On the valuation of compositions in Lévy term structure models

We derive explicit valuation formulae for an exotic path-dependent interest rate derivative, namely an option on the composition of LIBOR rates. The formulae are based on Fourier transform methods for option pricing. We consider two models for the evolution of interest rates: an HJM-type forward rate model and a LIBOR-type forward price model. Both models are driven by a time-inhomogeneous Lévy process.     

Time-changed Lévy LIBOR market model: Pricing and joint estimation of the cap surface and swaption cube

We propose a novel time-changed Lévy LIBOR (London Interbank Offered Rate) market model for jointly pricing of caps and swaptions. The time changes are split into three components. The first component allows matching the volatility term structure, the second generates stochastic volatility, and the third accommodates for stochastic skew. The parsimonious model is flexible enough to accommodate the behavior of both caps and swaptions. For the joint estimation we use a comprehensive data set spanning the financial crisis of 2007–2010. We find that, even during this period, neither market is as fragmented as suggested by the previous literature.     

Portfolio optimisation with jumps: Illustration with a pension accumulation scheme

In this paper, we address portfolio optimisation when stock prices follow general Lévy processes in the context of a pension accumulation scheme. The optimal portfolio weights are obtained in quasi-closed form and the optimal consumption in closed form. To solve the optimisation problem, we show how to switch back and forth between the stochastic differential and standard exponentials of the Lévy processes. We apply this procedure to both the Variance Gamma process and a Lévy process whose arrival rate of jumps exponentially decreases with size. We show through a numerical example that when jumps, and therefore asymmetry and leptokurtosis, are suitably taken into account, then the optimal portfolio share of the risky asset is around half that obtained in the Gaussian framework.     

A Lévy HJM multiple-curve model with application to CVA computation

We consider the problem of valuation of interest rate derivatives in the post-crisis set-up. We develop a multiple-curve model, set in the HJM framework and driven by a Lévy process. We proceed with joint calibration to OTM swaptions and co-terminal ATM swaptions of different tenors, the calibration to OTM swaptions guaranteeing that the model correctly captures volatility smile effects and the calibration to co-terminal ATM swaptions ensuring an appropriate term structure of the volatility in the model. To account for counterparty risk and funding issues, we use the calibrated multiple-curve model as an underlying model for CVA computation. We follow a reduced-form methodology through which the problem of pricing the counterparty risk and funding costs can be reduced to a pre-default Markovian BSDE, or an equivalent semi-linear PDE. As an illustration, we study the case of a basis swap and a related swaption, for which we compute the counterparty risk and funding adjustments.     

Exponential moments for HJM models with jumps

General HJM models driven by a Lévy process are considered. Necessary moment conditions for the discounted bond prices to be local martingales are derived. Under these moment conditions, it is proved that the discounted bond prices are local martingales if and only if a generalized HJM condition holds. Research supported in part by Polish KBN Grant P03A 034 29 “Stochastic evolution equations driven by Lévy noise”.     

Multivariate models for operational risk

Böcker and Klüppelberg [Risk Mag., 2005, December, 90–93] presented a simple approximation of OpVaR of a single operational risk cell. The present paper derives approximations of similar quality and simplicity for the multivariate problem. Our approach is based on the modelling of the dependence structure of different cells via the new concept of a Lévy copula.     

What is the Natural Scale for a Lévy Process in Modelling Term Structure of Interest Rates?

This paper gives examples of explicit arbitrage-free term structure models with Lévy jumps via the state price density approach. By generalizing quadratic Gaussian models, it is found that the probability density function of a Lévy process is a “natural” scale for the process to be the state variable of a market.      

Monte Carlo Option Pricing for Tempered Stable (CGMY) Processes

Lévy processes are popular models for stock price behavior since they allow to take into account jump risk and reproduce the implied volatility smile. In this paper, we focus on the tempered stable (also known as CGMY) processes, which form a flexible 6-parameter family of Lévy processes with infinite jump intensity. It is shown that under an appropriate equivalent probability measure a tempered stable process becomes a stable process whose increments can be simulated exactly. This provides a fast Monte Carlo algorithm for computing the expectation of any functional of tempered stable process. We use our method to price European options and compare the results to a recent approximate simulation method for tempered stable process by Madan and Yor (CGMY and Meixner Subordinators are absolutely continuous with respect to one sided stable subordinators, 2005).     

