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Academic Commons Search Resultsen-usStochastic Networks: Modeling, Simulation Design and Risk Control
http://academiccommons.columbia.edu/catalog/ac:189655
Li, Juanhttp://dx.doi.org/10.7916/D88P5ZV3Mon, 28 Sep 2015 12:09:02 +0000This dissertation studies stochastic network problems that arise in various areas with important industrial applications. Due to uncertainty of both external and internal variables, these networks are exposed to the risk of failure with certain probability, which, in many cases, is very small. It is thus desirable to develop efficient simulation algorithms to study the stability of these networks and provide guidance for risk control.
Chapter 2 models equilibrium allocations in a distribution network as the solution of a linear program (LP) which minimizes the cost of unserved demands across nodes in the network. Assuming that the demands are random (following a jointly Gaussian law), we study the probability that the optimal cost exceeds a large threshold, which is a rare event. Our contribution is the development of importance sampling and conditional Monte Carlo algorithms for estimating this probability. We establish the asymptotic efficiency of our algorithms and also present numerical results that demonstrate the strong performance of our algorithms.
Chapter 3 studies an insurance-reinsurance network model that deals with default contagion risks with a particular aim of capturing cascading effects at the time of defaults. We capture these effects by finding an equilibrium allocation of settlements that can be found as the unique optimal solution of an optimization problem. We are able to obtain an asymptotic description of the most likely ways in which the default of a specific group of participants can occur, by solving a multidimensional Knapsack integer programming problem. We also propose a class of strongly efficient Monte Carlo estimators for computing the expected loss of the network conditioned on the failure of a specific set of companies.
Chapter 4 discusses control schemes for maintaining low failure probability of a transmission system power line. We construct a stochastic differential equation to describe the temperature evolution in a line subject to stochastic exogenous factors such as ambient temperature, and present a solution to the resulting stochastic heat equation. A number of control algorithms designed to limit the probability that a line exceeds its critical temperature are provided.Operations research, Engineering, Financejl3035Operations Research, Industrial Engineering and Operations ResearchDissertationsTwo Essays in Financial Engineering
http://academiccommons.columbia.edu/catalog/ac:189643
Yang, Linanhttp://dx.doi.org/10.7916/D8K35T1MMon, 28 Sep 2015 00:00:00 +0000This dissertation consists of two parts. In the first part, we investigate the potential impact of wrong-way risk on calculating credit valuation adjustment (CVA) of a derivatives portfolio. A credit valuation adjustment (CVA) is an adjustment applied to the value of a derivative contract or a portfolio of derivatives to account for counterparty credit risk. Measuring CVA requires combining models of market and credit risk. Wrong-way risk refers to the possibility that a counterparty's likelihood of default increases with the market value of the exposure. We develop a method for bounding wrong-way risk, holding fixed marginal models for market and credit risk and varying the dependence between them. Given simulated paths of the two models, we solve a linear program to find the worst-case CVA resulting from wrong-way risk. We analyze properties of the solution and prove convergence of the estimated bound as the number of paths increases. The worst case can be overly pessimistic, so we extend the procedure for a tempered CVA by penalizing the deviation of the joint model of market and credit risk from a reference model. By varying the penalty for deviations, we can sweep out the full range of possible CVA values for different degrees of wrong-way risk. Here, too, we prove convergence of the estimate of the tempered CVA and the joint distribution that attains it. Our method addresses an important source of model risk in counterparty risk measurement. In the second part, we study investors' trading behavior in a model of realization utility. We assume that investors' trading decisions are driven not only by the utility of consumption and terminal wealth, but also by the utility burst from realizing a gain or a loss. More precisely, we consider a dynamic trading problem in which an investor decides when to purchase and sell a stock to maximize her wealth utility and realization utility with her reference points adapting to the stock's gain and loss asymmetrically. We study, both theoretically and numerically, the optimal trading strategies and asset pricing implications of two types of agents: adaptive agents, who realize prospectively the reference point adaptation in the future, and naive agents, who fail to do so. We find that an adaptive agent sells the stock more frequently when the stock is at a gain than a naive agent does, and that the adaptive agent asks for a higher risk premium for the stock than the naive agent does in equilibrium. Moreover, compared to a non-adaptive agent whose reference point does not change with the stock's gain and loss, both the adaptive and naive agents sell the stock less frequently, and the naive agent requires the same risk premium as the non-adaptive agent does.Operations research, Financely2220Operations Research, Industrial Engineering and Operations Research, BusinessDissertationsMethods for Pricing Pre-Earnings Equity Options and Leveraged ETF Options
http://academiccommons.columbia.edu/catalog/ac:186986
Santoli, Marcohttp://dx.doi.org/10.7916/D86Q1W99Thu, 07 May 2015 00:17:52 +0000In this thesis, we present several analytical and numerical methods for two financial engineering problems: 1) accounting for the impact of an earnings announcement on the price and implied volatility of the associated equity options, and 2) analyzing the price dynamics of leveraged exchange-traded funds (LETFs) and valuation of LETF options. Our pricing models capture the main characteristics of these options, along with jumps and stochastic volatility in the underlying asset. We illustrate our results through numerical implementation and calibration using market data.
