Real options ambiguity risk and insurance

REAL OPTIONS, AMBIGUITY, RISK AND INSURANCE Studies in Probability, Optimization and Statistics Volume Published previously in this series Vol Vol Vol Vol H.K Koo (Ed.), New Trends in Financial Engineering – Works Under the Auspices of the World Class University Program of Ajou University A Bensoussan, Dynamic Programming and Inventory Control J.W Cohen, Analysis of Random Walks R Syski, Passage Times for Markov Chains ISSN 0928-3986 (print) ISSN 2211-4602 (online) R Options Real O s, Amb biguity y, Risk k and Insuraance Wo orld Class University U Program in n Financiaal Engineerring, versity, Vo olume Two o Ajou Univ Edited by y Alain n Bensou ussan Ashbel Smith S Chair Professor P Naveen Jindal School of Management, thee University of Texas at D Dallas P of Risk R and Deccision Analyssis, City Univversity of Hoong Kong Chair Professor WCU U Distinguish hed Professor, Ajou Univversity S Shige Pen ng Professsor of Matheematics Distinguisheed Professor of the Ministry of Educa D ation of China a School of Mathematics, M , Shandong University, U J Jinan, China and Jaeeyoung Su ung Disting guished Proffessor of Fin nance, Deparrtment of Fin nancial Enginneering School of Business Administratiion, Ajou Un niversity S Suwon, Koreea Amstterdam • Berrlin • Tokyo ã Washington, DC â 2013 The authors and IOS Press All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without prior written permission from the publisher ISBN 978-1-61499-237-0 (print) ISBN 978-1-61499-238-7 (online) Library of Congress Control Number: 2013935544 Publisher IOS Press BV Nieuwe Hemweg 6B 1013 BG Amsterdam Netherlands fax: +31 20 687 0019 e-mail: order@iospress.nl Distributor in the USA and Canada IOS Press, Inc 4502 Rachael Manor Drive Fairfax, VA 22032 USA fax: +1 703 323 3668 e-mail: iosbooks@iospress.com LEGAL NOTICE The publisher is not responsible for the use which might be made of the following information PRINTED IN THE NETHERLANDS Real Options, Ambiguity, Risk and Insurance A Bensoussan et al (Eds.) IOS Press, 2013 © 2013 The authors and IOS Press All rights reserved v Preface Alain BENSOUSSAN, Shige PENG and Jaeyoung SUNG This book is the second volume in the WCU financial engineering series by the financial engineering program of Ajou University, supported by the Korean Government under the world-class-university (WCU) project grant Ajou University is the unique recipient of the grant in Korea to establish a world class university in financial engineering The main objective of the series is to disseminate, faster than textbooks, recent developments of important issues in financial engineering to graduate students and researchers, providing surveys or pedagogical expositions of published important papers in broad perspectives, or analyses of recent important financial news on financial-engineering research, practices or regulations The first volume was published by the IOS press in 2011 under the title of “New Trends in Financial Engineering”, containing articles to introduce recent topics in financial engineering, contributed by WCU-project participants This volume focuses on important topics in financial engineering such as ambiguity, real options, and credit risk and insurance, and has 12 chapters organized in three parts These chapters are contributed by globally recognized active researchers in mathematical finance mostly outside the WCU-project participants Part I consists of five chapters Real options analysis addresses the issue of investment decisions in complex, innovative, risky projects This approach extends considerably the traditional NPV approach, much too limited to deal with the complexity of real situations In preparing the investment decision, a project manager should determine which project to choose, when to choose it, and in what scale He/She should incorporate flexibility in order to benefit from acquiring later on important information about all aspects of uncertainties related to the investment Consequently, during the project life, the manager still faces further decisions on how to manage, contract, expand, or abandon and to meet