Drift-free simulation and libor market models

[Abstract] In this work, an efficient procedure to simulate the stochastic dynamics of Libor Market Model that avoids the use of the path dependent drifts in Monte Carlo simulation is proposed. For this purpose, we follow a Drift-Free Simulation methodology, by first simulating certain martingales and then obtaining the involved forward Libor rates in terms of them. More precisely, we propose a particular parameterization of those martingales so that the desired properties of the continuous models can be maintained after the discretization procedure when using either any intermediate forward probability measure or the spot one. Thus, the need of using the terminal probability measure to maintain the desired properties can be overcome. Some numerical results concerning caplets pricing illustrate that the proposed method outperforms other ones existing in the literature. We also explain how this methodology can be adapted to the case of Swap Market Model or any generic market model, and we extend it to the recently appeared multicurve setting. We also present how the proposed technique can be applied in the cross-markets context to price cross-currency, commodity or in ation derivatives, for example. Finally we place the here presented methodology into a graph theoretical framework.[Resumen] En este trabajo presentamos un procedimiento eficaz para simular las dinámicas estoc ásticas del Modelo de Mercado del Libor, procedimiento que evita el uso de los términos de deriva en la simulación de Monte Carlo. Para este propósito seguimos una metodología de Simulación Sin Derivas simulando en primer lugar ciertas martingalas y obteniendo después los tipos implícitos forward Libor en términos de ellas. En concreto, proponemos una parametrización particular de estas martingalas de modo que las propiedades que posee el modelo continuo se mantengan tras el procedimiento de discretización, tanto bajo cualquier medida de probabilidad forward intermedia como bajo la medida de probabilidad spot. De este modo, se supera la necesidad de usar la medida de probabilidad terminal para mantener esas propiedades deseables. Algunos resultados numéricos relativos a la valoración de caplets ilustran que el método propuesto supera a otros existentes en la literatura. Explicamos también cómo esta metodología puede ser adaptada para el Modelo de Mercado del Swap o cualquier modelo de mercado genérico, y la extendemos para el caso multicurva. Exponemos también cómo la técnica propuesta puede ser aplicada en el contexto de dos economías para valorar derivados tanto de dos monedas, como de ciertas mercancías, como de in- ación, entre otros. Finalmente enmarcamos toda la metodología presentada durante el trabajo dentro del ámbito de la teoría de grafos.[Resumo] Neste traballo presentamos un procedemento eficaz para simular as dinámicas estocásticas do Modelo de Mercado do Libor, procedemento que evita o uso dos termos de deriva na simulación de Monte Carlo. Para este propósito seguimos unha metodoloxía de Simulación Sen Derivas simulando nun primer lugar certas martingalas e obtendo despois os tipos implícitos forward Libor en termos delas. En concreto, propo~nemos unha parametrización particular destas martingalas de xeito que as propiedades que posee o modelo continuo sexan mantidas tras o procedemento de discretizaci ón, tanto baixo calquera medida de probabilidade forward intermedia como baixo a medida de probabilidade spot. De este xeito, se supera a necesidade de facer uso da medida de probabilidade terminal para manter esas propiedades desexables. Alguns resultados numéricos relativos á valoración de caplets ilustran que o método proposto supera a outros xa existentes na literatura. Explicamos tamén cómo esta metodoloxía pode ser adaptada para o Modelo de Mercado do Swap ou para calquera modelo de mercado xenérico, e exténdese para o caso multicurva. Amosamos tamén cómo a técnica proposta pode ser aplicada no contexto de dúas economías para valorar derivados tanto de dúas moedas, como de certas mercancías, como de in ación, entre outros. Finalmente enmarcamos toda a metodoloxía presentada durante o traballo dentro do campo da teoría de grafos

Global Optimization for Automatic Model Points Selection in Life Insurance Portfolios

