June 24-28, 2012
The 2012 Annual Meeting of the Canadian Applied and Industrial Mathematics Society

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  Mathematical Finance
Theme Organizers: Matt Davison and Adam Metzler (UWO)

This thematic session will focus on the application of quantitative methods to problems of financial regulation. Specific topics include, but are not limited to, the burgeoning fields of financial networks (helpful in identifying institutions that are too big to fail), systemic risk measures (valuable in designing more appropriate capital regulations) and contingent capital (a possible market-based solution to enforcing discipline and reducing the burden of costly financial bailouts). Our panel will be truly interdisciplinary, showcasing speakers with such diverse backgrounds as physics, mathematics, engineering, economics and business. In addition, we plan to host at least one session showcasing Ph.D. students in financial mathematics.

Our theme will consist of three major parts
1. Invited talks.
2. Round-table discussion on contingent capital
3. Minisymposium showcasing PhD students in mathematical finance

The confirmed invited speakers for the first part are:
1. Dilip Madan (plenary), Robert H. Smith School of Business, University of Maryland at College Park.
2. George Pennacchi, College of Business, University of Illinois.
3. Matheus Grasselli, Department of Mathematics and Statistics, McMaster University.
4. Frank Milne, Department of Economics, Queen's University.
5. Kay Giesecke, Department of Management Science and Engineering. Stanford University.
6. Michael Gordy, Senior Economist (Risk Analysis Section, Division of Research and Statistics), Board of Governors of the Federal Reserve System.
7. Bhaskar DasGupta, Department of Computer Science, University of Illinois at Chicago.
8. David Saunders, Department of Statistics and Actuarial Science, University of Waterloo.

For the minisymposium we are expecting approximately eleven doctoral students from the following schools/departments
1. University of Toronto. Department of Statistics and Actuarial Science, Department of Computer Science.
2. York University. Department of Mathematics and Statistics.
3. McMaster University. Department of Mathematics and Statistics.
4. University of Calgary. Haskayne School of Business.

For the round-table discussion we have confirmed participants from the Bank of Canada, as well as the Office of the Superintendent of Financial Institutions.

Speaker Abstracts

Monday, June 25

George Pennacchi, University of Illinois
A Structural Model of Contingent Bank Capital

This paper develops a structural credit risk model of a bank that issues short-term deposits, shareholders equity, and xed- or oating-coupon contingent capital (CoCos). The model assumes that bank assets follow a jump-diffusion process, interest rates are stochastic, and capital ratios are mean-reverting. Allowing for sudden declines in asset val- ues, as occur during nancial crises, has distinctive implications. CoCo credit spreads are higher when: the capital conversion trigger is lower; the conversion write-down is greater; and conversion awards a xed, rather than variable, number of shares. Dual price trigger CoCos are more similar to nonconvertible subordinated debt. Issuing CoCos can create a debt overhang problem and a moral hazard incentive for the bank to raise its asset risk, but these problems are often less than if the bank issued a similar amount of subordinated debt. In general, incentive problems are least when contract terms minimize CoCos credit

David Saunders
, University of Waterloo
Calculating Regulatory Capital for Credit Risk: Mathematical and Computational Issues

The inadequacies of methods for calculating credit risk capital, particularly in the trading book, in the lead-up to the global nancial crisis have led to a reevaluation of regulatory capital, resulting in the new Basel III requirements. I will discuss mathematical and computational problems that arise when computing the new capital requirements for credit risk in the trading book

Michael B. Gordy, Board of Governors of the Federal Reserve System
Counterparty Credit Risk and Interconnectedness in CDS Trade Repository Data
Authors: Celso Brunetti and Michael Gordy

As evidenced during the financial crisis, OTC derivative markets can
be an important pathway for the transmission of systemic risk. The default of a large market participant can impose significant direct losses on its counterparties, which may cascade to the counterparties of the defaulted firms' counterparties. Though more difficult to quantify, a distressed firm's interconnectedness in OTC markets may be no less a concern. If the firm plays a significant role in intermediation, then the normal functioning of the OTC market may be disrupted even if the firm has balanced positions with all significant counterparties.

