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In this chapter we discuss the definition, construction, interpolation and application of . We will discuss discount curves, a tool for the valuation of deterministic cash-flows and forward curves, a tool for the valuation of linear cash-flows of an index. A curve is mainly a tool to interpolate certain basic financial products (zero coupon bonds, FRAs) with respect to maturity date and fixing date, such that it can be used to value products, which can be represented as linear functions of possibly interpolated values of a discount or forward curve. For this, the chosen interpolation method and interpolation entity plays an important role. Distinguishing forward curves from discount curves (representing the collateralization of the forward) motivates an alternative interpolation method, namely interpolation of the forward value (the product of the forward and the discount factor). In addition, treating forward curves as native curves (instead of representing them by pseudo-discount curves) will avoid other problems, like that of overlapping instruments. Besides the interpolation, we discuss the calibration of the curves for which we give a generic object-oriented implementation in Fries (Curve calibration. Object-oriented reference implementation, 2010–2015, []). We give some numerical results, which have been obtained using this implementation and conclude with a remark on how to define term-structure models (analog to a LIBOR market model) based on the definition of the performance index of an accrual account associated with a discount curve.

We present a detailed analysis of interest rate derivatives valuation under credit risk and collateral modeling. We show how the credit and collateral extended valuation framework in Pallavicini et al. (2011) can be helpful in defining the key market rates underlying the multiple interest rate curves that characterize current interest rate markets. We introduce the collateralized valuation measures and formulate a consistent realistic dynamics for the rates emerging from our analysis. We point out limitations of multiple curve models with deterministic basis considering valuation of particularly sensitive products such as basis swaps.

The intensity of a default time is obtained by assuming that the default indicator process has an absolutely continuous compensator. Here we drop the assumption of absolute continuity with respect to the Lebesgue measure and only assume that the compensator is absolutely continuous with respect to a general -finite measure. This allows for example to incorporate the Merton-model in the generalized intensity-based framework. We propose a class of generalized Merton models and study absence of arbitrage by a suitable modification of the forward rate approach of Heath–Jarrow–Morton (1992). Finally, we study affine term structure models which fit in this class. They exhibit stochastic discontinuities in contrast to the affine models previously studied in the literature.

The purpose of this article is to give a closed Fourier-based valuation formula for a caplet in the framework of the Lévy forward process model which was introduced in Eberlein and Özkan, Financ. Stochast. 9:327-348, 2005, []. Afterwards, we compute Greeks by two approaches which come from totally different mathematical fields. The first is based on the integration-by-parts formula, which lies at the core of the application of the Malliavin calculus to finance. The second consists in using Fourier-based methods for pricing derivatives as exposed in Eberlein, Quantitative Energy Finance, 2014, []. We illustrate the results in the case where the jump part of the underlying model is driven by a time-inhomogeneous Gamma process and alternatively by a Variance Gamma process.

The paper considers a no-arbitrage setting for pricing and relative value analysis of risky sovereign bonds. The typical case of an emerging market country (EM) that has bonds outstanding both in foreign hard currency (Eurobonds) and local soft currency (treasuries) is inspected. The resulting two yield curves give rise to a credit and currency spread that need further elaboration. We discuss their proper measurement and also derive and analyze the necessary no-arbitrage conditions that must hold. Then we turn attention to the CDS-Bond basis in this multi-curve environment. For EM countries the concept shows certain specifics both in theoretical background and empirical performance. The paper further focuses on analyzing these peculiarities. If the proper measurement of the basis in the standard case of only hard currency debt being issued is still problematic, the situation is much more complicated in a multi-curve setting when a further contingent claim on the sovereign risk in the face of local currency debt curve appears. We investigate the issue and provide relevant theoretical and empirical input.

In this paper we employ a one-factor Lévy model to determine basket option prices. More precisely, basket option prices are determined by replacing the distribution of the real basket with an appropriate approximation. For the approximate basket we determine the underlying characteristic function and hence we can derive the related basket option prices by using the Carr–Madan formula. We consider a three-moments-matching method. Numerical examples illustrate the accuracy of our approximations; several Lévy models are calibrated to market data and basket option prices are determined. In the last part we show how our newly designed basket option pricing formula can be used to define implied Lévy correlation by matching model and market prices for basket options. Our main finding is that the implied Lévy correlation smile is flatter than its Gaussian counterpart. Furthermore, if (near) at-the-money option prices are used, the corresponding implied Gaussian correlation estimate is a good proxy for the implied Lévy correlation.

The asset management business is driven by fee structures. In the context of hedge funds, fees have usually been a hybrid combination of two different types, which has coined a well-known business term of “2 and 20”. As an attempt to provide better alignment with their investors, in a new context of low interest rates and lukewarm performance, a new type of fund fees has been introduced in the last few years that offers a more symmetric payment structure, which we will refer to as . In this framework, in return for receiving performance fees, the fund manager provides some downside protection against losses to the investors. We show that the position values of the investor and the hedge fund manager can be formulated as portfolios of options, and discuss issues regarding pricing and fairness of the fee rates, and incentives for both investors and hedge fund managers. In particular, we will be able to show that, from a present value perspective, these fee structures can be set up as being favorable either to the hedge fund manager or to the investor. The paper is based on an arbitrage-free pricing framework. However, if one is to take into account the value to the business that investor capital brings to a fund, which is not part of our framework, it is possible to create a situation where both investors as well as asset managers win.

Investing into a bond and at the same time buying CDS protection on the same bond is known as buying a basis package. Loosely speaking, if the bond pays more than the CDS protection costs, the position has an allegedly risk-free positive payoff known as “negative basis”. However, several different mathematical definitions of the negative basis are present in the literature. The present article introduces an innovative measurement, which is demonstrated to fit better into arbitrage pricing theory than existing approaches. This topic is not only interesting for negative basis investors. It also affects derivative pricing in general, since the negative basis might act as a liquidity spread that contributes as a net funding cost to the value of a transaction; see Morini and Parampolini (Risk, 58–63, 2011, []).

Contingent convertible bonds (CoCos) are new hybrid capital instruments that have a loss absorbing capacity which is enforced either automatically via the breaching of a particular CET1 level or via a regulatory trigger. The price performance of outstanding CoCos, after a new CoCo issue is announced by the same issuer, is investigated in this paper via two methods. The first method compares the returns of the outstanding CoCos after an announcement of a new issue with some overall CoCo indices. This method does not take into account idiosyncratic movements and basically compares with the general trend. A second model-based method compares the actual market performance of the outstanding CoCos with a theoretical model. The main conclusion of the investigation of 24 cases of new CoCo bond issues is a moderated negative effect on the outstanding CoCos.

In this work we explore the implications of cointegration in a system of commodity prices on the premiums of options written on various spreads on the futures prices of these commodities. We employ a parsimonious, yet comprehensive model for cointegration in a system of commodity prices. The model has an exponential affine structure and is flexible enough to allow for an arbitrary number of cointegration relationships. We conduct an extensive simulation study on pricing spread options. We argue that cointegration creates an upward sloping term structure of correlation, that in turn lowers the volatility of spreads and consequently the price of options on them.