Risk Management and Simulation K11622_FM.indd 5/13/13 3:08 PM K11622_FM.indd 5/13/13 3:08 PM Risk Management and Simulation Aparna Gupta K11622_FM.indd 5/13/13 3:08 PM MATLAB® is a trademark of The MathWorks, Inc and is used with permission The MathWorks does not warrant the accuracy of the text or exercises in this book This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2014 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed on acid-free paper Version Date: 20130509 International Standard Book Number-13: 978-1-4398-3594-4 (Hardback) This book contains information obtained from authentic and highly regarded sources Reasonable 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Contents I Risk and Regulation Defining Risk 1.1 Types of Risk 1.1.0.1 Pure Risk 1.1.0.2 Speculative Risk 1.1.1 Classification of Pure Risk 1.1.2 Classification of Speculative Risk 1.2 Getting Started with Modeling Risk 1.2.1 Random Variable and Probability 1.2.1.1 Summarizing Random Variables 1.2.1.2 Several Random Variables and Correlation 1.2.1.3 Conditional Probability 1.2.2 Specific Models of Risk 1.2.2.1 Normal Distribution 1.2.2.2 Uniform Distribution 1.2.2.3 Central Limit Theorem 1.2.2.4 Binomial Distribution 1.2.2.5 Poisson Distribution 1.2.2.6 Exponential Distribution 1.2.2.7 Weibull Distribution 1.2.2.8 Lognormal Distribution 1.2.2.9 Chi-Square Distribution 1.2.2.10 Gamma Distribution 1.3 MATLAB Tools for Distributions 1.4 Summary 1.5 Questions and Exercises 5 12 12 14 16 17 19 19 20 21 23 23 24 26 27 27 29 30 30 31 Framework for Risk Management 2.1 How to Handle Risk 2.1.1 The Risk Management Framework 2.1.2 Risk Preference vs Risk Aversion 2.1.2.1 Normative vs Behavioral Choice 2.1.3 Risk Measures 2.1.4 Risk Management 2.1.5 Elements of the Framework 35 36 37 40 43 45 48 50 vii viii Contents 2.1.5.1 Avoid 2.1.5.2 Mitigate 2.1.5.3 Transfer 2.1.5.4 Keep Example Contexts to Apply the Framework 2.2.1 Analysis Using Central Measures 2.2.2 Tail Analysis 2.2.3 Scenario Analysis 2.2.4 Stress Testing MATLAB Tools for Risk Measures Summary Questions and Exercises 51 51 53 54 54 55 56 58 59 60 61 61 Banking 65 66 67 71 73 74 80 84 86 86 Principles of Simulation and Generating Random Variates 4.1 Principles of Simulation 4.1.1 What Is Simulation? 4.2 Random Number Generation 4.2.1 Linear Congruential Generator 4.2.2 Lagged Fibonacci Generator 4.3 Generation of Discrete Random Variates 4.3.1 n-Outcome Random Variate 4.3.2 Poisson Random Variate 4.4 Generation of Continuous Random Variates 4.4.1 Inverse Transform Method 4.4.2 Acceptance-Rejection Method 4.4.3 Normal Random Variate 4.4.3.1 Box-Muller Method 4.4.3.2 Polar-Marsaglia Method 4.4.3.3 Generation of Multi-Variate Normal 4.4.4 Chi-Square and Other Random Variates 4.5 Testing Random Variates 4.5.1 Testing for Independence of Random Numbers 93 93 94 96 97 97 98 98 99 100 100 101 103 104 104 106 107 107 108 2.2 2.3 2.4 2.5 Regulations and Risk Management 3.1 Regulations Overview 3.1.1 Regulatory Evolution for Banking 3.1.2 Regulatory Evolution for Investment 3.1.3 Regulatory Evolution for Insurance 3.2 Regulations and Banking 3.3 Regulations and Investment Banking 3.4 Regulations and Insurance 3.5 Summary 3.6 Questions and Exercises II Modeling and Simulation of Risk Contents 4.5.1.1 Shuffling Procedure Testing for Correctness of Distribution 4.5.2.1 The χ2 Goodness of Fit Test 4.5.2.2 Kolmogorov-Smirnov Test 4.6 Validation of Model 4.6.1 Techniques for Model Verification 4.6.2 Techniques for Model Validation 4.7 Output Analysis 4.7.1 Descriptive Output Analysis 4.7.1.1 Designing Simulation Run by Properties of Estimators 4.7.2 Inferential Output Analysis 4.8 MATLAB Tools for