A multiple-curve Lévy forward rate model in a two-price economy

An advanced Heath–Jarrow–Morton forward rate model driven by time-inhomogeneous Lévy processes is presented which is able to handle the recent development to multiple curves and negative interest rates. It is also able to exploit bid and ask price data. In this approach in order to model spreads between curves for different tenors, credit as well as liquidity risk is taken into account. Deterministic conditions are derived to ensure the positivity of spreads and thus the monotonicity of the curves for the various tenors. Valuation formulas for standard interest rate derivatives such as caps, floors, swaptions and digital options are established. These formulas can be evaluated numerically very fast using Fourier-based valuation methods. In order to exploit bid and ask prices we develop this approach in the context of a two-price economy. Explicit formulas for bid as well as ask prices of the derivatives are stated. A specific model framework based on normal inverse Gaussian and Gamma processes is proposed which allows for calibration to market data. Calibration results are presented based on multiple-curve bootstrapping and cap market quotes. We use data from September 2013 as well as September 2016. The latter is of particular interest since rates were deep in negative territory at that time.     

Calibration to American options: numerical investigation of the de-Americanization method

American options are the reference instruments for the model calibration of a large and important class of single stocks. For this task, a fast and accurate pricing algorithm is indispensable. The literature mainly discusses pricing methods for American options that are based on Monte Carlo, tree and partial differential equation methods. We present an alternative approach that has become popular under the name de-Americanization in the financial industry. The method is easy to implement and enjoys fast run-times (compared to a direct calibration to American options). Since it is based on ad hoc simplifications, however, theoretical results guaranteeing reliability are not available. To quantify the resulting methodological risk, we empirically test the performance of the de-Americanization method for calibration. We classify the scenarios in which de-Americanization performs very well. However, we also identify the cases where de-Americanization oversimplifies and can result in large errors.     

Simulating Ruin Probabilities for a Class of Semimartingales by Importance Sampling Methods

We consider the problem of finding the probability of ruin when the risk process is assumed to be a special semimartingale with absolutely continuous characteristics. We show how the generalized Girsanov theorem can be used in connection with Monte Carlo simulation to obtain estimates of the ruin probabilities. It is shown by both analytical and numerical examples that these methods can be significantly better than ordinary simulations provided the new measure is chosen with some care.     

First passage time for compound Poisson processes with diffusion: ruin theoretical and financial applications

In this paper, we propose to revisit Kendall’s identity (see, e.g. Kendall (1957)) related to the distribution of the first passage time for spectrally negative Lévy processes. We provide an alternative proof to Kendall’s identity for a given class of spectrally negative Lévy processes, namely compound Poisson processes with diffusion, through the application of Lagrange’s expansion theorem. This alternative proof naturally leads to an extension of this well-known identity by further examining the distribution of the number of jumps before the first passage time. In the process, we generalize some results of Gerber (1990 Gerber, H. U. (1990). When does the surplus reach a given target? Insurance: Mathematics and Economics 9, 115–119.  [Google Scholar]) to the class of compound Poisson processes perturbed by diffusion. We show that this main result is particularly relevant to further our understanding of some problems of interest in actuarial science. Among others, we propose to examine the finite-time ruin probability of a dual Poisson risk model with diffusion or equally the distribution of a busy period in a specific fluid flow model. In a second example, we make use of this result to price barrier options issued on an insurer’s stock price.     

Assessing Selectivity and Market Timing Performance of Mutual Funds for an Emerging Market: The Case of Turkey

This paper derives and analyzes the selectivity and market timing performance of the mutual funds for the Turkish economy for the financial crisis period by employing high-frequency data. The determinants of these derived abilities are investigated within a regression analysis. The results suggest weak evidence about selection ability and some evidence about superior market timing quality. They also indicate that management fees are negatively correlated with the ability measure, which is quite surprising. Experience emerges as an important factor, especially for market timing ability.     

Monetary policy and financial stability: What role for the futures market?

This paper examines interactions between monetary policy and financial stability. There is a general view that central banks smooth interest rate changes to enhance the stability of financial markets. But might this induce a moral hazard problem, and induce financial institutions to maintain riskier portfolios, the presence of which would further inhibit active monetary policy? Hedging activities of financial institutions, such as the use of interest rate futures and swap markets to reduce risk, should further protect markets against consequences of unforeseen interest rate changes. Thus, smoothing may be both unnecessary and undesirable. The paper shows by a theoretical argument that smoothing interest rates may lead to indeterminacy of the economy's rational expectations equilibrium. Nevertheless, our empirical analysis supports the view that the Federal Reserve smoothes interest rates and reacts to interest rate futures. We add new evidence on the importance for policy of alternative indicators of financial markets stress.