In the first part, we model the pricing of equity options around an earnings announcement (EA). Empirical studies have shown that an earnings announcement can lead to an immediate price shock to the company stock. Since many companies also have options written on their stocks, the option prices should reflect the uncertain price impact of an upcoming EA before expiration. To represent the shock due to earnings, we incorporate a random jump on the announcement date in the dynamics of the stock price. We consider different distributions of the scheduled earnings jump as well as different underlying stock price dynamics before and after the EA date. Our main contributions include analytical option pricing formulas when the underlying stock price follows the Kou model along with a double-exponential or Gaussian EA jump on the announcement date. Furthermore, we derive analytic bounds and asymptotics for the pre-EA implied volatility under various models. The calibration results demonstrate adequate fit of the entire implied volatility surface prior to an announcement. The comparison of the risk-neutral distribution of the EA jump to its historical counterpart is also discussed. Moreover, we discuss the valuation and exercise strategy of pre-EA American options, and present an analytical approximation and numerical results.
The second part focuses on the analysis of LETFs. We start by providing a quantitative risk analysis of LETFs with an emphasis on the impact of leverage ratios and investment horizons. Given an investment horizon, different leverage ratios imply different levels of risk. Therefore, the idea of an {admissible range of leverage ratios} is introduced. For an admissible leverage ratio, the associated LETF satisfies a given risk constraint based on, for example, the value-at-risk (VaR) and conditional VaR. Moreover, we discuss the concept of {admissible risk horizon} so that the investor can control risk exposure by selecting an appropriate holding period. The intra-horizon risk is calculated, showing that higher leverage can significantly increase the probability of an LETF value hitting a lower level. This leads us to evaluate a stop-loss/take-profit strategy for LETFs and determine the optimal take-profit given a stop-loss risk constraint. In addition, the impact of volatility exposure on the returns of different LETF portfolios is investigated.
In the last chapter, we study the pricing of options written on LETFs. Since LETFs on the same reference index share the same source of risk, it is important to consider a consistent pricing methodology of these options. In addition, LETFs can theoretically experience a loss greater than 100\%. In practice, some LETF providers design the fund so that the daily returns are capped both downward and upward. We incorporate these features and model the reference index by a stochastic volatility model with jumps. An efficient numerical algorithm based on transform methods to value options under this model is presented. We illustrate the accuracy of our pricing algorithm by comparing it to existing methods. Calibration using empirical option data shows the impact of leverage ratio on the implied volatility. Our method is extended to price American-style LETF options.Finance, Operations researchIndustrial Engineering and Operations ResearchDissertationsOptimal Multiple Stopping Approach to Mean Reversion Trading
http://academiccommons.columbia.edu/catalog/ac:186941
Li, Xinhttp://dx.doi.org/10.7916/D88K781SFri, 24 Apr 2015 18:33:24 +0000This thesis studies the optimal timing of trades under mean-reverting price dynamics subject to fixed transaction costs. We first formulate an optimal double stopping problem whereby a speculative investor can choose when to enter and subsequently exit the market. The investor's value functions and optimal timing strategies are derived when prices are driven by an Ornstein-Uhlenbeck (OU), exponential OU, or Cox-Ingersoll-Ross (CIR) process. Moreover, we analyze a related optimal switching problem that involves an infinite sequence of trades. In addition to solving for the value functions and optimal switching strategies, we identify the conditions under which the double stopping and switching problems admit the same optimal entry and/or exit timing strategies. A number of extensions are also considered, such as incorporating a stop-loss constraint, or a minimum holding period under the OU model.
A typical solution approach for optimal stopping problems is to study the associated free boundary problems or variational inequalities (VIs). For the double optimal stopping problem, we apply a probabilistic methodology and rigorously derive the optimal price intervals for market entry and exit. A key step of our approach involves a transformation, which in turn allows us to characterize the value function as the smallest concave majorant of the reward function in the transformed coordinate. In contrast to the variational inequality approach, this approach directly constructs the value function as well as the optimal entry and exit regions, without a priori conjecturing a candidate value function or timing strategy. Having solved the optimal double stopping problem, we then apply our results to deduce a similar solution structure for the optimal switching problem. We also verify that our value functions solve the associated VIs.
