducted in Ferland & Watier (2008) on optimal strategies in markets driven by Brownian noise with random drift and volatility parameters.


Active and Passive Portfolio Management with Latent Factors    [PDF]
with S. Jaimungal

We address a portfolio selection problem that combines active (outperformance) and passive (tracking) objectives using techniques from convex analysis. We assume a general semimartingale market model where the assets' growth rate processes are driven by a latent factor. Using techniques from convex analysis we obtain a closed-form solution for the optimal portfolio and provide a theorem establishing its uniqueness. The motivation for incorporating latent factors is to achieve improved growth rate estimation, an otherwise notoriously difficult task. To this end, we focus on a model where growth rates are driven by an unobservable Markov chain. The solution in this case requires a filtering step to obtain posterior probabilities for the state of the Markov chain from asset price information, which are subsequently used to find the optimal allocation. We show the optimal strategy is the posterior average of the optimal strategies the investor would have held in each state assuming the Markov chain remains in that state. Finally, we implement a number of historical backtests to demonstrate the performance of the optimal portfolio.

Other Projects

Solving Nonlinear and High-Dimensional Partial Differential Equations via Deep Learning    [PDF]
with A. Correia, D. Naiff, G. Jardim and Y. Saporito

In this work we present a methodology for numerically solving a wide class of partial differential equations (PDEs) and PDE systems using deep neural networks. The PDEs we consider are related to various applications in quantitative finance including option pricing, optimal investment and the study of mean field games and systemic risk. The numerical method is based on the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) with modifications made depending on the application of interest.

Winning entry for the inaugural Financial Mathematics Team Challenge 2018 in Brazil.


Stochastic portfolio theory: Literature review    [PDF]

A high-level overview of a number of papers in the stochastic portfolio theory literature.

Completed for the Winter 2016 session of the Research Topics in Mathematical Finance course taught by Prof. Sebastian Jaimungal at the University of Toronto. A much more in-depth overview of SPT can be found in "Stochastic Portfolio Theory: An Overview" by Fernholz & Karatzas (2008).


Stochastic Optimal Control of Pairs Trading Strategies with Absolute and Relative Inventory Penalties    [PDF]    [GitHub]

Statistical arbitrage (StatArb) trading strategies are a class of algorithmic trading strategies predicated on the notion that the movements of portfolios of multiple assets can be more accurately predicted than those of individual assets. Strategies of this sort attempt to exploit the mean-reverting properties of certain portfolios of assets to make a profit. In this paper, we apply techniques from stochastic control theory to derive the optimal trading rules for a pair of cointegrated assets. The agent trades in the two assets over a fixed investment horizon, and then liquidates her position at the terminal date with the objective of maximizing the expected proceeds. Additionally, we assume the presence of temporary price impacts which force the agent to complete her transactions at less favorable prices when increasing her trading activity. Finally, the agent also incorporates two non-financial penalties into her objective function: absolute and relative inventory running penalties. These penalties dissuade the agent from remaining in a large position in either asset for too long and force the strategy to be self-financing by penalizing the injection of additional cash (or holding a surplus of cash for too long).

Completed for the Winter 2015 session of the Applied Stochastic Control: High Frequency & Algorithmic Trading course taught by Prof. Sebastian Jaimungal at the University of Toronto.


Optimal portfolio choice with inflation-linked minimum consumption constraints   

This paper addresses the problem of determining the optimal investment and consumption policies for a utility-maximizing investor whose consumption is bounded below, where the lower bound is assumed to be driven by inflation and is therefore stochastic. A motivating example that illustrates this problem is to consider an investor who is free to consume any amount on "wants" but must first buy a certain basket of basic goods representing "needs" at all times (e.g. rent, food, etc.). The price of this basket of goods represents the lower bound of their consumption. However, we must take into account that the price of this basket of goods evolves along with inflation, so that at any given point in time the nominal amount of minimum consumption is uncertain in the future. In addition, the investor has access to certain assets that they can allocate a portion of their wealth to. The goal then is to determine the optimal allocation and consumption decisions in order to maximize expected utility of consumption while ensuring that the stochastic minimum consumption constraint is satisfied at all times. Optimal policies are found by applying standard stochastic control techniques, bearing in mind that the introduction of inflation leads to the addition of a state variable in the formulation of the stochastic control problem.

Completed as part of a Supervised Research Course in Winter 2015 and supervised by Prof. Sebastian Jaimungal.


Stochastic control in asset allocation: Literature review    [PDF]

A high-level overview of a number of papers addressing portfolio selection problems via optimal stochastic control with a focus on habit formation and minimum consumption constraints.

This work was completed as part of a Supervised Research Course at the University of Toronto in Winter 2015 and supervised by Prof. Sebastian Jaimungal.


Functional data analysis of AIDS clinical trials data using principal analysis by conditional expectation    [PDF]    [GitHub]

Functional data analysis (FDA) and functional principal components (FPC) techniques are employed to determine the efficacy of a treatment regimen for individuals infected with HIV. In particular, we apply the principal analysis through conditional expectation (PACE) approach described in "Functional Data Analysis for Sparse Longitudinal Data" by Yao et al. (2005) - which extends FPC analysis to cases with sparse and irregularly spaced longitudinal data.

Completed for the Winter 2015 session of the Functional Data Analysis course taught by Prof. Fang Yao at the University of Toronto.


Credit scoring via logisitic regression    [PDF]    [GitHub]

We use a logistic regression model to predict the creditworthiness of a number of bank customers using a number of predictors related to their personal status and financial history. Model adequacy and robustness checks are performed to ensure that the model is being properly fitted and interpreted.

Completed for the Winter 2014 session of the Methods of Applied Statistics II course taught by Prof. Nancy Reid at the University of Toronto.


Stock return predictability
with G. Qian and P. Zhang

This essay provides an overview on the topic of stock return predictability. The first section focuses on the theoretical aspects of predictability, namely establishing a working definition of return predictability, and positioning it alongside two other foundational topics in financial theory: market efficiency and factor models. In the second section we provide a survey of the multitude of factors used in the prediction of stock returns, with a more in-depth examination of price-to-dividend ratio as a predictor. The following section presents specialized areas of research related to stock return predictability: statistical issues involved in tests of predictability (especially with respect to long horizon predictability), the notion of time-varying predictability, and nonlinear prediction approaches. Finally, we present an empirical study where we investigate whether or not aggregate stock market returns can be predicted from industry portfolios, and the extent of this predictability at different levels of granularity and at different horizons.

Completed for the Fall 2014 session of the Asset Pricing course taught by Prof. Peter Cziraki at the University of Toronto.