We have "snapshots" of the credit default swap market on
two dates in 2010 as captured by the CDS trade repository. To identify
counterparty exposures of potential concern, we sift the data for the
largest bilateral and multilateral positions at three levels of market aggregation. We apply network methods to characterize and quantify patterns of interconnectedness. Broadly speaking, our aim is to identify firms crucial to the transfer of risk from end-buyers to end-sellers, and to assess the resilience of the trading network to the loss of one or more crucial

(Tentative) Stochastic Time-Change of Default Intensity Models: Pricing and Estimation
Joint with Ovidiu Costin, Min Huang, and Pawel Szerszen

We introduce stochastic time change to default intensity models of credit risk as a parsimonious way to account for stochastic volatility in credit spreads. We derive two series solutions for the survival probability function, and show that both methods are applicable when the intensity follows the widely-used basic affine process. This leads to straightforward and efficient solutions to bond prices and CDS spreads. We then estimate the time-changed model on panels of CDS spreads (across maturity and observation time) using Bayesian MCMC meth-
ods. We find strong evidence of stochastic time change.

Justin Sirignano, Stanford University
Large Portfolio Asymptotics for Loss From Default
Joint with Kostas Spiliopoulos (Brown), Richard Sowers (Illinois), and Kay Giesecke (Stanford)

We prove a law of large numbers for the loss from default and use it for approximating the distribution of the loss from default in large, potentially heterogenous portfolios. The density of the limiting measure is shown to solve a non-linear stochastic PDE, and certain moments of the limiting measure are shown to satisfy an in nite system of SDEs. The solution to this system leads to the distribution of the limiting portfolio loss, which we propose as an approximation to the loss distribution for a large portfolio. Numerical tests illustrate the accuracy of the approximation, and highlight its computational advantages over a direct Monte Carlo simulation of the original stochastic system.

Bruno Rémillard, HEC Montréal
Optimal hedging in continuous time

In this talk I will cover the problem of mean-variance optimal hedging for some continuous time models: regime-switching geometric Levy processes and stochastic volatility models. It is also shown that the continuous time solution can be approximated by discrete time Markov models processes. In some cases, the optimal prices corresponds to prices under an equivalent martingale measure, making that measure a natural choice for pricing. However, even if the optimal hedging strategy is not the usual delta hedging, it can be easily computed by Monte Carlo methods.

Tuesday, June 26

Frank Milne, Queen's University
The Anatomy of Systemic Risk
Joint with John Crean (University of Toronto)

Systemic risk arises almost entirely on credit exposures to real sectors of the economy. Data in the paper show that such risks are concentrated in a few systemically important real sectors (SIRS). Typical firms in all potential SIRS share common characteristics: high fixed costs, low marginal costs of production, heavy competition and high leverage. Downturns in such sectors are spasmodic and deep. The particular sectors that cause systemic risk change from recession to recession. The paper constructs a dynamic theory that reflects these characteristics. The model is initially structured without short term bank deposits. The model generates several conclusions. Credit crises in SIRS generate the key macroeconomic phenomena of systemic crises even in the absence of short term funding runs. In such crises, insolvencies among firms and banks spread unexpectedly. Effects extend outside the SIRS. Complaints of re-sale pricing and credit restrictions are widespread. The introduction of short term deposits deepens downturns. Liquidity runs on particular banks cannot be adequately forecast without an explicit analysis of the SIRS and other credit risks of particular banks. The paper explains why standard models have difficulty in predicting major credit and liquidity events. The model outlines the taxonomy of systemic risk in a manner that enables such risk to be identified exante. It therefore has important implications for structuring efficient stress tests.

Matheus R. Grasselli
, McMaster University
An Agent-Based Computational Model for Bank Formation and Inter-
bank Networks

Joint with Omneia R. H. Ismail (McMaster University)

We introduce a simple framework where banks emerge as a response to a natural need in a society of individuals with heterogeneous liquidity preferences. We examine bank failures and the conditions for
an interbank market is to be established. We start with an economy consisting of a group of individuals arranged in a 2-dimensional cellular automaton and two types of assets available for investment. Because of uncertainty, individuals might change their investing preferences and accordingly seek their surroundings neighbours as trading partners to satisfy their new preferences. We demonstrate that the individual uncertainty regarding preference shocks coupled with the possibility of not finding a suitable trading partners when needed give rise to banks as liquidity providers. Using a simple learning process, individuals decide whether or not to join the banks, and through a feedback mechanism we illustrate how banks get established in the society. We then show how the same uncertainty in individual investing preferences that gave rise to banks also causes bank failures. In the second level of our analysis, in a similar fashion, banks are treated as agents and use their own learning process to avoid failures and create an interbank market. In addition to providing a bottom up model for the formation of banks and inter-bank markets, our model allows us to address under what conditions bank oligopolies and frequent banks failures are to be observed, and when an interbank market leads to a more stable system with fewer failures and less concentrated market players.