Among our results, we find that under OU or CIR price dynamics, the optimal stopping problems admit the typical buy-low-sell-high strategies. However, when the prices are driven by an exponential OU process, the investor generally enters when the price is low, but may find it optimal to wait if the current price is sufficiently close to zero. In other words, the continuation (waiting) region for entry is disconnected. A similar phenomenon is observed in the OU model with stop-loss constraint. Indeed, the entry region is again characterized by a bounded price interval that lies strictly above the stop-loss level. As for the exit timing, a higher stop-loss level always implies a lower optimal take-profit level. In all three models, numerical results are provided to illustrate the dependence of timing strategies on model parameters.Operations research, FinanceIndustrial Engineering and Operations ResearchDissertationsHigh-Dimensional Portfolio Management: Taxes, Execution and Information Relaxations
http://academiccommons.columbia.edu/catalog/ac:185815
Wang, Chunhttp://dx.doi.org/10.7916/D8M043JJMon, 07 Jul 2014 00:00:00 +0000Portfolio management has always been a key topic in finance research area. While many researchers have studied portfolio management problems, most of the work to date assumes trading is frictionless. This dissertation presents our investigation of the optimal trading policies and efforts of applying duality method based on information relaxations to portfolio problems where the investor manages multiple securities and confronts trading frictions, in particular capital gain taxes and execution cost. In Chapter 2, we consider dynamic asset allocation problems where the investor is required to pay capital gains taxes on her investment gains. This is a very challenging problem because the tax to be paid whenever a security is sold depends on the tax basis, i.e. the price(s) at which the security was originally purchased. This feature results in high-dimensional and path-dependent problems which cannot be solved exactly except in the case of very stylized problems with just one or two securities and relatively few time periods. The asset allocation problem with taxes has several variations depending on: (i) whether we use the exact or average tax-basis and (ii) whether we allow the full use of losses (FUL) or the limited use of losses (LUL). We consider all of these variations in this chapter but focus mainly on the exact and average-cost tax-basis LUL cases since these problems are the most realistic and generally the most challenging. We develop several sub-optimal trading policies for these problems and use duality techniques based on information relaxations to assess their performances. Our numerical experiments consider problems with as many as 20 securities and 20 time periods. The principal contribution of this chapter is in demonstrating that much larger problems can now be tackled through the use of sophisticated optimization techniques and duality methods based on information-relaxations. We show in fact that the dual formulation of exact tax-basis problems are much easier to solve than the corresponding primal problems. Indeed, we can easily solve dual problem instances where the number of securities and time periods is much larger than 20. We also note, however, that while the average tax-basis problem is relatively easier to solve in general, its corresponding dual problem instances are non-convex and more difficult to solve. We therefore propose an approach for the average tax-basis dual problem that enables valid dual bounds to still be obtained. In Chapter 3, we consider a portfolio execution problem where a possibly risk-averse agent needs to trade a fixed number of shares in multiple stocks over a short time horizon. Our price dynamics can capture linear but stochastic temporary and permanent price impacts as well as stochastic volatility. In general it's not possible to solve even numerically for the optimal policy in this model, however, and so we must instead search for good sub-optimal policies. Our principal policy is a variant of an open-loop feedback control (OLFC) policy and we show how the corresponding OLFC value function may be used to construct good primal and dual bounds on the optimal value function. The dual bound is constructed using the recently developed duality methods based on information relaxations. One of the contributions of this chapter is the identification of sufficient conditions to guarantee convexity, and hence tractability, of the associated dual problem instances. That said, we do not claim that the only plausible models are those where all dual problem instances are convex. We also show that it is straightforward to include a non-linear temporary price impact as well as return predictability in our model. We demonstrate numerically that good dual bounds can be computed quickly even when nested Monte-Carlo simulations are required to estimate the so-called dual penalties. These results suggest that the dual methodology can be applied in many models where closed-form expressions for the dual penalties cannot be computed. In Chapter 4, we apply duality methods based on information relaxations to dynamic zero-sum games. We show these methods can easily be used to construct dual lower and upper bounds for the optimal value of these games. In particular, these bounds can be used to evaluate sub-optimal policies for zero-sum games when calculating the optimal policies and game value is intractable.Operations research, FinanceIndustrial Engineering and Operations ResearchDissertationsFrom Continuous to Discrete: Studies on Continuity Corrections and Monte Carlo Simulation with Applications to Barrier Options and American Options
http://academiccommons.columbia.edu/catalog/ac:171186