industrial competition, not to mention performing basic managerial functions and making financial decisions Towards the end of the project, the manager faces closure decisions such as sale, reorganization or liquidation Flexibility is not the unique characteristics of real options One additional idea is to take advantage of valuation techniques developed in context of financial products, in order to define properly the value of industrial projects This is more and more possible in the context of commodities with an organized market The energy sector is an important example An important difference between real and financial options concerns the issue of competition For complex investment projects, there are generally few possible players For financial products, the number of players is very large and therefore each of them does not change dramati1 Graduate Department of Financial Engineering, Ajou University The research culminated in this book was supported by WCU(World Class University) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R31-2009-000-20007-0) vi cally the context (it may be possible of course) The decision making with competition introduces challenging problems Villeneuve and Décamps examine the optimal investment policy for a cashconstrained firm which has no access to external financing, and show that an increase in the volatility of the underlying asset can actually decrease the value of the growth option value Huisman, Kort and Plasmans apply the real option theory to analyze a real life case, and show that negative NPV projects are optimally undertaken (when discount rates are high and technology progresses fast) in the hope of new opportunities or growth options for the firm Thijssen enriches real options analysis by introducing industrial competition into standard real option problems and argues competition can be bad for welfare in a dynamic setting Hugonnier and Morellec consider a real options problem for a risk averse decision maker with undiversifiable risks and show that the risk aversion can make him/her delay investment, reducing the (market) value of the project Finally, Bensoussan and Chevalier-Roignant consider capital budgeting decisions on not only timing but also scale of a project and show how optimal trigger policy integrates the two aspects Part II has three chapters on ambiguity We believe that the notion of ambiguity is one of major breakthroughs in the expected utility theory Ambiguity arises as uncertainties cannot be precisely described in the probability space The objective is to understand rational decision making behaviors of an economic agent when his decision making environment is subject to ambiguity Mathematics underlying those economics problems can be very challenging, imposing great obstacles to the economic analysis of the problems Chen, Tian and Zhao survey recent developments on problems of optimal stopping under ambiguity, and develop the theory of optimal stopping under ambiguity in a general framework Ji and Wei review the principal-agent literature in continuous time, and apply to the optimal insurance design problem in the presence of ambiguity Shige Peng provides a survey of recent significant and systematic progress in the area of G-expectations: new central limit theorems under sublinear expectations, Brownian motions under ambiguity (G-Brownian motions), its related stochastic calculus of Itô’s types and some typical pricing models He further shows that prices of contingent claims in the world of ambiguity can be expressed as g-expectation (nonlinear expectation) of future claims, and that the method of the nonlinear expectation turns out to be powerful in characterizing these prices in general In Part III, four chapters are devoted to risk and insurance In particular, this part covers mutual insurance for non-traded risks, downside risk management, and credit risk in fixed income markets Liu, Taksar and Yuan introduce mutual insurance which can be viewed as a mutual reserve system for homogeneous mutual members, such as P&I Clubs in marine mutual insurance and Federal Reserve reserve banks in the U.S., and explain