Simulation 4.9 Summary 4.10 Questions and Exercises 4.5.2 ix 109 110 110 112 113 114 115 117 118 119 120 121 122 122 Modeling Risk Evolving over Time 127 5.1 Stochastic Processes 127 5.2 Discrete-Time Evolution of Risk 128 5.2.1 Discrete-Time Markov Chains 129 5.2.2 Simple Random Walk 133 5.2.3 Geometric Random Walk 135 5.3 Continuous-Time Evolution of Risk 136 5.3.1 Continuous-Time Markov Chains 136 5.3.2 Poisson Process 138 5.3.3 Birth-Death Process 140 5.3.4 Markov Process 141 5.3.5 Gaussian Process 142 5.3.6 Brownian Motion 144 5.3.6.1 Approximating Brownian Motion by a Random Walk 145 5.3.6.2 Convergence of Random Variables 146 5.3.6.3 Properties of the Wiener Process 147 5.3.7 Brownian Motion with Drift and Geometric Brownian Motion 149 5.3.8 Additional Concepts for Stochastic Processes 150 5.4 Modeling Correlation 152 5.4.1 Correlated Brownian Motion 152 5.4.2 Copulas for Correlation 153 5.5 MATLAB Tools for Modeling Risk Evolving over Time 156 5.6 Summary 156 5.7 Questions and Exercises 157 482 Bibliography [44] Y.M Kaniovski and G.Ch Pflug Risk assessment for credit portfolios: A coupled Markov chain model Journal of Banking & Finance, 31:2303– 2323, 2007 [45] I Karatzas and S.E Shreve Brownian Motion and Stochastic Calculus (Graduate Texts in Mathematics) Springer, 1991 [46] S Karlin and H.M Taylor A First Course in Stochastic Processes, Second Edition Academic Press, 1975 [47] S Karlin and H.M Taylor A Second Course in Stochastic Processes Academic Press, 1981 [48] P.E Kloeden and E Platen Numerical Solution of Stochastic Differential Equations Springer, 2000 [49] F.H Knight Risk, Uncertainty, and Profit The Riverside Press, 1921 [50] D Knuth The Art of Computer Programming, Vol Addison-Wesley, 2000 [51] A.J Koning and L Peng Goodness-of-fit tests for a heavy tailed distribution Journal of Statistical Planning and Inference, 138(12):3960–3981, 2008 [52] H Konno Portfolio optimization of small scale fund using mean-absolute deviation model International 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and S Priaulet Fixed-Income Securities: Valuation, Risk Management and Portfolio Strategies The Wiley Finance Series, 2003 [60] A Mas-Colell, M.D Winston, and J.R Green Microeconomic Theory Oxford University Press, New York, 1995 [61] MATLAB MATLAB Documentation MathWorks Inc., 2009 [62] A Melino and S.M Turnbull Pricing foreign currency options with stochastic volatility Journal of Econometrics, 45:239–265, 1990 [63] E.L Melnick and B.S Everitt Encyclopedia of Quantitative Risk Analysis and Assessment Wiley, 2008 [64] F Modigliani and M.H Miller The cost of capital, corporation finance, and the theory of investment American Economic Review, 48:261–297, 1958 [65] N Mora The cost of capital, corporation finance, and the theory of investment Economic Review - Federal Reserve Bank of Kansas City, 2, 2012 [66] D.R Nance, C.W Smith Jr., and C.W Smithson On the determinants of corporate hedging The Journal of Finance, 48(1):267–284, 1993 [67] B.L Nelson Stochastic Modeling, Analysis and 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W Schoutens Levy Processes in Finance: Pricing Financial Derivatives Wiley Series in Probability and Statistics, 2003 [78] E Schwartz The stochastic behavior of commodity prices: Implication for valuation and hedging Journal of Finance, 51(3):923–973, 1997 [79] S.G Sharma Over-the-counter derivatives: A new era of financial regulation Law and Business Review of the Americas, 17(2):279–315, 2011 [80] W.F Sharpe Mean-absolute-deviation characteristic lines for securities and portfolios Management Science, 18(2), 1971 [81] S.E Shreve Stochastic Calculus for Finance II: Continuous-Time