Cao, Menghuihttp://dx.doi.org/10.7916/D8PG1PS1Fri, 28 Feb 2014 00:00:00 +0000This dissertation 1) shows continuity corrections for first passage probabilities of Brownian bridge and barrier joint probabilities, which are applied to the pricing of two-dimensional barrier and partial barrier options, and 2) introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American options. The joint distribution of Brownian motion and its first passage time has found applications in many areas, including sequential analysis, pricing of barrier options, and credit risk modeling. There are, however, no simple closed-form solutions for these joint probabilities in a discrete-time setting. Chapter 2 shows that, discrete two-dimensional barrier and partial barrier joint probabilities can be approximated by their continuous-time probabilities with remarkable accuracy after shifting the barrier away from the underlying by a factor. We achieve this through a uniform continuity correction theorem on the first passage probabilities for Brownian bridge, extending relevant results in Siegmund (1985a). The continuity corrections are applied to the pricing of two-dimensional barrier and partial barrier options, extending the results in Broadie, Glasserman & Kou (1997) on one-dimensional barrier options. One interesting aspect is that for type B partial barrier options, the barrier correction cannot be applied throughout one pricing formula, but only to some barrier values and leaving the other unchanged, the direction of correction may also vary within one formula. In Chapter 3 we introduce new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American-style options. For simulation algorithms that compute lower bounds of American option values, we apply martingale control variates and introduce the local policy enhancement, which adopts a local simulation to improve the exercise policy. For duality-based upper bound methods, specifically the primal-dual simulation algorithm (Andersen and Broadie 2004), we have developed two improvements. One is sub-optimality checking, which saves unnecessary computation when it is sub-optimal to exercise the option along the sample path; the second is boundary distance grouping, which reduces computational time by skipping computation on selected sample paths based on the distance to the exercise boundary. Numerical results are given for single asset Bermudan options, moving window Asian options and Bermudan max options. In some examples the computational time is reduced by a factor of several hundred, while the confidence interval of the true option value is considerably tighter than before the improvements.Operations research, FinanceIndustrial Engineering and Operations ResearchDissertationsPricing, Trading and Clearing of Defaultable Claims Subject to Counterparty Risk
http://academiccommons.columbia.edu/catalog/ac:169814
Kim, Jinbeomhttp://dx.doi.org/10.7916/D8319SWWMon, 03 Feb 2014 00:00:00 +0000The recent financial crisis and subsequent regulatory changes on over-the-counter (OTC) markets have given rise to the new valuation and trading frameworks for defaultable claims to investors and dealer banks. More OTC market participants have adopted the new market conventions that incorporate counterparty risk into the valuation of OTC derivatives. In addition, the use of collateral has become common for most bilateral trades to reduce counterparty default risk. On the other hand, to increase transparency and market stability, the U.S and European regulators have required mandatory clearing of defaultable derivatives through central counterparties. This dissertation tackles these changes and analyze their impacts on the pricing, trading and clearing of defaultable claims. In the first part of the thesis, we study a valuation framework for financial contracts subject to reference and counterparty default risks with collateralization requirement. We propose a fixed point approach to analyze the mark-to-market contract value with counterparty risk provision, and show that it is a unique bounded and continuous fixed point via contraction mapping. This leads us to develop an accurate iterative numerical scheme for valuation. Specifically, we solve a sequence of linear inhomogeneous partial differential equations, whose solutions converge to the fixed point price function. We apply our methodology to compute the bid and ask prices for both defaultable equity and fixed-income derivatives, and illustrate the non-trivial effects of counterparty risk, collateralization ratio and liquidation convention on the bid-ask prices. In the second part, we study the problem of pricing and trading of defaultable claims among investors with heterogeneous risk preferences and market views. Based on the utility-indifference pricing methodology, we construct the bid-ask spreads for risk-averse buyers and sellers, and show that the spreads widen as risk aversion or trading volume increases. Moreover, we analyze the buyer's optimal static trading position under various market settings, including (i) when the market pricing rule is linear, and (ii) when the counterparty -- single or multiple sellers -- may have different nonlinear pricing rules generated by risk aversion and belief heterogeneity. For defaultable bonds and credit default swaps, we provide explicit formulas for the optimal trading positions, and examine the combined effect of heterogeneous risk aversions and beliefs. In particular, we find that belief heterogeneity, rather than the difference in risk aversion, is crucial to trigger a trade. Finally, we study the impact of central clearing on the credit default swap (CDS) market. Central clearing of CDS through a central counterparty (CCP) has been proposed as a tool for mitigating systemic risk and counterpart risk in the CDS market. The design of CCPs involves the implementation of margin requirements and a default fund, for which various designs have been proposed. We propose a mathematical model to quantify the impact of the design of the CCP on the incentive for clearing and analyze the market equilibrium. We determine the minimum number of clearing participants required so that they have an incentive to clear part of their exposures. Furthermore, we analyze the equilibrium CDS positions and their dependence on the initial margin, risk aversion, and counterparty risk in the inter-dealer market. Our numerical results show that minimizing the initial margin maximizes the total clearing positions as well as the CCP's revenue.Operations research, Financejk3071Industrial Engineering and Operations ResearchDissertationsStochastic Models of Limit Order Markets
http://academiccommons.columbia.edu/catalog/ac:161685