why many mutual insurance companies, which were once quite popular in the financial markets, are either disappeared or converted to non-mutual ones The importance of downside risk minimization has attracted lots of attention from both practitioners and academics in light of recent experience of the Subprime Mortgage Crisis Nagai discusses the large deviation estimates of the probability of falling below a given target growth rate for controlled semi-martingales, in relation to certain ergodic risk-sensitive stochastic control problems in the risk averse case Portfolio insurance techniques are related to the downside risk minimization problem Sekine reviews several dynamic portfolio insurance techniques such as generalized CPPI (Constant Propor- vii tion Portfolio Insurance) methods, American OBPI (Option-Based Portfolio Insurance) method, and DFP (Dynamic Fund Protection) method, and applies these techniques to solve the long-term risk-sensitive growth rate maximization problem subject to the floor constraint or the generalized drawdown constraint Credit risk is also an important topic for both practitioners and academics, being particularly important to the determination of subprime mortgage rates Ahn and Sung provide a pedagogical review of literature focusing on determinants of credit risk spreads with emphasis on methodological aspects of structural models This broad spectrum of concepts and methods shows the richness of the domain of mathematical/engineering finance We hope this volume will be useful to both graduate students and researchers in understanding relatively new areas in economics and finance and challenging aspects of mathematics In this manner, we think contributing to the expectations of the WCU project viii Acknowledgement This book was supported by WCU(World Class University) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R31-2009-000-20007-0) We, together with the other contributing authors, are grateful for support from the staff of the IOS press, especially, Maureen Twaig, and Kim Willems and for assistance of Xiaoyan Chen, Sanghyun Cho and Gang Geun Lee ix Contents Preface Alain Bensoussan, Shige Peng and Jaeyoung Sung v Part Real Options Optimal Investment Under Liquidity Constraints Jean-Paul Décamps and Stéphane Villeneuve Introduction Optimal Investment in Perfect Capital Markets 2.1 The Benchmark Model 2.2 Discussion Optimal Stopping for a Cash-Constrained Firm 3.1 The Model 3.2 Value of the Firm with No Growth Option 3.3 Value of the Firm with a Growth Option 3.3.1 A Verification Theorem 3.3.2 Solution to Optimal Stopping Problem φ 3.3.3 φ as a Super Solution to HJB Equation (3.13) Future Works Investment in High-Tech Industries: An Example from the LCD Industry Kuno J.M Huisman, Peter M Kort and Joseph E.J Plasmans Introduction The Investment Model with Geometric Brownian Motion Investment in LCD Industry 3.1 Industry 3.2 Production Process 3.3 Data and Estimations Industry Analysis Conclusion A Appendix A.1 Proof of Proposition A.2 Proof of Proposition Game Theoretic Real Options and Competition Risk Jacco J.J Thijssen Introduction General Set-Up of a Real Option Duopoly Preemption Games Markov Perfect Equilibrium Firm Value and Welfare Implications Conclusion 3 4 7 10 11 17 18 20 20 22 24 24 25 25 27 29 29 29 31 33 33 35 37 38 43 49 268 C Ahn and J Sung / Credit Risk Models: A Review Q sup (σZs + νs) ≤ x = Q 0≤s≤T =N sup (Zs + νσ −1 s) ≤ xσ −1 0≤s≤T x − νT √ σ T − e2νσ −2 x −x − νT √ σ T N Since the process −Zt is also a standard Brownian motion under Q, it holds that for any x < 0, Q inf (σZs + νs) ≥ x = Q 0≤s≤T =N sup (σZs − νs) ≤ −x 0≤s≤T −x + νT √ σ T − e2νσ −2 x N x + νT √ σ T Since F (T ) is derived as Q(γ ≤ T ), F (T ) = Q(γ ≤ T ) = Q(γ < T ) = − Q(γ ≥ T ) =1−Q =1−N =N inf Ys ≥ 0≤s≤T y0 + νT √ σ T −y0 − νT √ σ T =1−Q −2 + e−2νσ + e−2νσ −2 y0 y0 N N inf Xs ≥ −y0 0≤s≤T −y0 + νT √ σ T −y0 + νT √ σ T where y0 = ln(V /VB ) and ν = r − δ − 12 σ Integrating the first term of Eq.