Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

“Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.

SET-Valued Analysis

We present a pricing method based on Fourier-cosine expansions for early-exercise and discretely-monitored barrier options. The method works well for exponential Levy asset price models. The error convergence is exponential for processes characterized by very smooth ($${{\rm{C}}^{\infty}[a,b]\in\mathbb {R}}$$) transitional probability density functions. The computational complexity is O((M − 1)N log N) with N a (small) number of terms from the series expansion, and M, the number of early-exercise/monitoring dates. This paper is the follow-up of (Fang and Oosterlee in SIAM J Sci Comput 31(2):826–848, 2008) in which we presented the impressive performance of the Fourier-cosine series method for European options.

https://mpra.ub.uni-muenchen.de/9248/1/MPRA_paper_9248.pdf

Pricing early-exercise and discrete barrier options by fourier-cosine series expansions

Credit Risk: Pricing, Measurement, and Management, by Darrell Duffie and Kenneth J Singleton, 2003, Princeton, NJ: Princeton University Press Credit risk is the major challenge for risk managers and market regulators International regulation of banks' credit risk was put in place in 1988 and since that time there has been no consensus on how to improve that regulatory framework Part of the explanation resides in the complexity of this risk Banks, regulators, and central banks do not agree on how to measure credit risk and, more particularly, on how to compute the optimal capital that is necessary for protecting the different partners that share this risk For example, what proportion of yield spreads on corporate bonds is explained by credit risk? Is it 30 percent, 50 percent, or even 90 percent? Is the credit risk proportion of the observed spreads solely a function of variations in the default probability or is it also explained by variations in the recovery rate over time or across cycles? Are macroeconomic cycles themselves or default risk premia, market liquidity, and even market risk significant determinants of yield spreads? These questions are important because some models such as CreditMetrics use the entire yield spread to compute the capital for credit risk If credit risk explains only a small fraction of yield spreads, these models compute too much capital for regulation and even for credit risk management (Dionne et al, 2004 and references therein) Asking banks to keep too much capital in reserve to cover credit risk can be a source of market distortion in risk management behavior (Allen and Gale, 2003; Dionne and Harchaoui, 2003) For example, it may generate some asset substitution activities that increase the risky position of banks, in order to set the level of risk at its optimal rather than regulatory level All these issues arise in part because credit risk is not well understood So the book by Duffie and Singleton will be welcomed by the academics, regulators, and practitioners who consult it The book has 13 chapters, three appendices (two on affine processes), a comprehensive list of references, and an index (authors and subjects) It covers all subjects related to credit risk It is designed for three broad audiences: academics and graduate students; those involved in the measurement and control of financial risks; and those involved in trading and marketing products with significant credit risk The main focus is modeling credit risk: measuring portfolio credit risk and pricing different securities exposed to credit risk The focus on credit risk management is less important in the book The introduction (indeed the entire book) is very well written and presents the subjects treated with clarity Credit risk is distinguished from other sources of risk such as market risk, liquidity risk, operational risk, systemic risk, and regulatory and legal risk The distinctions take many dimensions such as time horizon, liquidity, the parties implicated, methodology, and information asymmetries However, the authors insist on the fact that this does not mean that all these different risks should be managed separately These different risks may be correlated over time, so integrated frameworks for measuring and pricing them are necessary, particularly for market, credit, and liquidity risks For example, factors underlying changes in credit risk are often correlated with those underlying market risk and changes in liquidity risk can be viewed as a component of market risk and may generate credit risk The last chapter proposes an original way of integrating credit and market risks in a portfolio model The introduction also provides an overview of the book The chapters are organized to highlight the major topics related to credit risk, such as "Definition and Management" (Chapter 2), "Default and Transition" (Chapters 3 and 4), "Valuation" (including valuation of credit derivatives, Chapters 5-9), "Default Correlation" and "Portfolio Valuation" (Chapters 10 and 11), "Credit Risk in OTC Derivatives Positions" and "Portfolio Risk Measurement" (Chapters 12 and 13) …

Credit Risk: Pricing, Measurement, and Management

This paper presents a novel method to price discretely monitored single- and double-barrier options in Levy process-based models. The method involves a sequential evaluation of Hilbert transforms of the product of the Fourier transform of the value function at the previous barrier monitoring date and the characteristic function of the (Esscher transformed) Levy process. A discrete approximation with exponentially decaying errors is developed based on the Whittaker cardinal series (Sinc expansion) in Hardy spaces of functions analytic in a strip. An efficient computational algorithm is developed based on the fast Hilbert transform that, in turn, relies on the FFT-based Toeplitz matrix–vector multiplication. Our method also provides a natural framework for credit risk applications, where the firm value follows an exponential Levy process and default occurs at the first time the firm value is below the default barrier on one of a discrete set of monitoring dates.

https://papers.ssrn.com/sol3/Delivery.cfm/mafi_338.pdf?abstractid=1144911&mirid=1&type=2