Kukanov, Arseniyhttp://hdl.handle.net/10022/AC:P:20511Thu, 30 May 2013 00:00:00 +0000During the last two decades most stock and derivatives exchanges in the world transitioned to electronic trading in limit order books, creating a need for a new set of quantitative models to describe these order-driven markets. This dissertation offers a collection of models that provide insight into the structure of modern financial markets, and can help to optimize trading decisions in practical applications. In the first part of the thesis we study the dynamics of prices, order flows and liquidity in limit order markets over short timescales. We propose a stylized order book model that predicts a particularly simple linear relation between price changes and order flow imbalance, defined as a difference between net changes in supply and demand. The slope in this linear relation, called a price impact coefficient, is inversely proportional in our model to market depth - a measure of liquidity. Our empirical results confirm both of these predictions. The linear relation between order flow imbalance and price changes holds for time intervals between 50 milliseconds and 5 minutes. The inverse relation between the price impact coefficient and market depth holds on longer timescales. These findings shed a new light on intraday variations in market volatility. According to our model volatility fluctuates due to changes in market depth or in order flow variance. Previous studies also found a positive correlation between volatility and trading volume, but in order-driven markets prices are determined by the limit order book activity, so the association between trading volume and volatility is unclear. We show how a spurious correlation between these variables can indeed emerge in our linear model due to time aggregation of high-frequency data. Finally, we observe short-term positive autocorrelation in order flow imbalance and discuss an application of this variable as a measure of adverse selection in limit order executions. Our results suggest that monitoring recent order flow can improve the quality of order executions in practice. In the second part of the thesis we study the problem of optimal order placement in a fragmented limit order market. To execute a trade, market participants can submit limit orders or market orders across various exchanges where a stock is traded. In practice these decisions are influenced by sizes of order queues and by statistical properties of order flows in each limit order book, and also by rebates that exchanges pay for limit order submissions. We present a realistic model of limit order executions and formalize the search for an optimal order placement policy as a convex optimization problem. Based on this formulation we study how various factors determine investor's order placement decisions. In a case when a single exchange is used for order execution, we derive an explicit formula for the optimal limit and market order quantities. Our solution shows that the optimal split between market and limit orders largely depends on one's tolerance to execution risk. Market orders help to alleviate this risk because they execute with certainty. Correspondingly, we find that an optimal order allocation shifts to these more expensive orders when the execution risk is of primary concern, for example when the intended trade quantity is large or when it is costly to catch up on the quantity after limit order execution fails. We also characterize the optimal solution in the general case of simultaneous order placement on multiple exchanges, and show that it sets execution shortfall probabilities to specific threshold values computed with model parameters. Finally, we propose a non-parametric stochastic algorithm that computes an optimal solution by resampling historical data and does not require specifying order flow distributions. A numerical implementation of this algorithm is used to study the sensitivity of an optimal solution to changes in model parameters. Our numerical results show that order placement optimization can bring a substantial reduction in trading costs, especially for small orders and in cases when order flows are relatively uncorrelated across trading venues. The order placement optimization framework developed in this thesis can also be used to quantify the costs and benefits of financial market fragmentation from the point of view of an individual investor. For instance, we find that a positive correlation between order flows, which is empirically observed in a fragmented U.S. equity market, increases the costs of trading. As the correlation increases it may become more expensive to trade in a fragmented market than it is in a consolidated market. In the third part of the thesis we analyze the dynamics of limit order queues at the best bid or ask of an exchange. These queues consist of orders submitted by a variety of market participants, yet existing order book models commonly assume that all orders have similar dynamics. In practice, some orders are submitted by trade execution algorithms in an attempt to buy or sell a certain quantity of assets under time constraints, and these orders are canceled if their realized waiting time exceeds a patience threshold. In contrast, high-frequency traders submit and cancel orders depending on the order book state and their orders are not driven by patience. The interaction between these two order types within a single FIFO queue leads bursts of order cancelations for small queues and anomalously long waiting times in large queues. We analyze a fluid model that describes the evolution of large order queues in liquid markets, taking into account the heterogeneity between order submission and cancelation strategies of different traders. Our results show that after a finite initial time interval, the queue reaches a specific structure where all orders from high-frequency traders stay in the queue until execution but most orders from execution algorithms exceed their patience thresholds and are canceled. This "order crowding" effect has been previously noted by participants in highly liquid stock and futures markets and was attributed to a large participation of high-frequency traders. In our model, their presence creates an additional workload, which increases queue waiting times for new orders. Our analysis of the fluid model leads to waiting time estimates that take into account the distribution of order types in a queue. These estimates are tested against a large dataset of realized limit order waiting times collected by a U.S. equity brokerage firm. The queue composition at a moment of order submission noticeably affects its waiting time and we find that assuming a single order type for all orders in the queue leads to unrealistic results. Estimates that assume instead a mix of heterogeneous orders in the queue are closer to empirical data. Our model for a limit order queue with heterogeneous order types also appears to be interesting from a methodological point of view. It introduces a new type of behavior in a queueing system where one class of jobs has state-dependent dynamics, while others are driven by patience. Although this model is motivated by the analysis of limit order books, it may find applications in studying other service systems with state-dependent abandonments.Operations research, Finance, Statisticsak2870Industrial Engineering and Operations ResearchDissertationsFinancial Portfolio Risk Management: Model Risk, Robustness and Rebalancing Error