(11) by parts and computing yields d(V ; VB , T ) = c(T ) c(T ) + e−rT p(T ) − [1 − F (T )] r r + ρ(T )VB − c(T ) G(T ), r where F (T ) = N G(T ) = −b − aσ T √ σ T V VB −a+z N and where r − δ − σ /2 , σ2 V b = ln , VB a= z= (aσ )2 + 2rσ σ2 + V VB −2a −b − zσ T √ σ T N + −b + aσ T √ , σ T V VB −a−z N −b + zσ T √ , σ T 269 C Ahn and J Sung / Credit Risk Models: A Review and N (·) is the cumulative standard normal distribution Defining x = a + z, note that as T → ∞, V VB c(∞) 1− d(V ; VB , T ) → r −x + ρ(∞)VB V VB −x which is the same equation as Leland (1994) derived for infinite-horizon risky debt when ρ(∞) = (1 − α), where α is the fraction of asset value lost in bankruptcy, c(∞) is the coupon paid by the infinite-maturity bond, and (V /VB )−x represents the present value of one dollar contingent on future bankruptcy Consider an environment where the firm continuously sells a constant (principal) amount of new debt with maturity of T years from issuance, which (if solvent) it will redeem at par upon maturity New bond principal is issued at a rate p = (P/T ) per year, where P is the total principal value of all outstanding bonds The same amount of principal will be retired when the previously-issued bonds mature As long as the firm remains solvent, at any time s the total outstanding debt principal will be P , and have a uniform distribution of principal over maturities in the interval (s, s+T ) Without loss of generality, we define the current time s = Bonds with principal p pay a constant coupon rate c = (C/T ) per year, implying the total coupon paid by all outstanding bonds is C per year Total debt service payments are therefore time-independent and equal to (C + P/T ) per year Later we shall show that this environment is consistent with a constant VB Now we assume this to be the case Let D(V ; VB , T ) denote the total value of debt, when debt of maturity T is issued The fraction of firm asset value lost in bankruptcy is α The remaining value (1 − α)VB is distributed to bond holders so that the sum of all fractional claims ρ(t) for debt of all maturities outstanding equals (1 − α) For simplicity we will assume that ρ(t) = ρ/T per year for all t This in turn implies ρ = (1 − α) We can now determine the value of all outstanding bonds: T D(V ; VB , T ) = d(V ; VB , t)dt t=0 = C C + P− r r − e−rT C − I(T ) + (1 − α)VB − rT r J(T ) where I(T ) = (G(T ) − e−rT F (T )) rT J(T ) = − + √ V VB √ V VB zσ T zσ T −a+z N −b − zσ t −b − zσ t √ √ σ t σ t N −b + zσ t −b + zσ t √ √ σ t σ t −a−z The total market value of the firm, v, equals the asset value plus the value of tax benefits, less the value of bankruptcy costs, over the infinite horizon Tax benefits accrue at rate τ C per year as long as V > VB , where τ is the corporate tax rate Following Leland (1994), the total firm value is given by 270 C Ahn and J Sung / Credit Risk Models: A Review v(V ; VB ) = V + τC 1− r V VB −x − αVB V VB −x (14) The value of equity is given by E(V ; VB , T ) = v(V ; VB ) − D(V ; VB , T ) To determine the equilibrium default boundary VB endogenously, one can use the smooth-pasting condition that has the property of maximizing (with respect to VB ) the value of the firm, subject to the limited liability of equity (E(V ) ≥ for all V ≥ VB ), which also implies EV V (VB ) = ∂ E(VB )/∂V ≥ VB solves the equation ∂E(V ; VB , T ) |V =VB = ∂V We can solve equation for VB : VB = (C/r)(A/(rT ) − B) − AP/(rT ) − τ Cx/r + αx − (1 − α)B (15) where √ A = 2ae−rT N (aσ T ) √ √ 2e−rT − √ n(zσ T ) + √ n(aσ T ) + (z − a) σ T σ T √ √ 2 B = − 2z + N (zσ T ) − √ n(zσ T ) + (z − a) + zσ T zσ T σ T and n(·) denotes the standard normal density function When T → ∞, it can be shown that VB → (1 − τ )(Cx/r)/(1 + x), as in Leland (1994) Leland and Toft (1996) utilize Eq.(15) and Eq.(14) to make a number of interesting claims based on their numerical examples As is the case with the optimal capital structure decision discussed in Leland (1994), the optimal maturity decision is also a tradeoff between tax benefits, bankruptcy costs and agency costs Short term debt reduces agency costs but it also reduces tax benefits Thus riskier firms issue shorter-term debt in addition to using less debt Firms with high bankruptcy costs prefer long term debt Firms with high growth opportunities use shorter term debt, as tax benefits of long term debt can be reduced The authors further argue that credit spreads increase with maturities up to 20 years at the optimal leverage ratio Like the other structural models that we have thus far discussed, credit spreads in Leland and Toft also fundamentally depend on the loss distribution of debt as seen in the valuation Eq.