Pricing discretely monitored barrier options and defaultable bonds in lévy process models: a fast hilbert transform approach

We propose a new computational method for the valuation of options in jump-diffusion models. The option value function for European and barrier options satisfies a partial integrodifferential equation (PIDE). This PIDE is commonly integrated in time by implicit-explicit (IMEX) time discretization schemes, where the differential (diffusion) term is treated implicitly, while the integral (jump) term is treated explicitly. In particular, the popular IMEX Euler scheme is first-order accurate in time. Second-order accuracy in time can be achieved by using the IMEX midpoint scheme. In contrast to the above approaches, we propose a new high-order time discretization scheme for the PIDE based on the extrapolation approach to the solution of ODEs that also treats the diffusion term implicitly and the jump term explicitly. The scheme is simple to implement, can be added to any PIDE solver based on the IMEX Euler scheme, and is remarkably fast and accurate. We demonstrate our approach on the examples of Merton's and Kou's jump-diffusion models, the diffusion-extended variance gamma model, as well as the two-dimensional Duffie-Pan-Singleton model with correlated and contemporaneous jumps in the stock price and its volatility. By way of example, pricing a one-year double-barrier option in Kou's jump-diffusion model, our scheme attains accuracy of 10-5 in 72 time steps (in 0.05 seconds). In contrast, it takes the first-order IMEX Euler scheme more than 1.3 million time steps (in 873 seconds) and the second-order IMEX midpoint scheme 768 time steps (in 0.49 seconds) to attain the same accuracy. Our scheme is also well suited for Bermudan options. Combining simplicity of implementation and remarkable gains in computational efficiency, we expect this method to be very attractive to financial engineering modelers.

http://users.iems.northwestern.edu/~linetsky/extrapolation.pdf

Pricing Options in Jump-Diffusion Models: An Extrapolation Approach

We present a fast and accurate method to compute exponential moments of the discretely observed maximum of a Levy process. The method involves a sequential evaluation of Hilbert transforms of expressions involving the characteristic function of the (Esscher-transformed) Levy process. It can be discretized with exponentially decaying errors of the form O(exp (−aM
 b
 )) for some a,b>0, where M is the number of discrete points used to compute the Hilbert transform. The discrete approximation can be efficiently implemented using the Toeplitz matrix–vector multiplication algorithm based on the fast Fourier transform, with total computational cost of O(NMlog (M)), where N is the number of observations of the maximum. The method is applied to the valuation of European-style discretely monitored floating strike, fixed strike, forward start and partial lookback options (both newly written and seasoned) in exponential Levy models.

Computing exponential moments of the discrete maximum of a Lévy process and lookback options

This paper presents a set of schemes for the fast and accurate inversion of analytic characteristic functions. The schemes are based on sinc expansion approximation of functions that are analytic in a horizontal strip in the complex plane. A function in this class can be reconstructed highly accurately from its values on a uniform grid along a horizontal line in the strip. The discretization error decays exponentially in terms of 1/h, where h is the step size of the grid. Consequently, transforms and integrals of such functions can be approximated using very simple schemes with remarkable accuracy. These schemes lead to high performance numerical inversion of analytic characteristic functions. Probability densities are approximated by evaluating the corresponding inverse Fourier transform integrals. The trapezoidal rule is highly accurate with exponentially decaying discretization errors. Cumulative distribution functions and expectations involving indicator functions can be represented by Hilbert transforms, which can again be evaluated highly accurately. Numerical results exhibit that the proposed schemes are fast, accurate, and robust to extreme inputs. The schemes we present can be used in statistics, applied probability, engineering, economics, and finance, where the inversion of analytic characteristic functions often arises.

Inverting Analytic Characteristic Functions and Financial Applications

This paper presents a Hilbert transform method for pricing Bermudan options in Levy process models. The corresponding optimal stopping problem can be solved using a backward induction, where a sequence of inverse Fourier and Hilbert transforms need to be evaluated. Using results from a sinc expansion based approximation theory for analytic functions, the inverse Fourier and Hilbert transforms can be approximated using very simple rules. The approximation errors decay exponentially with the number of terms used to evaluate the transforms for many popular Levy process models. The resulting discrete approximations can be efficiently implemented using the fast Fourier transform. The early exercise boundary is obtained at the same time as the price. Accurate American option prices can be obtained by using Richardson extrapolation.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID2306566_code827262.pdf?abstractid=2306566

Liming Feng

Papers

Pricing discretely monitored barrier options and defaultable bonds in lévy process models: a fast hilbert transform approach

Pricing Options in Jump-Diffusion Models: An Extrapolation Approach

Computing exponential moments of the discrete maximum of a Lévy process and lookback options

Inverting Analytic Characteristic Functions and Financial Applications

Pricing Bermudan Options in Lévy Process Models