Models Springer, 2004 [82] T.K Siu, H Yang, and J.W Lau Pricing currency options under twofactor Markov-modulated stochastic volatility models Insurance: Mathematics and Economics, 43(3):295–302, 2008 [83] C.W Smith and R.M Stulz The determinants of firms’ hedging policies The Journal of Financial and Quantitative Analysis, 20(4):391–405, 1985 [84] S Sundaresan Fixed Income Markets and Their Derivatives SouthWestern Pub, 1996 [85] M Tenenbaum and H Pollard Ordinary Differential Equations Dover, 2001 [86] L.C Thomas Consumer Credit Models, Pricing, Profit and Portfolios Oxford University Press, 2009 [87] L.C Thomas Consumer finance: Challenges for operational research Journal of the Operational Research Society, 61:41–52, 2010 [88] A Tversky and D Kahneman Loss aversion in riskless choice: A reference-dependent model Quarterly Journal of Economics, 106(4):1039–1061, 1991 [89] H.R Varian Microeconomic Theory W.W Norton and Company, Inc., New York, 1992 [90] P Wilmott Paul Wilmott on Quantitative Finance Wiley, January 2000 [91] G Wright Behavioral Decision Making Plenum Press, New York, N.Y., 1985 [92] G Zhou Beyond Black-Litterman: Letting the data speak Journal of Portfolio Management, 36(1):36–45, 2009 Index σ-algebra, 13 k-th order moments, 16 nth -to-default swap, 358 absolute error criterion, 182 absolute risk aversion, 42 acceptance-rejection method, 103 Accuracy Ratio (AR), 337 Adams-Bashford method, 187 Advanced Measurement Approach (AMA), 396 Almost Sure Convergence, 146 alternative risk transfer (ART), 374 Altman’s Z-Score model, 348 American option, 233 analytical solutions, 174 antithetic variates, 447 antitrust, 65 arbitrage, 215 asset liquidity risk, 11 asset-backed securities (ABSs), 358 asset-liability management, 381 assets reprice before liabilities, 386 at-the-money, 211 attachment point, 360 attributes, 333 axioms of choice, 40 balance sheet risks, 381 Bank for International Settlements (BIS), 79 Bank Holding Company Act of 1956, 69 barrier function, 459 barrier parameter, 460 Basel Committee on Banking Supervision, 35 basis risk, 293 Bayesian learning, 209 bear spread, 250 behavioral scoring, 331 behavioral utility function, 44 Bernoulli distribution, 23 bias, 120 bimodal distribution, 55 binomial distribution, 23 binomial tree, 220 biologically-inspired population-based optimization algorithms, 467 birth-death process, 140 Black-Litterman portfolio framework, 209 Black-Scholes partial differential equation (PDE), 216 Box-Muller method, 104 Brownian motion, 144 bull spread, 250 business disruption and system failures, 394 business intelligence and analytics, 379 business models, 378 butterfly spread, 251 canonical optimization problem, 456 cap, 291 capital budgeting, 374 caption, 291 Central Limit Theorem, 21 central moments, 16 CEV model, 237 characteristics, 333 485 486 Chi-square distribution, 28 classical mean-variance framework, 205 classical portfolio theory, 201 classification and regression tree (CART) analysis, 333 classification of pure risk, Clayton Antitrust Act in 1914, 72 coherent risk measure, 47 collateral, 360 Collateralized Bond Obligations (CBOs), 358 Collateralized Debt Obligations (CDOs), 358 Collateralized Loan Obligations (CLOs), 358 Collateralized Mortgage Obligations (CMOs), 358 commodity, 294 Commodity Futures Trading Commission (CFTC), 82 commodity swaps, 305 competition risk, 379 conditional expectation, 18 conditional probabilities, 17 Conditional Value-at-Risk (CVaR), 46, 284, 441 confidence interval, 118, 442 confidence level, 442 constrained optimization problem, 456 consumer lending, 327 contingent claims approach (CCA), 349 continuity, 458 continuous random variable, 14 continuous-time Markov chain, 136 continuous-time stochastic processes, 136 control variate, 443 control variate estimator, 444 convenience yield, 305, 376 Convergence in Distribution, 146 Convergence in Mean, 146 Convergence in Probability, 146 Index convex optimization problem, 459 convex risk measure, 47 convexity, 267, 283 convexity hedge, 292 copula, 153, 362 corporate tax-structure, 373 correlated Wiener processes, 152 correlation, 17 cost of capital, 373 cost of carry, 304 cost to fix, 394 costs of bankruptcy, 374 counterparty risk, 325 coupon, 266 covariance, 16 covered call strategy, 249 credit default swaps (CDS), 355 credit derivatives, 351 credit rating system (CRS), 341 credit risk, credit risk analytics, 379 credit scorecards, 333 credit scoring, 332 credit scoring model, 333 Credit Value-at-Risk (Credit VaR), 344 CreditMetricsT M , 344 cross-hedge, 293 Cumulative Accuracy Profile (CAP), 337 cumulative density (distribution) (CDF) function, 15 cumulative gap, 384 cumulative gap analysis, 387 cumulative mass function, 14 cut-off score, 334 cyclicality, 297 damage to physical assets, 394 data analytics, 379 debt migration matrix, 342 deductible, 425 default correlation, 362 default hazard rate, 339 default intensity, 351 Index delta, 227 delta hedging, 256 delta-gamma hedging, 256 delta-gamma-vega hedging, 257 delta-vega hedging, 257 demand risk, 379 descriptive theory of choice, 43 detachment point, 360 differentiability, 458 diffusion coefficient, 142 diffusion process, 142 direction of descent, 463 discontinuous, 469 discrete random variable, 14 discrete-time risk model, 128 discriminant analysis (DA), 330 diversification, 201 Dodd-Frank Wall Street Reform and Consumer Protection Act of 2010, 70 downside risk, 56 drift, 142 duration, 267, 281 duration gap, 384 duration gap analysis, 388 DV01, 281 dynamic gap, 384 dynamic gap analysis, 388 dynamic investment strategies, 203 efficient frontier, 206 employment practices and workplace safety, 394 enterprise risk management (ERM), 406 equity risk, 200 Equivalent Martingale Measure (EMM), 219 Ergodicity, 132 error analysis, 180 estimator, 220, 444 Euler scheme, 175 events, 12 evolutionary strategy, 468 Exit Times, 151 487 exotic options, 229, 287, 305 exponential distribution, 25 exposure at default (EAD), 329 Exposure Indicator (EI), 396 external fraud, 394 factor models, 298 Fair Isaac Corp (FICO), 336 Federal Deposit Insurance Corporation (FDIC), 68, 77 Federal Reserve Act in 1913, 72 Federal Reserve System, 68, 74 Feynman-Kac Theorem, 217 FICO scores, 336 financial crisis of 2008, 381 Financial Industry Regulatory Authority (FINRA), 83 first-to-default swap, 358 fixed income instruments, 266, 277 fixed-to-fixed currency swap, 311 floor, 291 floortion, 291 foreign exchange risk, 10 fortuitous losses, 410 forward curve, 269 forward rates, 269 forward spread, 357 frequency of loss, 395 fundamental analysis, 199 fundamental risk, 409 funding liquidity risk, 11 future net cash flow, 373 gamma, 227 gamma distribution, 29 gap analysis, 385 gap convexity analysis, 388 GARCH(p,q) models, 244 Gaussian distribution, 19 Gaussian process, 142 genetic algorithm, 468 geometric Brownian motion, 174 geometric random walk, 136 Girsanov Theorem, 219 Glass-Steagall Act of 1933, 69 488 global discretization error, 181 gradient, 463 Gramm-Leach-Bliley Act of 1999, 70 Greeks, 226 hard commodities, 294 hazard, 409 health risk, 426 heavy-tailed or fat-tailed distribution, 16 hedged portfolio, 292 hedging strategies, 442 Hessian, 463 heuristic strategies, 464 hitting times, 151 idiosyncratic risk, immunization, 286 implied volatility, 224 importance sampling, 455 in-the-money, 