http://academiccommons.columbia.edu/catalog/ac:161415
Xu, Xingbohttp://hdl.handle.net/10022/AC:P:20382Mon, 20 May 2013 00:00:00 +0000Risk management has always been in key component of portfolio management. While more and more complicated models are proposed and implemented as research advances, they all inevitably rely on imperfect assumptions and estimates. This dissertation aims to investigate the gap between complicated theoretical modelling and practice. We mainly focus on two directions: model risk and reblancing error. In the first part of the thesis, we develop a framework for quantifying the impact of model error and for measuring and minimizing risk in a way that is robust to model error. This robust approach starts from a baseline model and finds the worst-case error in risk measurement that would be incurred through a deviation from the baseline model, given a precise constraint on the plausibility of the deviation. Using relative entropy to constrain model distance leads to an explicit characterization of worst-case model errors; this characterization lends itself to Monte Carlo simulation, allowing straightforward calculation of bounds on model error with very little computational effort beyond that required to evaluate performance under the baseline nominal model. This approach goes well beyond the effect of errors in parameter estimates to consider errors in the underlying stochastic assumptions of the model and to characterize the greatest vulnerabilities to error in a model. We apply this approach to problems of portfolio risk measurement, credit risk, delta hedging, and counterparty risk measured through credit valuation adjustment. In the second part, we apply this robust approach to a dynamic portfolio control problem. The sources of model error include the evolution of market factors and the influence of these factors on asset returns. We analyze both finite- and infinite-horizon problems in a model in which returns are driven by factors that evolve stochastically. The model incorporates transaction costs and leads to simple and tractable optimal robust controls for multiple assets. We illustrate the performance of the controls on historical data. Robustness does improve performance in out-of-sample tests in which the model is estimated on a rolling window of data and then applied over a subsequent time period. By acknowledging uncertainty in the estimated model, the robust rules lead to less aggressive trading and are less sensitive to sharp moves in underlying prices. In the last part, we analyze the error between a discretely rebalanced portfolio and its continuously rebalanced counterpart in the presence of jumps or mean-reversion in the underlying asset dynamics. With discrete rebalancing, the portfolio's composition is restored to a set of fixed target weights at discrete intervals; with continuous rebalancing, the target weights are maintained at all times. We examine the difference between the two portfolios as the number of discrete rebalancing dates increases. We derive the limiting variance of the relative error between the two portfolios for both the mean-reverting and jump-diffusion cases. For both cases, we derive ``volatility adjustments'' to improve the approximation of the discretely rebalanced portfolio by the continuously rebalanced portfolio, based on on the limiting covariance between the relative rebalancing error and the level of the continuously rebalanced portfolio. These results are based on strong approximation results for jump-diffusion processes.Operations research, Finance, Mathematicsxx2126Industrial Engineering and Operations Research, BusinessDissertationsContingent Capital: Valuation and Risk Implications Under Alternative Conversion Mechanisms
http://academiccommons.columbia.edu/catalog/ac:152933
Nouri, Behzadhttp://hdl.handle.net/10022/AC:P:14800Fri, 28 Sep 2012 00:00:00 +0000Several proposals for enhancing the stability of the financial system include requirements that banks hold some form of contingent capital, meaning equity that becomes available to a bank in the event of a crisis or financial distress. Specific proposals vary in their choice of conversion trigger and conversion mechanism, and have inspired extensive scrutiny regarding their effectivity in avoiding costly public rescues and bail-outs and potential adverse effects on market dynamics. While allowing banks to leverage and gain a higher return on their equity capital during the upturns in financial markets, contingent capital provides an automatic mechanism to reduce debt and raise the loss bearing capital cushion during the downturns and market crashes; therefore, making it possible to achieve stability and robustness in the financial sector, without reducing efficiency and competitiveness of the banking system with higher regulatory capital requirements. However, many researchers have raised concerns regarding unintended consequences and implications of such instruments for market dynamics. Death spirals in the stock price near the conversion, possibility of profitable stock or book manipulations by either the investors or the issuer, the marketability and demand for such hybrid instruments, contagion and systemic risks arising from the hedging strategies of the investors and higher risk taking incentives for issuers are among such concerns. Though substantial, many of such issues are addressed through a prudent design of the trigger and conversion mechanism. In the following chapters, we develop multiple models for pricing and analysis of contingent capital under different conversion mechanisms. In Chapter 2 we analyze the case of contingent capital with a capital-ratio trigger and partial and on-going conversion. The capital ratio we use is based on accounting or book value to approximate the regulatory ratios that determine capital requirements for banks. The conversion process is partial and on-going in the sense that each time a bank's capital ratio reaches the minimum threshold, just enough debt