The aim of this article is to provide a systematic analysis of the conditions such that Fourier transform valuation formulas are valid in a general framework; i.e. when the option has an arbitrary payoff function and depends on the path of the asset price process. An interplay between the conditions on the payoff function and the process arises naturally. We also extend these results to the multi-dimensional case and discuss the calculation of Greeks by Fourier transform methods. As an application, we price options on the minimum of two assets in Levy and stochastic volatility models.Option valuation, Fourier transform, semimartingales, Levy processes, stochastic volatility models, options on several assets,
This paper is devoted to obtaining a wellposedness result for multidimensional BSDEs with possibly unbounded random time horizon and driven by a general martingale in a filtration only assumed to satisfy the usual hypotheses, i.e. the filtration may be stochastically discontinuous. We show that for stochastic Lipschitz generators and unbounded, possibly infinite, time horizon, these equations admit a unique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven by discrete-time approximations of general martingales.2010 Mathematics Subject Classification. 60G48, 60G55, 60G57, 60H05. Key words and phrases. BSDEs, processes with jumps, stochastically discontinuous martingales, random time horizon, stochastic Lipschitz generator. We thank Martin Schweizer, two anonymous referees and the associate editor for their comments that have resulted in a significant improvement of the manuscript. Alexandros Saplaouras gratefully acknowledges the financial support from the DFG Research Training Group 1845 "Stochastic Analysis with Applications in Biology, Finance and Physics". Dylan Possamaï gratefully acknowledges the financial support of the ANR project PACMAN, ANR-16-CE05-0027. Moreover, all authors gratefully acknowledge the financial support from the PROCOPE project "Financial markets in transition: mathematical models and challenges". 1 The authors are indebted to Saïd Hamadène for pointing out this reference. The published version of [51] states that the article was received on October 27, 1971. It is also present in the bibliography of [21], though it is never referred to in the text. 2 We emphasize that the references given below are just the tip of the iceberg, though most of them are, in our view, among the major ones of the field. Nonetheless, we do not make any claim about comprehensiveness of the following list. 1 2 A. PAPAPANTOLEON, D. POSSAMAÏ, AND A. SAPLAOURAS mention El Karoui and Huang [60], Bender and Kohlmann [18], Wang, Ran and Chen [138] as well as Briand and Confortola [27]. The first results going beyond the linear growth assumption in z, which assumed quadratic growth, were obtained independently by Kobylanski [97, 98, 99] and Dermoune, Hamadène and Ouknine [56], for bounded ξ and f Lipschitz in y. These results were then further studied by Eddhabi and Ouknine [59], and improved by Lepeltier and San Martín [107, 108], Briand, Lepeltier and San Martín [35] and revisited by Briand and Élie [31], but still for bounded ξ. Wellposedness in the quadratic case when ξ has sufficiently large exponential moments was then investigated by Briand and Hu [33, 34], followed by Delbaen, Hu and Richou [54, 55], Essaky and Hassani [66], and Briand and Richou [36]. A specific quadratic setting with only square integrable terminal conditions has been considered recently by Bahlali, Eddahbi and Ouknine [6, 7], while a result with logarithmic growth was also obtained by Bahlali and El Asri [8], and Bahlali, Kebiri, Khelfallah and Moussaoui [13]. The ca...
We provide a general and flexible approach to LIBOR modeling based on the class of affine factor processes. Our approach respects the basic economic requirement that LIBOR rates are nonnegative, and the basic requirement from mathematical finance that LIBOR rates are analytically tractable martingales with respect to their own forward measure. Additionally, and most importantly, our approach also leads to analytically tractable expressions of multi-LIBOR payoffs. This approach unifies therefore the advantages of well-known forward price models with those of classical LIBOR rate models. Several examples are added and prototypical volatility smiles are shown. We believe that the CIR process-based LIBOR model might be of particular interest for applications, since closed form valuation formulas for caps and swaptions are derived. KEY WORDS: LIBOR rate models, forward price models, affine processes, analytically tractable models. 627 628 M. KELLER-RESSEL, A. PAPAPANTOLEON, AND J. TEICHMANN THE AFFINE LIBOR MODELS 629 AXIOMSLet us denote by L(t, T) the time-t forward LIBOR rate that is settled at time T and received at time T + δ; here T denotes some finite time horizon. The LIBOR rate is related to the prices of zero coupon bonds, denoted by B(t, T), and the forward price, denoted by F(t, T, T + δ), by the following equations:One postulates that the LIBOR rate should satisfy the following axioms, motivated by economic theory, arbitrage pricing theory, and applications.AXIOM 2.1. The LIBOR rate should be nonnegative, i.e., L(t, T) ≥ 0 for all 0 ≤ t ≤ T.AXIOM 2.2. The LIBOR rate process should be a martingale under the corresponding forward measure, i.e., L(·, T) ∈ M(P T+δ ). AXIOM 2.3. The LIBOR rate process, i.e., the (multivariate) collection of all LIBOR rates, should be analytically tractable with respect to as many forward measures as possible. Minimally, closed-form or semi-analytic valuation formulas should be available for the most liquid interest rate derivatives, i.e., caps and swaptions, so that the model can be calibrated to market data in reasonable time.Furthermore we wish to have rich structural properties: that is, the model should be able to reproduce the observed phenomena in interest rate markets, e.g., the shape of the implied volatility surface in cap markets or the implied correlation structure in swaption markets. EXISTING APPROACHESThere are several approaches to LIBOR modeling developed in the literature attempting to fulfill the axioms and practical requirements discussed in the previous section. We describe later the two main approaches, namely the LIBOR market models and the forward price model, and comment on their ability to fulfill them. We also briefly discuss Markov-functional models. forward LIBOR rate is modeled as an exponential Brownian motion under its corresponding forward measure. This model provides a theoretical justification for the common market practice of pricing caplets according to Black's futures formula (Black 1976), i.e., assuming that the forward LIBOR rate is log-no...
Abstract. We introduce a multiple curve framework that combines tractable dynamics and semi-analytic pricing formulas with positive interest rates and basis spreads. Negatives rates and positive spreads can also be accommodated in this framework. The dynamics of OIS and LIBOR rates are specified following the methodology of the affine LIBOR models and are driven by the wide and flexible class of affine processes. The affine property is preserved under forward measures, which allows us to derive Fourier pricing formulas for caps, swaptions and basis swaptions. A model specification with dependent LIBOR rates is developed, that allows for an efficient and accurate calibration to a system of caplet prices.
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