211 indemnification, 410 independence, 17 individual and group health insurance, 415 informational asymmetry, 377 insurability of pure risk, 414 insurer, 407 intensity matrix, 137 interest rate risk, internal fraud, 393 International Association of Insurance Supervisors (IAIS), 74 international Fisher effect, 306, 309 inverse transform method, 101 irrelevance of risk management, 372 Issue-Specific Credit Ratings, 342 Issuer Credit Ratings, 341 Ito calculus, 171 Ito formula, 171 Ito integral, 167, 168 Ito isometry, 169 joint centrality measure, 16 Index joint cumulative distribution function, 16 jump-diffusion process, 245 Karush-Kuhn-Tucker (KKT) conditions, 460 Kolmogorov-Smirnov (KS) test, 57 kurtosis, 20 Lagged Fibonacci Generator, 97 Lagrangian function, 460 Latin hypercube method, 454 legal risk, 393 level of loss, 395 liabilities repricing before assets, 386 liability risk, 408 life insurance, 414 Linear Congruential Generator (LCG), 97 Lines of Business (LoB), 396 liquidity contingency plan, 381 liquidity management, 381 liquidity risk, 380 local discretization error, 181 local optima, 467 logistic regression, 333 lognormal distribution, 27 Long-Term Asset/Liability Ratio, 384 Long-term Value at Risk (LT-VaR), 389 loss frequency, 419 loss given default (LGD), 329 loss given events (LGE), 397 loss prevention, 425 loss severity, 419 marginal distribution function, 17 market price of risk, 243, 274 market risk, market risk analytics, 379 Markov chain, 129, 342 Markov process, 141 Markovian property, 133, 136 martingale, 147 maximum likelihood method, 189 Index Maximum Possible Loss, 420 Maximum Probable Loss, 420 mean, 15 mean error, 184 mean-absolute deviation, 55 mean-reversion, 241, 271, 297, 300 method of moments, 192 minimum variance hedge, 303 minimum variance unbiased estimator, 119 mitigate, 51 model, 12, 94 model risk, 393 modified duration, 267, 282 modified trapezoidal method, 186 Modigliani and Miller propositions, 48, 376 moment generating function, 149 moneyness, 224 moral hazard, 409 mortality index, 432 mortality risk, 432 mortgage-backed securities (MBSs), 358 multi-modal, 469 multi-name credit default swap, 358 National Association of Insurance Commissioners (NAIC), 73 negative gap, 386 neighborhood size, 466 net interest income, 385 non-differentiable, 469 non-insurance transfers, 418 non-life insurance, 414 non-linear equality constraints, 456 non-linear inequality constraints, 456 nonseasonal factor, 300 normal distribution, 19 normative theory of choice, 43 objective probability, objective risk, 408 obligors, 340 Office of Comptroller of the Currency (OCC), 68, 76 489 on-peak demand, 300 operational control risk, 393 operational loss model, 395 operational risk, 418 operational Value-at-Risk, 396 optimal hedge framework, 303 optimal hedge ratio, 254 Ornstein-Uhlenbeck process, 143 out-of-the-money, 211 parallelization, 469 particle swarm optimization algorithm, 470 particular risk, 409 path-dependent options, 446 path-wise approximation, 181 penalty function, 459 penalty parameter, 459 people risk, 393 perils, 409 Personal risks, 407 plain-vanilla European option, 210, 446 Poisson distribution, 23 Poisson process, 138 pooling of losses, 410 portfolio optimization problems, 441 positive definite, 464 positive gap, 386 price of the bond, 267 principal, 266 Principal Component Analysis, 292 probability density function, 15 probability measure, 13 probability of default, 329 probability space, 13 process risk, 393 properties for the probability measure, 14 property and casualty insurance, 415 protective put strategy, 247 pseudo-random numbers, 96 purchasing power parity, 309 pure risk, 6, 406 