(11) Unlike others, Leland and Toft explain how the loss distribution of debt can be related to important corporate financial decisions C Ahn and J Sung / Credit Risk Models: A Review 271 2.4 Collin-Dufresne and Goldstein model The use of debt in the capital structure of the firm affects the riskiness of the whole firm, as the probability of bankruptcy increases In spite of elegant results proposed by Leland (1994) and Leland and Toft (1996), incorporating optimal capital structure decisions into more realistic credit risk models still remains to be an interesting issue Collin-Dufresne and Goldstein (2001) extend Longstaff and Schwartz’s (1995) model by allowing leverage ratios to to be dynamically adjusted, and argue that this additional feature can result in increases in credit spreads for risky debt Moreover, unlike other structural models, default boundaries in Collin-Dufresne and Goldstein are stochastic Assume the firm-value process follows a geometric Brownian motion under the risk neutral measure: dVt = (r − δ)Vt dt + σVt dzt where r is the risk-free rate, δ is the payout ratio, and σ is the volatility It is convenient to define y = log V Then we have dyt = r−δ− σ2 dt + σdzt The authors also assume that the dynamics of the log of default threshold, kt , can be expressed as dkt = λ(yt − ν − kt )dt When kt is less than (yt − ν), the firm acts to increase kt , and vice-versa This model captures that firms tend to issue debt when their leverage ratio falls below some target, and are more hesitant to replace maturing debt when their leverage ratio is above that target Define the log-leverage lt = kt −yt From Itˆo’s lemma, lt follows the one-factor Markov dynamics: dlt = dkt − dyt = λ(¯l − lt )dt − σdzt , where ¯l ≡ −r + δ + σ /2 − ν λ Define τ¯ as the random time at which l(t) reaches zero for the first time, triggering default Assume that a risky discount bond with maturity T receives one dollar at T if τ > T , or (1 − ω) at time T if τ ≤ T The price of this risky discount bond can be written as P T (l0 ) = e−rT E I{¯τ >T } + (1 − ω)I{¯τ ≤T } = e−rT [1 − ωQ(l0 , T )] , 272 C Ahn and J Sung / Credit Risk Models: A Review where E denotes risk-neutral conditional expectation at time and Q(l0 , T ) is the risk-neutral probability that default occurs before time T given that the leverage ratio is l0 at time Using the integral equation, it provides an implicit formula for the first hitting-time density: Πf (T |l0 , 0) = T g(lt = 0, t|l0 , 0)Πf (T |lt = 0, t)dt, where g(lt = 0, t|l0 , 0) is the probability density that the first hitting time is at time t, and Πf (T |ls , s) is the date-s conditional probability that lT > To obtain an explicit function for the first passage density, this model discretize time into n equal intervals and numerically find to approximate the density function Therefore, hitting-time density is estimated by j Q(l0 , tj ) = j = 2, 3, · · · , n qi , i=1 q1 = qi = N (a1 ) N (b(1/2) ) ⎡ ⎣N (ai ) − N (b(1/2) ) i−1 ⎤ qj N (bi−j+1/2 )⎦ i = 2, 3, · · · , n j=1 where tj = jT /n ≡ jΔt for j ∈ {1, 2, · · · , n} and N (·) denotes the cumulative standard normal distribution function and = M (iΔt) S(iΔt) bi = L(iΔt) S(iΔt) M (t) = l0 e−λt + ¯l(1 − e−λt ) L(t) = ¯l(1 − e−λt ) S (t) = σ2 (1 − e−2λt ) 2λ Collin-Dufresne and Goldstein generalize their two-factor model by allowing interest rates rt to be stochastic as in the Vasicek model: drt = κ(θ − rt )dt + ηdzt where dzt dzt = ρdt The generalized version assumes dynamics of the log-default threshold as follows: for some φ ≥ 0, dkt = λ[yt − ν − φ(rt − θ) − kt ]dt Defining as before the log-leverage ratio lt = kt − Yt , and applying Itˆo’s lemma, we obtain 273 C Ahn and J Sung / Credit Risk Models: A Review dlt = λ(¯l(rt ) − lt )dt − σdzt , where ¯l(r) ≡ δ + σ /2 − ν + φθ − r λ +φ λ They assume that a risky discount bond pays − ω of the principal at the maturity if default occurs prior to maturity We define QT (r0 , l0 , T ) ≡ E0T [I{¯τ T } = EtQ exp − T T (r(u) + λ(u))du t where I is the indicator function Intuitively, exp − t λ(u)du can be viewed as the risk-neutral survival probability of the bond The reduced-form approach can also incorporate the notion of the recovery rate that is the payoff to debt holders in case of default of the firm For instance, the price of a defaultable bond with maturity T and recovery rate δt is given by 276 C Ahn and J Sung / Credit Risk Models: A Review d(t, T ) = EtQ = EtQ I{τ >T } + I{τ ≤T } δτ exp − T exp − T r(u)du t λ(u)du (1 − δτ ) + δτ t T exp − r(u)du t The recovery rate is modeled using various stochastic processes See Bakshi, Madan and Zhang (2001) The reduced-form models can be useful for computing values of credit derivatives, particularly when corresponding structural models are too complex to deal with economics of default timings and payoffs Various aspects of reduced-form approach are discussed in Duffie and Singleton(1999), Lando(1998) and Jarrow and Turnbull(1995) Empirical analysis-Structural models Huang and Huang (2003) calibrate structural models such as Merton (1974), Longstaff and Schwartz (1995), Leland and Toft (1996), and Collin-Dufresne and Goldstein (2001) according to historical default and recovery rates, and find that all generate credit spreads which are well below historical levels Eom, Helwege and Huang (2004) empirically compare abilities of corporate bond pricing models to predict corporate bond spreads under similar assumptions, using a sample of noncallable bonds belonging to firms with simple capital structures between 1986 and 1997 The authors find that Merton (1974), Longstaff and Schwartz (1995), Leland and Toft (1996), and Collin-Dufresne and Goldstein (2001) have substantial spread prediction errors, but their errors differ sharply in both sign and magnitude The two-factor models of Longstaff and Schwartz (1995), and Collin-Dufresne and Goldstein (2001) incorporate stochastic interest rates and a correlation between firm value and interest rates Eom at al find that the correlation is not very important empirically Stochastic interest rates raise the average predicted spreads, but the results are rather sensitive to the interest rate volatility estimates from the Vasicek model The Collin-Dufresne and Goldstein (2001) model might alleviate the problem of excessive dispersion in predicted spreads if the most underprediction occurs among firms with leverage ratios below their targets and the overpredicted spreads belong to bonds with unusually high leverage ratios The Collin-Dufresne and Goldstein’s (2001) model helps somewhat in this regard, but it tends toward overestimation of credit risk overall Conclusion We have reviewed methodological aspects of five of representative structural models in the literature on credit risk: Merton (1974), Longstaff and Schwartz (1995), Leland and Toft (1996), Collin-Dufresne and Goldstein (2001), and Chen, CollinDufresne and Goldstein (2009) The first four models try to explain credit spreads C Ahn and J Sung / Credit Risk Models: A Review 277 of corporate bonds based on the distribution of losses in case of default Regarding the loss distribution, Merton assumes default can only occur at maturity, Longstaff and Schwartz allow default to occur before maturity and interest rates to be stochastic, Leland and Toft consider effects of strategic default decisions, and Collin-Dufresne and Goldstein extend Longstaff and Schwartz with stochastic default boundaries Chen, Collin-Dufresne and Goldstein point out that credit spread can be affected not only by the loss distribution, but by correlation of loss payoffs with the market, and argue that credit risk spreads predicted by their model are consistent with historical observations The availability of good structural models is important not only for academics but practitioners Because of their inability to explain observed credit spreads in real life, structural models have not been fully utilized in