173 research outputs found
Essays in Quantitative Finance
This thesis contributes to the quantitative finance literature and consists of four research papers.Paper 1. This paper constructs a hybrid commodity interest rate market model with a stochastic local volatility function that allows the model to simultaneously fit the implied volatility of commodity and interest rate options. Because liquid market prices are only available for options on commodity futures (not forwards), a convexity correction formula is derived to account for the difference between forward and futures prices. A procedure for efficiently calibrating the model to interest rate and commodity volatility smiles is constructed. Finally, the model is fitted to an exogenously given cross-correlation structure between forward interest rates and commodity prices. When calibrating to options on forwards (rather than futures), the fitting of cross-correlation preserves the (separate) calibration in the two markets (interest rate and commodity options), whereas in the case of futures, a (rapidly converging) iterative fitting procedure is presented. The cross-correlation fitting is reduced to finding an optimal rotation of volatility vectors, which is shown to be an appropriately modified version of the “orthonormal Procrustes” problem. The calibration approach is demonstrated on market data for oil futures.Paper 2. This paper describes an efficient American Monte Carlo approach for pricing Bermudan swaptions in the LIBOR market model using the Stochastic Grid Bundling Method (SGBM) which is a regression-based Monte Carlo method in which the continuation value is projected onto a space in which the distribution is known. We demonstrate an algorithm to obtain accurate and tight lower–upper bound values without the need for the nested Monte Carlo simulations that are generally required for regression-based methods.Paper 3. The credit valuation adjustment (CVA) for over-the-counter derivatives are computed using the portfolio’s exposure over its lifetime. Usually, future exposure is approximated by Monte Carlo simulations. For derivatives that lack an analytical approximation for their mark-to-market (MtM) value, such as Bermudan swaptions, the standard practice is to use the regression functions from the least squares Monte Carlo method to approximate their simulated MtMs. However, such approximations have significant bias and noise, resulting in an inaccurate CVA charge. This paper extend the SGBM to efficiently compute expected exposure, potential future exposure, and CVA for Bermudan swaptions. A novel contribution of the paper is that it demonstrates how different measures, such as spot and terminal measures, can simultaneously be employed in the SGBM framework to significantly reduce the variance and bias.Paper 4. This paper presents an algorithm for simulation of options on Lévy driven assets. The simulation is performed on the inverse transition matrix of a discretised partial differential equation. We demonstrate how one can obtain accurate option prices and deltas on the variance gamma (VG) and CGMY model through finite element-based Monte Carlo simulations
Modelling energy spot prices by volatility modulated Levy-driven Volterra processes
This paper introduces the class of volatility modulated L\'{e}vy-driven
Volterra (VMLV) processes and their important subclass of L\'{e}vy
semistationary (LSS) processes as a new framework for modelling energy spot
prices. The main modelling idea consists of four principles: First,
deseasonalised spot prices can be modelled directly in stationarity. Second,
stochastic volatility is regarded as a key factor for modelling energy spot
prices. Third, the model allows for the possibility of jumps and extreme spikes
and, lastly, it features great flexibility in terms of modelling the
autocorrelation structure and the Samuelson effect. We provide a detailed
analysis of the probabilistic properties of VMLV processes and show how they
can capture many stylised facts of energy markets. Further, we derive forward
prices based on our new spot price models and discuss option pricing. An
empirical example based on electricity spot prices from the European Energy
Exchange confirms the practical relevance of our new modelling framework.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ476 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
The asymptotic behavior of the term structure of interest rates
In this dissertation we investigate long-term interest rates, i.e. interest rates with maturity
going to infinity, in the post-crisis interest rate market. Three different concepts of long-term
interest rates are considered for this purpose: the long-term yield, the long-term simple rate,
and the long-term swap rate. We analyze the properties as well as the interrelations of these
long-term interest rates. In particular, we study the asymptotic behavior of the term structure of
interest rates in some specific models. First, we compute the three long-term interest rates in the
HJM framework with different stochastic drivers, namely Brownian motions, LĂ©vy processes,
and affine processes on the state space of positive semidefinite symmetric matrices. The HJM
setting presents the advantage that the entire yield curve can be modeled directly. Furthermore,
by considering increasingly more general classes of drivers, we were able to take into account
the impact of different risk factors and their dependence structure on the long end of the yield
curve. Finally, we study the long-term interest rates and especially the long-term swap rate in
the Flesaker-Hughston model and the linear-rational methodology
Pricing Chinese rain: a multi-site multi-period equilibrium pricing model for rainfall derivatives
Many industries are exposed to weather risk which they can transfer on financial markets via weather derivatives. Equilibrium models based on partial market clearing became a useful tool for pricing such kind of financial instruments. In a multi-period equilibrium pricing model agents rebalance their portfolio of weather bonds and a risk free asset in each period such that they maximize the expected utility of their incomes constituted by possibly weather dependent profits and payoffs of portfolio positions. We extend the model to a multisite version and apply it to pricing rainfall derivatives for Chinese provinces. By simulating realistic market conditions with two agent types, farmers with profits highly exposed to weather risk and a financial investor diversifying her financial portfolio, we obtain equilibrium prices for weather derivatives on cumulative monthly rainfall. Dynamic portfolio optimization under market clearing and utility indifference of these representative agents determines equilibrium quantity and price for rainfall derivatives.rainfall derivatives, equilibrium pricing, space-time Markov model
Consistent Valuation Across Curves Using Pricing Kernels
The general problem of asset pricing when the discount rate differs from the
rate at which an asset's cash flows accrue is considered. A pricing kernel
framework is used to model an economy that is segmented into distinct markets,
each identified by a yield curve having its own market, credit and liquidity
risk characteristics. The proposed framework precludes arbitrage within each
market, while the definition of a curve-conversion factor process links all
markets in a consistent arbitrage-free manner. A pricing formula is then
derived, referred to as the across-curve pricing formula, which enables
consistent valuation and hedging of financial instruments across curves (and
markets). As a natural application, a consistent multi-curve framework is
formulated for emerging and developed inter-bank swap markets, which highlights
an important dual feature of the curve-conversion factor process. Given this
multi-curve framework, existing multi-curve approaches based on HJM and
rational pricing kernel models are recovered, reviewed and generalised, and
single-curve models extended. In another application, inflation-linked,
currency-based, and fixed-income hybrid securities are shown to be consistently
valued using the across-curve valuation method.Comment: 56 page
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