put-call parity, 216 490 quadratic programming problem, 206 quadratic variation, 149 quality of collateral, 361 Quasi-Maximum Likelihood method, 191 quasi-Newton’s method, 464 random number generator, 97 random numbers, 96 random search algorithm, 465 random variable, 12 random walk, 133 rate of convergence, 180 real options, 375 real-world measure, 272 realization, 128 recovery rate, 329 regime-switching, 298 regulation, 65 regulatory structure, 66 relative risk aversion, 42 repo, 278 resolution expense, 394 response surface, 469 retail banking, 327 reverse-repo, 278 reward-risk measures, 202 Riegle-Neal Interstate Banking and Branching Efficiency Act of 1994, 70 Riemann integral, 165 risk, risk aversion, 41 risk identification, 420 risk management framework, 39, 50 risk management process, 37 risk measure, 45 risk neutral, 41 risk preference, 41 risk-neutral measure, 273 risk-neutral probabilities, 223 Runge-Kutta method, 187 sample path, 128 sample space, 12 Index sampling distribution, 120 scatter search, 467 scenario analysis, 58, 316, 390 seasonal factor, 300 seasonality, 297 Secretary of the Treasury, 67 Securities Act of 1933, 72 Securities and Exchange Commission (SEC), 72, 81 securitization, 351, 432 security holder welfare, 373 seniority structure, 359 Sherman Antitrust Act of 1890, 72, 73 short rate model, 270 simulated annealing, 465 simulation, 93 simulation-based optimization, 209 single factor interest models, 274 single-name CDS, 355 skewness, 16 soft commodities, 294 Special Purpose Entity (SPE), 360 Special Purpose Vehicle (SPV), 360 speculative risk, spread option, 357 standard deviation, 15 standard normal distribution, 19 standard symmetric random walk, 133 static gap, 383 stationary probability vector, 132 stationary process, 343 statistical classification methods, 330 steepest descent or gradient method, 463 stochastic calculus, 168 stochastic differential equation, 167 stochastic optimization problem, 456 stochastic process, 127 stochastic volatility model, 241 stop loss strategy, 254 straddle, 251 strangle, 251 stratified sampling, 450 Index Stratonovich integral, 167 stratum, 450 stress testing, 59, 316, 363, 390 stressed VaR, 316 STRIPs, 278 strong law of large numbers, 411 structural gap, 382 structure a simulation study, 94 structured liabilities or debt, 377 subjective probabilities, subjective risk, 408 submartingale, 148 subprime crisis of 2007, 73 supermartingale, 148 systems and technology risk, 393 tabu search, 466 tails of a distribution, 56 taxonomy of risk, 11 technical analysis, 200 term structure of interest rates, 268 theta, 227 time-discretization, 175 trajectory, 128 tranche definitions, 361 tranches, 359 transaction risk, 393 transfer, 53 transition probabilities, 131 transition probability density, 141 trapezoidal method, 186 uncertainty, unconstrained optimization problem, 456 unexpected losses, 398 uniform distribution, 20 unimodal distribution, 55 upper attachment point, 360 utility function, 41 value of the firm, 373 Value-at-Risk (VaR), 46, 284, 313, 441 vanilla currency swap, 311 variance, 15 491 variance of an estimator, 119 variance reduction, 442 vega, 227 volatility smile, 224 volatility smirk, 240 waterfall of cashflows, 359 weak law of large numbers, 410 weather derivatives, 301, 431 Weibull distribution, 26 Wiener process, 144 write-downs, 394 yield curve, 267 yield spread, 357, 361 yield to maturity (YTM), 268 [...]