[Abstract] Starting from an original portfolio of life insurance policies, in this article we propose a methodology to select model points portfolios that reproduce the original one, preserving its market risk under a certain measure. In order to achieve this goal, we first define an appropriate risk functional that measures the market risk associated to the interest rates evolution. Although other alternative interest rate models could be considered, we have chosen the LIBOR (London Interbank Offered Rate) market model. Once we have selected the proper risk functional, the problem of finding the model points of the replicating portfolio is formulated as a problem of minimizing the distance between the original and the target model points portfolios, under the measure given by the proposed risk functional. In this way, a high-dimensional global optimization problem arises and a suitable hybrid global optimization algorithm is proposed for the efficient solution of this problem. Some examples illustrate the performance of a parallel multi-CPU implementation for the evaluation of the risk functional, as well as the efficiency of the hybrid Basin Hopping optimization algorithm to obtain the model points portfolio.This research has been partially funded by EU H2020 MSCA-ITN-EID-2014 (WAKEUPCALL Grant Agreement 643045), Spanish MINECO (Grant MTM2016-76497-R) and by Galician Government with the grant ED431C2018/033, both including FEDER financial support. A.F., J.G. and C.V. also acknowledge the support received from the Centro de Investigación de Galicia “CITIC”, funded by Xunta de Galicia and the European Union (European Regional Development Fund- Galicia 2014-2020 Program), by grant ED431G 2019/01Xunta de Galicia; ED431C2018/03Xunta de Galicia; ED431G 2019/0

Pricing Inflation and Interest Rates Derivatives with Macroeconomic Foundations

I develop a model to price inflation and interest rates derivatives using continuous-time dynamics linked to monetary macroeconomic models: in this approach the reaction function of the central bank, the bond market liquidity, and expectations play an important role. The model explains the effects of non-standard monetary policies (like quantitative easing or its tapering) on derivatives pricing. A first adaptation of the discrete-time macroeconomic DSGE model is proposed, and some changes are made to use it for pricing: this is respectful of the original model, but it soon becomes clear that moving to continuous time brings significant benefits. The continuous-time model is built with no-arbitrage assumptions and economic hypotheses that are inspired by the DSGE model. Interestingly, in the proposed model the short rates dynamics follow a time-varying Hull-White model, which simplifies the calibration. This result is significant from a theoretical perspective as it links the new theory proposed to a well-established model. Further, I obtain closed forms for zero-coupon and year-on-year inflation payoffs. The calibration process is fully separable, which means that it is carried out in many simple steps that do not require intensive computation. The advantages of this approach become apparent when doing risk analysis on inflation derivatives: because the model explicitly takes into account economic variables, a trader can assess the impact of a change in central bank policy on a complex book of fixed income instruments, which is not straightforward when using standard models. The analytical tractability of the model makes it a candidate to tackle more complex problems, like inflation skew and counterparty/funding valuation adjustments (known by practitioners as XVA): both problems are interesting from a theoretical and an applied point of view, and, given their computational complexity, benefit from a tractable model. In both cases the results are promising.Open Acces

Calibration and Simulation of the Gaussian Two-Additive-Factor Interest Rate Model

Calibrazione e simulazione del modello G2++ con alcuni esempi di prezzaggi

Structural pricing of XVA metrics for energy commodities OTC trades

The aim of the present Chapter is to improve of the structural rst-passage framework built in Chapter 1 along several directions as well as test its robustness. Since typically commodity trades are not clearable under Central Clearing Counterparts (CCPs), it is worthy to assess the eect of bilateral Collateral Support Annex (CSA) agreements on CVA/DVA metrics. Moreover I introduce within my CCR modelling, the impact of state-dependent stochastic recovery rates. Furthermore, in order to stress-test my framework, I investigate the eects on CCR measures of multiplicative shocks to the two major drivers in the game: credit and volatility. Finally I propose an alternative balance-sheet calibration based on hybrid market/accounting data which is well suited in the commodity context in the light of small and medium size of corporations usually operating in the EU commodity derivatives market for risk-management purposes.The global nancial crisis revealed that no economic entity can be considered default-free any more. Because of that, both banks and corporations have to deal with bilateral Counterparty Credit Risk (CCR) in their OTC derivatives trades. Such evidence implies the fair pricing of these risks, namely the Credit Valuation Adjustment (CVA) and its counterpart, the Debt Valuation Adjustment (DVA). Despite the more commonly used reduced-form approach, in this work the random default time is addressed via a structural approach a la Black and Cox (1976), so that the bankruptcy of a given rm is modelled as the rst-passage time of its equity value from a predetermined lower barrier. As in Ballotta et al. (2015), I make use of a time-changed Levy process as underlying source of both market and credit risk. The main advantage of this setup relies on its superior capability to replicate non null short-term default probabilities, unlike pure diusion models. Moreover, a numerical computation of the valuation adjustments for bilateral CCR in the context of energy commodities OTC derivatives contracts has been performed.The global nancial crisis revealed that no economic entity can be considered default-free any more, so that both banks and corporate rms have to cope with bilateral Counterparty Credit Risk (CCR) when negotiating OTC derivatives. Since the mainstream approach typically used in practical settings is to evaluate derivatives in terms of the cost of their respective hedging strategies, the pricing of CCR metrics implicitly relates to the way these strategies are nanced. Within the numerical section of the present work, the valuation adjustments for CCR have been computed. Moreover, the role played by funding costs and their impact in widening bid-ask spreads have been assessed. A similar reasoning has been applied for the investigation of the cost of funding Initial Margins (IM), typically eective on top of Variation Margins (VM) when trading under Central Clearing Counterparties (CCPs). As the Initial Margin Valuation Adjustment (MVA) is concerned, it is here showed that, dierently from what can happen for FVAs, no osetting eect can materialize. As a consequence, in aggregate terms IMs can cause systemic liquidity eects. The computed XVA metrics are relative to energy commodities OTC derivative trades