Bhaskar DasGupta, University of Illinois at Chicago
Global Stability of Banking Networks Against Financial Contagion: Mea-
sures, Evaluations and Implications

Instabilities of major nancial institutions during the recent financial crisis of 2007 and later have generated renewed interests in evaluating the stabilities (or, lack thereof) of banking networks among economists, regulatory authorities and other relevant segments of the population. In particular, one reason of such type of vulnerabilities to the so-called financial contagion process in which failures of few individual banks propagate through the "web of banking dependencies" to affect a significant part of the entire global banking system. We initiate a systematic scientific investigation of defining and evaluating a global stability measure for the nancial contagion process for several classes of banking
networks, and discuss some interesting implications of our evaluations of this stability measure.

2:30 pm
(C)Meng Han
, University of Toronto
Approximations to Loss Probabilities of Loan Portfolios
Co-authors: Ken Jackson, Alex Kreinin
Credit risk analysis and management at the portfolio level are challenging problems for financial institutions due to their portfolios' large size, heterogeneity and complex correlation structure. The conditional independence framework is widely used to calculate loss probabilities of credit portfolios. The existing computational approaches within this framework fall into two categories: (1) simulation-based approximations and (2) asymptotic approximations. The simulation-based approximations often involve a two-level Monte Carlo method, which is extremely time-consuming, while the asymptotic approximations, which are typically based on the Law of Large Number (LLN), are not accurate enough for tail probabilities, especially for heterogeneous portfolios. We give a more accurate asymptotic approximation based on the Central Limit Theorem (CLT), and we discuss its convergence and when it can be applied. To further increase accuracy for lumpy portfolios, we also propose a hybrid approximation, which combines the simulation-based approximation and the asymptotic approximation. We test our approximations with some artificial and real portfolios. Numerical examples show that, for a similar computational cost, the CLT approximation is more accurate than the LLN approximation for both homogeneous and heterogeneous portfolios, while the hybrid approximation is even more accurate than the CLT approximation. Moreover, the hybrid approximation significantly reduces the computing time for comparable accuracy compared to simulation-based approximations.
3:00 pm.
(C)Lung Kwan Tsui
Efficient Calculation of Economic and Regulatory Capital for Structured Credit Instruments
Computing the economic capital of a portfolio containing CDO and CLO tranche is a challenging practical problem faced by financial institutes holding these kinds of securities. Firstly, pricing CDOs and CLOs for a large number of scenarios is computational intensive. Furthermore, we have to take into account both the default risk and credit migration risk embedded in these instruments. We provide a efficient simulation methodology to compute VaR and CVaR of a portfolio containing CDOs and CLOs. We first approximate the tranche pricing function by matching the moments of the distribution of the survival probabilities of the entities in the collaterals of a CDO or CLO. We then compute the prices on a sparse multi-dimension grid which is used for interpolation. This methodology significantly enhances the computational efficiency of the Monte Carlo simulation required for computing the VaR and CVaR.
3:30 pm
(C)Daniel Hackmann,
York University
The optimal dividend problem for two families of meromorphic Levy processes
A recent paper develops a solution to De Finetti's optimal dividend problem by finding an explicit expression for the value function in cases when the underlying wealth model is a spectrally negative Lévy process. The value function is presented in terms of a so-called scale function which is implicitly defined using the Laplace transform. For two recently introduced families of Lévy Processes (the beta and theta families), which can be modified through an appropriate choice of parameters to have paths of infinite activity and infinite variation, one can find manageable series expressions for the scale function. This provides an opportunity to evaluate the value function through standard numerical methods. In this talk I will discuss the techniques used to derive an accurate approximation and compare the results to the value function of the well known Cramér- Lundberg process with exponentially distributed jumps. The comparison shows that unless the starting capital of the company is close to zero, the simpler Cramér-Lundberg model gives nearly identical results to those calculated for the more complicated beta and theta processes. Additionally, I will discuss the empirical distribution of the time of ruin under the optimal reflection strategy when the wealth model is the Cramér-Lundberg process. The results of several Monte Carlo simulations show that an interesting avenue of further research is to consider the optimal dividend problem when a penalty, in the form of a function of the time of ruin, is imposed.
(C)Stephen Tse
, University of Waterloo
Comparison between the Mean Variance optimal and the Mean Quadratic Variation optimal trading strategies
Coauthors: Peter Forsyth, Shannon Kennedy, Heath Windcliff
We compare optimal liquidation policies in continuous time in the presence of trading impacts by numerical solutions of Hamilton Jacobi Bellman (HJB) partial differential equations (PDE). We show quantitatively that the mean-quadratic-variation strategy can be significantly suboptimal in terms of mean-variance efficiency and that the mean-variance strategy can be significantly suboptimal in terms of mean-quadratic-variation efficiency. Moreover, the mean-quadratic-variation strategy is on average more suboptimal than the mean-variance strategy, in the above sense. In the semi-Lagrangian discretization used for solving the HJB PDEs, we show that interpolating along the semi-Lagrangian characteristics results in significant improvement in accuracy over standard interpolation while still guaranteeing convergence to the viscosity solution.