is converted to equity to meet the capital requirement, so long as the contingent capital has not been depleted. In Chapter 3 we simplify the design to all-at-once conversion however we perform the analysis through a much richer model which incorporates tail risk in terms of jumps, endogenous optimal default policy and debt rollover. We also investigate the case of bail-in debt, where at default the original shareholders are wiped out and the converted investors take control of the firm. In the case of contingent convertibles the conversion trigger is assumed as a contractual term specified by market value of assets. For bail-in debt the trigger is where the original shareholders optimally default. We study incentives of shareholders to change the capital structure and how CoCo's affect risk incentives. Several researchers have advocated use of a market based trigger which is forward looking, continuously updated and readily available, while some others have raised concerns regarding unintended consequences of a market based trigger. In Chapter 4 we investigate one of these issues, namely the existence and uniqueness of equilibrium when the conversion trigger is based on the stock price.Finance, Operations researchbn2164Industrial Engineering and Operations Research, BusinessDissertationsRisk Premia and Optimal Liquidation of Defaultable Securities
http://academiccommons.columbia.edu/catalog/ac:139526
Leung, Siu Tang; Liu, Penghttp://hdl.handle.net/10022/AC:P:11331Mon, 03 Oct 2011 00:00:00 +0000This paper studies the optimal timing to liquidate defaultable securities in a general intensity-based credit risk model under stochastic interest rate. We incorporate the potential price discrepancy between the market and investors, which is characterized by risk-neutral valuation under different default risk premia specifications. To quantify the value of optimally timing to sell, we introduce the delayed liquidation premium which is closely related to the stochastic bracket between the market price and a pricing kernel. We analyze the optimal liquidation policy for various credit derivatives. Our model serves as the building block for the sequential buying and selling problem. We also discuss the extensions to a jump-diffusion default intensity model as well as a defaultable equity model.Finance, Economic theorytl2497Industrial Engineering and Operations ResearchArticlesAccounting for Risk Aversion in Derivatives Purchase Timing
http://academiccommons.columbia.edu/catalog/ac:138783
Leung, Siu Tang; Ludkovski, Mikehttp://hdl.handle.net/10022/AC:P:11191Fri, 16 Sep 2011 00:00:00 +0000We study the problem of optimal timing to buy/sell derivatives by a risk-averse agent in incomplete markets. Adopting the exponential utility indifference valuation, we investigate this timing flexibility and the associated delayed purchase premium. This leads to a stochastic control and optimal stopping problem that combines the observed market price dynamics and the agent's risk preferences. Our results extend recent work on indifference valuation of American options, as well as the authors' first paper (Leung and Ludkovski, SIAM J. Fin. Math., 2011). In the case of Markovian models of contracts on non-traded assets, we provide analytical characterizations and numerical studies of the optimal purchase strategies, with applications to both equity and credit derivatives.Finance, Economic theorytl2497Industrial Engineering and Operations ResearchArticlesQuantitative Modeling of Credit Derivatives
http://academiccommons.columbia.edu/catalog/ac:131549
Kan, Yu Hanghttp://hdl.handle.net/10022/AC:P:10272Thu, 05 May 2011 00:00:00 +0000The recent financial crisis has revealed major shortcomings in the existing approaches for modeling credit derivatives. This dissertation studies various issues related to the modeling of credit derivatives: hedging of portfolio credit derivatives, calibration of dynamic credit models, and modeling of credit default swap portfolios. In the first part, we compare the performance of various hedging strategies for index collateralized debt obligation (CDO) tranches during the recent financial crisis. Our empirical analysis shows evidence for market incompleteness: a large proportion of risk in the CDO tranches appears to be unhedgeable. We also show that, unlike what is commonly assumed, dynamic models do not necessarily perform better than static models, nor do high-dimensional bottom-up models perform better than simpler top-down models. On the other hand, model-free regression-based hedging appears to be surprisingly effective when compared to other hedging strategies. The second part is devoted to computational methods for constructing an arbitrage-free CDO pricing model compatible with observed CDO prices. This method makes use of an inversion formula for computing the aggregate default rate in a portfolio from expected tranche notionals, and a quadratic programming method for recovering expected tranche notionals from CDO spreads. Comparing this approach to other calibration methods, we find that model-dependent quantities such as the forward starting tranche spreads and jump-to-default ratios are quite sensitive to the calibration method used, even within the same model class. The last chapter of this dissertation focuses on statistical modeling of credit default swaps (CDSs). We undertake a systematic study of the univariate and multivariate properties of CDS spreads, using time series of the CDX Investment Grade index constituents from 2005 to 2009. We then propose a heavy-tailed multivariate time series model for CDS spreads that captures these properties. Our model can be used as a framework for measuring and managing the risk of CDS portfolios, and is shown to have better performance than the affine jump-diffusion or random walk models for predicting loss quantiles of various CDS portfolios.Finance, Mathematicsyk2246Industrial Engineering and Operations ResearchDissertationsContagion and Systemic Risk in Financial Networks
http://academiccommons.columbia.edu/catalog/ac:131474