Hybrid multi-curve models with stochastic basis

The financial markets have changed radically since the start of the 2007 credit crisis. Following the bankruptcies of large financial institutions as well as bailouts of multiple banks and asset management institutions like Bear Sterns, Lehman Brothers, and AIG, the market participants recognised the serious credit and liquidity risks present in the widely traded interest rate derivatives. The effect of rising credit and liquidity risks was observed by the spike in the spreads between nearly risk-free OIS rates used for collateral and risky unsecured LIBOR loan rates. Most of the classical interest rate models used by mentioned market participants relied on the assumption that there exists a risk-free and unique LIBOR lending rate, which is no longer true. This has opened new ground for complex, hybrid models for interest rate derivatives. This PhD thesis presents my work on developing novel interest rate models which are mathematically and historically sound and can be used for pricing interest rate derivatives including stochastic basis spreads between unsecured LIBOR and OIS rates. This work is split into two problems: first we analyse the discrepancies between forward-LIBOR lending rates and their classic replication strategy with spot-LIBOR rates. For this problem, we propose an extension of a known LIBOR Panel Model, which enables us to jointly model OIS and spot- and forward-LIBOR rates with an error within the quoted bid-ask spreads. The second part of this thesis looks into the problem of pricing non-linear derivatives like caps linked to rates on multiple LIBOR tenors. We propose a novel hybrid credit-interest rate model, which allows to jointly model OIS and multi-tenor LIBOR rates and to price multi-tenor caps. The proposed hybrid short-rate model is intuitive, semi-analytically tractable and can be calibrated using liquid, available market data. We compare the market data fit with a benchmark model using fixed LIBOR-OIS spread assumption. The last chapter shows the impact of this model on credit value adjustments for interest rate trades

Delayed Forward-Backward stochastic PDE's driven by non Gaussian Lévy noise with application in finance

From the very first results, the mathematical theory of financial markets has undergone several changes, mostly due to financial crises who forced the mathematical-economical community to change the basic assumptions on which the whole theory is founded. Consequently a new mathematical foundation were needed. In particular, the 2007/2008 credit crunch showed the word that a new financial theoretical framework was necessary, since several empirical evidences emerged that aspects that were neglected prior to these years were in fact fundamental if one has to deal with financial markets. The goal of the present thesis goes in this direction; we aim at developing rigorous mathematical instruments that allow to treat fundamental problems in modern financial mathematics. In order to do so, the talk is thus divided into three main parts, which focus on three different topics of modern financial mathematics. The first part is concerned with delay equations. In particular, we will prove Feynman-Kac type result for BSDE's with time-delayed generator, as well as an ad hoc Ito formula for delay equations with jumps. The second part deal with infinite dimensional analysis and network models, focusing in particular on existence and uniqueness results for infinite dimensional SPDE's on networks with general non-local boundary conditions. The last part treats the topic of rigorous asymptotic expansions, providing a small noise asymptotic expansion for SDE with Lévy noise with several concrete application to financial models