Wednesday, June 27

Bruno Remillard (100 min tutorial)
Tutorial for graduate students in mathematical finance
Optimal hedging in discrete time

In this tutorial I will discuss the implementation of mean-variance optimal hedging for discrete time models. In particular, I will cover models with independent increments, HMM models and GARCH models.

2:30 pm (20 minute talks)

(C)Amir Memartoluie University of Waterloo
Counterparty Credit Risk, a Mass Transportation Approach

In this work, we propose a new approach for calculating the Conditional Value at Risk (CVaR) of a portfolio. One of the main issues that quantitative modellers face in this regard is estimating the joint distribution of credit risk factors and market risk factors. After describing the underlying Counterparty Credit Risk problem, we describe the risk measure which fits our model best. After that we adapt a new approach which is based on utilizing Transportation Problem for formulating our optimization problem. We finish by presenting our numerical results.

(C)Duy Minh Dang
(University of Waterloo)
An efficient numerical PDE approach for pricing foreign exchange interest rate hybrid derivatives
Coauthors: Duy Minh Dang, Christina Christara, Ken Jackson, and Asif Lakhany.

We discuss efficient pricing methods via a Partial Differential Equation (PDE) approach for long-dated foreign exchange (FX) interest rate hybrids under a three-factor multi-currency pricing model with FX volatility skew. The emphasis of the paper is on Power-Reverse Dual-Currency (PRDC) swaps with popular exotic features, namely knockout and FX Target Redemption (FX-TARN). Challenges in pricing these derivatives via a PDE approach arise from the high-dimensionality of the model PDE, as well as from the complexities in handling the exotic features, especially in the case of the FX-TARN provision, due to its path-dependency. Our proposed PDE pricing framework for FX-TARN PRDC swaps is based on partitioning the pricing problem into several independent pricing sub-problems over each time period of the swap's tenor structure, with possible communication at the end of the time period. Each of these pricing sub-problems can be viewed as equivalent to a knockout PRDC swap, and requires a solution of the model PDE, which, in our case, is a time-dependent parabolic PDE in three space dimensions. Finite difference schemes on non-uniform grids are used for the spatial discretization of the model PDE, and the Alternating Direction Implicit (ADI) timestepping methods are employed for its time discretization. Numerical examples illustrating the convergence properties and efficiency of the numerical methods are provided.

(C) 3:30
Zhenyu Cui
, University of Waterlo
Nearly Exact Option Price Simulation using Characteristic Functions
Coauthors: Carole Bernard (University of Waterloo), Don Mcleish (University of Waterloo)

This paper presents a new approach to perform a nearly unbiased simulation using inversion of the characteristic function. As an application we are able to give unbiased estimates of the price of forward starting options in the Heston model and of continuously monitored Parisian options in the Black-Scholes framework. This method of simulation can be applied to problems for which the characteristic functions are known but the corresponding probability density functions are complicated.