Moussa, Amalhttp://hdl.handle.net/10022/AC:P:10249Fri, 29 Apr 2011 00:00:00 +0000The 2007-2009 financial crisis has shed light on the importance of contagion and systemic risk, and revealed the lack of adequate indicators for measuring and monitoring them. This dissertation addresses these issues and leads to several recommendations for the design of an improved assessment of systemic importance, improved rating methods for structured finance securities, and their use by investors and risk managers. Using a complete data set of all mutual exposures and capital levels of financial institutions in Brazil in 2007 and 2008, we explore in chapter 2 the structure and dynamics of the Brazilian financial system. We show that the Brazilian financial system exhibits a complex network structure characterized by a strong degree of heterogeneity in connectivity and exposure sizes across institutions, which is qualitatively and quantitatively similar to the statistical features observed in other financial systems. We find that the Brazilian financial network is well represented by a directed scale-free network, rather than a small world network. Based on these observations, we propose a stochastic model for the structure of banking networks, representing them as a directed weighted scale free network with power law distributions for in-degree and out-degree of nodes, Pareto distribution for exposures. This model may then be used for simulation studies of contagion and systemic risk in networks. We propose in chapter 3 a quantitative methodology for assessing contagion and systemic risk in a network of interlinked institutions. We introduce the Contagion Index as a metric of the systemic importance of a single institution or a set of institutions, that combines the effects of both common market shocks to portfolios and contagion through counterparty exposures. Using a directed scale-free graph simulation of the financial system, we study the sensitivity of contagion to a change in aggregate network parameters: connectivity, concentration of exposures, heterogeneity in degree distribution and network size. More concentrated and more heterogeneous networks are found to be more resilient to contagion. The impact of connectivity is more controversial: in well-capitalized networks, increasing connectivity improves the resilience to contagion when the initial level of connectivity is high, but increases contagion when the initial level of connectivity is low. In undercapitalized networks, increasing connectivity tends to increase the severity of contagion. We also study the sensitivity of contagion to local measures of connectivity and concentration across counterparties --the counterparty susceptibility and local network frailty-- that are found to have a monotonically increasing relationship with the systemic risk of an institution. Requiring a minimum (aggregate) capital ratio is shown to reduce the systemic impact of defaults of large institutions; we show that the same effect may be achieved with less capital by imposing such capital requirements only on systemically important institutions and those exposed to them. In chapter 4, we apply this methodology to the study of the Brazilian financial system. Using the Contagion Index, we study the potential for default contagion and systemic risk in the Brazilian system and analyze the contribution of balance sheet size and network structure to systemic risk. Our study reveals that, aside from balance sheet size, the network-based local measures of connectivity and concentration of exposures across counterparties introduced in chapter 3, the counterparty susceptibility and local network frailty, contribute significantly to the systemic importance of an institution in the Brazilian network. Thus, imposing an upper bound on these variables could help reducing contagion. We examine the impact of various capital requirements on the extent of contagion in the Brazilian financial system, and show that targeted capital requirements achieve the same reduction in systemic risk with lower requirements in capital for financial institutions. The methodology we proposed in chapter 3 for estimating contagion and systemic risk requires visibility on the entire network structure. Reconstructing bilateral exposures from balance sheets data is then a question of interest in a financial system where bilateral exposures are not disclosed. We propose in chapter 5 two methods to derive a distribution of bilateral exposures matrices. The first method attempts to recover the balance sheet assets and liabilities "sample by sample". Each sample of the bilateral exposures matrix is solution of a relative entropy minimization problem subject to the balance sheet constraints. However, a solution to this problem does not always exist when dealing with sparse sample matrices. Thus, we propose a second method that attempts to recover the assets and liabilities "in the mean". This approach is the analogue of the Weighted Monte Carlo method introduced by Avellaneda et al. (2001). We first simulate independent samples of the bilateral exposures matrix from a relevant prior distribution on the network structure, then we compute posterior probabilities by maximizing the entropy under the constraints that the balance sheet assets and liabilities are recovered in the mean. We discuss the pros and cons of each approach and explain how it could be used to detect systemically important institutions in the financial system. The recent crisis has also raised many questions regarding the meaning of structured finance credit ratings issued by rating agencies and the methodology behind them. Chapter 6 aims at clarifying some misconceptions related to structured finance ratings and how they are commonly interpreted: we discuss the comparability of structured finance ratings with bond ratings, the interaction between the rating procedure and the tranching procedure and its consequences for the stability of structured finance ratings in time. These insights are illustrated in a factor model by simulating rating transitions for CDO tranches using a nested Monte Carlo method. In particular, we show that the downgrade risk of a CDO tranche can be quite different from a bond with same initial rating. Structured finance ratings follow path-dependent dynamics that cannot be adequately described, as usually done, by a matrix of transition probabilities. Therefore, a simple labeling via default probability or expected loss does not discriminate sufficiently their downgrade risk. We propose to supplement ratings with indicators of downgrade risk. To overcome some of the drawbacks of existing rating methods, we suggest a risk-based rating procedure for structured products. Finally, we formulate a series of recommendations regarding the use of credit ratings for CDOs and other structured credit instruments.Finance, Statisticsam2810Industrial Engineering and Operations Research, StatisticsDissertations