... of 2008 and euro crisis evolving through 2011-2012 (a) Profit and individual positions of a protective put (b) Profit and individual positions of a reverse protective put (a) Profit and individual positions of a covered call (b) Profit and individual positions of a reverse covered call (a) Profit and individual positions of a bull spread using call options (b) Profit and individual... is over the millennia, and especially in the past few centuries, we have created an increasingly complex and interdependent habitat Man no longer fends for himself and his small family unit by growing food and building shelter Instead we depend on people and their efforts across the globe for every single and simplest of our needs Therefore the happenings of the future at far and wide locations of the... combination of the individual stock helps mitigate the risk in the frontier Plot of mean and standard deviation of two stock returns The correlation of ρ = 1 and −1 define the right and left extents of the region, respectively Plot of mean and standard deviation space spanned by return on portfolio of stocks For a choice of expected portfolio return threshold,... manage the risks Impact and importance of risks and their management are significant to all - individuals, households, financial firms, non-financial corporations, business and commercial enterprises, not-for-profit enterprises, and governments Although the regulatory thrust and a large segment of risks, specifically pure risks, either require or maintain a defensive stance, an all-inclusive and a robust response... services industry In summary, this book is a unique and wonderful combination of risk management and financial simulation concepts that will set the reader on a strong footing for a rewarding career in risk management Drawing on her experience in teaching risk management and computational technique courses to undergraduate and graduate students for a decade and a half, the author offers an accessible book... individuals, firms, governments, and society Therefore, mastering risk management skills, with a comprehensive understanding of various types of risk, and the modeling and quantitative techniques for each of the risk types, as well as an awareness of present day challenges, are a must for any finance professional I wish you well in your efforts of learning and training using this book, and hope it opens doors... St =$35, σ = 23%, T − t = 1/2 year, and short-term interest rate of r = 2% The chart marks the dependence of European and American vanilla call and put option prices on parameters that determine the price Trajectories for valuation of a compound option Pay-offs of an up -and- out barrier call option and a down -and- in barrier call option ... and individual positions of a bear spread using put options (a) Profit and individual positions of a butterfly spread (a) Profit and individual positions of a straddle (b) Profit and individual positions of a strangle (a) Profit and individual positions of a strip (b) Profit and individual positions of a strap The points of time along the life of an... against them, and to limit losses, financial or otherwise, resulting from decisions with adverse consequences Our understanding of risk has been a slow process that picked up speed in the last few centuries, and rapidly accelerated in the last few decades Today, we understand the factors behind any adverse event with a lot more clarity than ever before We have developed analytical tools and risk management... to effectively address, and limit the losses of, any unfavorable outcome, from the minor and routine to the most devastating catastrophe in our realm of possibilities Of all the areas of human endeavor, risk is of paramount consequence to modern finance In today’s financial markets and financial institutions, risks and their prudent management have never been more important Risks and their management are