4:00 p.m.
(C)Nadia Saad
, University of Ottawa
Compound Wishart Matrices and Noisy Covariance Matrices: Risk Underestimation
Coauthors: B. Collins and D. McDonald

In finance, Covariance matrices are used to compute the weights and the risk of the optimal portfolio. Random Matrix Theory shows that Covariance matrices determined from empirical financial time series contain a high amount of noise. Using Random matrices techniques, we derive the asymptotic formula of the effect of this noise, resulting from estimating the Covariance matrix, on determining the risk of the Markowitz's problem and hence we get a perfect estimating of the risk of the optimal portfolio. The advantage of our result is that it deals not only with independent observations but also with correlated ones.

Thursday, June 28

(C)Dimby Ramarimbahoaka University of Calgary
A stochastic discount function modeled by a finite state Markov chain and the perpetual American option

Robert J.Elliott and John van der Hoek in 2010 investigated the theory of asset pricing using a stochastic discount function process where uncertainties in the economy are modeled by a Markov chain. Stock price models, futures pricing etc were derived. In a later paper (2011), in the same framework, they discussed finite maturity American options where prices are obtained as solutions of a finite dimensional variational inequality which is expressed in terms of a system of ordinary differential equation. With Robert J.Elliott, we give a discussion on the perpetual American option case.

Almas Nassem, University of Western Ontario
Analysis of Tax-deductible Interest Payments for Re-Advanceable Canadian Mortgages
Coauthors: Mark Reesor

According to Canadian tax law the interest on loans used for investment purposes is tax deductible while interest on personal mortgage loans is not. One way of transforming from non-tax deductible to tax deductible interest expenses is to borrow against home equity to make investments. A re-advanceable mortgage is a product specifically designed to take advantage of this tax discrepancy. Using simulation we study the risk associated with the re-advanceable mortgage strategy to provide a better description of the mortgagor’s position. We assume that the mortgagor invests the borrowings secured by home equity into a single risky asset (e.g., stock or mutual fund) whose evolution is described by geometric Brownian motion (GBM). With a re-advanceable mortgage we find that the average mortgage payoff time is less than the original mortgage term. However, there is considerable variation in the payoff times with a significant probability of a payoff time exceeding the original mortgage term. Higher income homeowners enjoy a payoff time distribution with both a lower average and a lower standard deviation than low-income homeowners. Thus this strategy is most beneficial to those with the highest income. We also find this strategy protects the homeowner in the event of job loss. This work is important to lenders, financial planners and homeowners to more fully understand the benefits and risk associated with this strategy.

Bernardo Reis Carneiro da Costa Lima
, McMaster University
Dynamical Model for an Economy with Credit Expansion, Asset Price Bubbles and Fragility

Steve Keen's mathematical formulation of Hyman Minsky's financial instability hypothe-
sis provides a framework to study the effect of credit expansion on the economy. Speculation is also studied in an enhanced model, demonstrating its destabilizing effect, much in line with Minsky's ideas. We propose a second extension of Keen's model, including a stock index price process that is partially driven by the level of speculation. Using jump-diffusion dynamics, we are able to capture the double-edged effect of Ponzi investors in the stock market. In turn, the cost of borrowing fluctuates inversely with the stock price, providing a feedback effect to the economy. I will discuss the stability properties of the first two models, along with interesting features of the proposed stochastic system.

(C)Adrian Walton,
University of Western Ontario
Market Composition and Price Dynamics
We derive a model from microeconomic principles that describes how an asset's price fluctuates around its fair value in continuous time. These dynamics depend on the relative market power and perceptions of different classes of traders such as value investors, high-frequency traders and hedgers. We show how our model is useful for assessing the impact of trading strategies on an asset's returns and volatility and present a mechanism for the formation of price bubbles.

Jason Ricci, University of Toronto
Calibration of the Generalized Hawkes Processes with Latent Point Types
It is well known that the classical Hawkes Process has modeling applications in many fields including biology, neuroscience, seismology, and finance. Motivated by high-frequency finance and algorithmic trading, we propose a larger class of marked point processes that may better represent the DGP for real-world, natural systems. In this class, points are classified as those that influence the underlying intensity process (influential) and those that do not (non-influential), where such classification is latent. Moreover, we provide efficient quasi-maximum-likelihood calibration methods that makes calibration of parameters in large data sets possible. Finally, modified Sequential Monte Carlo estimators are used for real-time estimation of the state of the corresponding intensity process.