pricing and managing credit-risk instruments in real life There is no doubt that the ultimate success of a structural model is judged by its practical applicability It would be interesting to implement recently successful credit-risk models to real world problems, and to assess their performance in managing credit risks References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] Amato, J.D and E.M Remolona (2003) The Credit Spread Puzzle, BIS Quarterly Review December, 51-63 Bakshi, F.,D Madan, and F Zhang (2001) Recovery in Default Risk Modeling: Theoretical Foundations and Empirical Applications, Working paper, Unversity of Maryland Black, F., and M Scholes (1973) The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81, 637654 Collin-Dufresne, P., and R Goldstein (2001) Do Credit Spreads Reflect Stationary Leverage Ratios?, Journal of Finance 56, 1929-1957 Collin-Dufresne, P., R Goldstein, and J Martin (2001) The Determinants of Credit Spread Changes, Journal of Finance 56, 2177-2208 Chen, L., Collin-Dufresne, P and R Goldstein (2008) On the Relation Between the Credit Spread Puzzle and the Equity Premium Puzzle, The Review of Financial Studies volume 22 number 2009, 3367-3409 Duffie, D., and K Singleton (1999) Modeling the Term Structure of Defaultable Bonds, Review of Financial Studies 12, 687-720 Elton, E J., M J Gruber, D Agrawal and C Mann (2001) Explaining the rate spread on corporate bonds, Journal of Finance vol LVI, no 1, February, pp 247-277 Eom, Y., J Helwege, and J Huang (2004) Structural Models of Corporate Bond Pricing, Review of Financial Studies 17, 499-544 Fortet, Robert (1943) Les fonctions al´ eatoires du type de Markoff associ´ ees a ` certaines e´quations line´ aires aux d´ eriv´ ees partielles du type parabolique”, Journal de Math´ ematiques Pures et Appliqu´ ees 22, 177-243 Helwege, J., and Christopher M Turner(1999), The slope of the 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(1995) A Simple Approach to Valuing Risky Fixed and Floating Rate Debt, Journal of Finance 50, 789-820 Malitz, I (2000) The Modern Role of Bond Covenants, The Research Foundation of the Institute of Chartered Financial Analysts Merton, R (1974) On the Pricing of Corporate Debt: The Risk Structure of Interest Rates, Journal of Finance 29, 449-470 Goldstein, R.(2010) Can Structural Models of Default Explain the Credit Spread Puzzle?, FRBSF Economic Letters February 22, 2010 Schultz, P (2001) Corporate Bond Trading Costs: A Peek Behind the Curtain, Journal of Finance vol LVI, no 2, April, 677-698 Vasicek, O (1977) An Equilibrium Characterization of the term structure, Journal of Financial Economics 5, 177-188 Zhang, B.Y., H Zhou and H Zhu (2009) Explaining Credit Default Swap Spreads with the Equity Volatility and Jump Risks of Individual Firms, Review of Financial Studies 5099-5131 Zhou, C (2001) The Term Structure of Credit Spreads with Jump Risk, Journal of Banking and Finance 25, 2015 - 2040 Real Options, Ambiguity, Risk and Insurance A Bensoussan et al (Eds.) IOS Press, 2013 © 2013 The authors and IOS Press All rights reserved 279 Author Index Ahn, C Bensoussan, A Chen, Z Chevalier-Roignant, B Décamps, J.-P Hugonnier, J Huisman, K.J.M Ji, S Kort, P.M Liu, J Morellec, E Nagai, H 255 v, 66 97 66 52 20 126 20 187 52 208 Peng, S Plasmans, J.E.J Sekine, J Sung, J Taksar, M Thijssen, J.J.J Tian, W Villeneuve, S Wei, Q Yuan, J Zhao, G v, 144 20 232 v, 255 187 33 97 126 187 97 This page intentionally left blank This page intentionally left blank This page intentionally left blank ... 35 37 38 43 49 x Real Options and Risk Aversion Julien Hugonnier and Erwan Morellec Introduction Model and Assumptions Real Options and Investment Timing 3.1 The Benchmark Case: Risk Neutrality... NETHERLANDS Real Options, Ambiguity, Risk and Insurance A Bensoussan et al (Eds.) IOS Press, 2013 © 2013 The authors and IOS Press All rights reserved v Preface Alain BENSOUSSAN, Shige PENG and. .. intentionally left blank Part Real Options This page intentionally left blank Real Options, Ambiguity, Risk and Insurance A Bensoussan et al (Eds.) IOS Press, 2013 © 2013 The authors and IOS Press All rights