Calibration, Simulation and Hedging in a Heston Libor Market Model with Stochastic Basis

LIBOR Market Model with Stochastic Volatility

The main objective of the present thesis is to provide efficient methods and tools for calibrating Libor Market Models (LMM) to market-prices of caps, swaptions and CMS spread options, and in this way to extract the volatility and correlation information implicitly contained in these products.

Pricing Path-Dependent Payoff Using Quadrature on Wiener Space

Solving SABR in Exact Form and Unifying it with LIBOR Market Model

On the Information in the Interest Rate Term Structure and Option Prices

In the more classical models for equities and interest rates evolution, constant volatility is usually assumed. However, in practice the volatilities are not constant in financial markets and different models allowing a varying local or stochastic volatility also appear in the literature. Particularly, here we consider the SABR model that has been first introduced in a paper by Hagan and coworkers, where an asymptotic closed-form formula for the implied volatility of European plain-vanilla options with short maturities is proposed. More recently, different works (Mercurio and Morini, Modeling Interest Rates: Advances in Derivatives Pricing, Risk Books 2009; Hagan and Lesniewski, LIBOR market model with SABR style stochastic volatility. Working Paper. http://lesniewski.us/papers/working/SABRLMM.pdf, 2008; Rebonato, A time-homogeneous SABR-consistent extension of the LMM. Risk, 2008) have extended the use of SABR model in the context of LIBOR market models for the evolution of forward rates (SABR-LMM). One drawback of these models in practice comes from the increase of computational cost, mainly due to the growth of model parameters to be calibrated. Additionally, sometimes either it is not always possible to compute an analytical approximation for the implied volatility or its expression results to be very complex, so that numerical methods (for example, Monte Carlo in the calibration process) have to be used. In this work we mainly review some recently proposed global optimization techniques based on Simulated Annealing (SA) algorithms and its implementation on Graphics Processing Units (GPUs) in order to highly speed up the calibration and pricing of different kinds of options and interest rate derivatives. Finally, we present some examples corresponding to real market data.KeywordsSABR volatility modelsSABR/LIBOR market modelsParallel simulated annealingGPUs

Neurons in the brain express intrinsic dynamic behavior which is known to be stochastic in nature. A crucial question in building models of neuronal excitability is how to be able to mimic the dynamic behavior of the biological counterpart accurately and how to perform simulations in the fastest possible way. The well-established Hodgkin-Huxley formalism has formed to a large extent the basis for building biophysically and anatomically detailed models of neurons. However, the deterministic Hodgkin-Huxley formalism does not take into account the stochastic behavior of voltage-dependent ion channels. Ion channel stochasticity is shown to be important in adjusting the transmembrane voltage dynamics at or close to the threshold of action potential firing, at the very least in small neurons. In order to achieve a better understanding of the dynamic behavior of a neuron, a new modeling and simulation approach based on stochastic differential equations and Brownian motion is developed. The basis of the work is a deterministic one-compartmental multi-conductance model of the cerebellar granule cell. This model includes six different types of voltage-dependent conductances described by Hodgkin-Huxley formalism and simple calcium dynamics. A new model for the granule cell is developed by incorporating stochasticity inherently present in the ion channel function into the gating variables of conductances. With the new stochastic model, the irregular electrophysiological activity of an in vitro granule cell is reproduced accurately, with the same parameter values for which the membrane potential of the original deterministic model exhibits regular behavior. The irregular electrophysiological activity includes experimentally observed random subthreshold oscillations, occasional spontaneous spikes, and clusters of action potentials. As a conclusion, the new stochastic differential equation model of the cerebellar granule cell excitability is found to expand the range of dynamics in comparison to the original deterministic model. Inclusion of stochastic elements in the operation of voltage-dependent conductances should thus be emphasized more in modeling the dynamic behavior of small neurons. Furthermore, the presented approach is valuable in providing faster computation times compared to the Markov chain type of modeling approaches and more sophisticated theoretical analysis tools compared to previously presented stochastic modeling approaches.

Event Abstract Back to Event Stochastic modeling of neuronal signaling Jukka Intosalmi1*, Tiina Manninen1, Keijo Ruohonen1 and Marja-Leena Linne1 1 Tampere University of Technology, Finland The time evolution of biochemical systems in neurons is traditionally modeled using deterministic ordinary differential equations (ODEs). Chemical reactions, however, are random in nature, and the deterministic approach is valid only for a restricted class of systems. Stochastic models take random fluctuations into account and are thus more realistic. Biochemical reactions can be modeled stochastically using numerous different methods. An ideal model would have the following three important properties. First, the model should be as realistic as possible, second, the mathematical method should be easily implementable as a computer algorithm, and third, the algorithm should be computationally effective. Naturally, all these three conditions cannot be fulfilled at the same time. Some realistic modeling approaches can be derived directly from chemical kinetics without making any approximations. Such approaches are called exact. A good example of an exact modeling approach is the stochastic simulation algorithm (SSA) developed by Gillespie. The SSA is applicable when the molecular populations in the system are small, but it becomes computationally inefficient when the numbers of molecules increase. In order to construct stochastic models that can be effectively simulated, new mathematical approaches have to be explored. As an approximate method also stochastic differential equations (SDEs) have been considered a promising way to model biochemical reactions stochastically. The SDE approach is attractive especially if we consider a system for which the SSA is computationally inefficient and the traditional deterministic ODE approach cannot be used as a good approximation. SDE models treat the chemical populations as real numbers and the construction of the model is based on the law of mass action, similarly as the construction of the traditional deterministic modeling approach. The SDE model, however, takes the random fluctuations into account by describing the time evolution of the system with the stochastic Itô process instead of the deterministic set of ODEs. The Itô process is basically a set of coupled stochastic differential equations. Although SDE modeling usually leads to equations that cannot be solved analytically, the solutions can be approximated numerically using different numerical integration methods. Thus, SDEs provide a mathematically rigorous way to construct models that can be simulated effectively. In order to investigate how well the SDE approach models biochemical systems, the results have to be compared with experimental data or with the simulation results from some exact simulation procedure. In this study, we use the methodology of spectral analysis to compare the simulation results of the SDE model and the SSA. As case studies, we consider e.g. calcium binding and protein kinase C (PKC) signal transduction pathway. We investigate the nature of noise in different modeling approaches and try to interpret the meaning of the noise in real biological systems. The main goal of our study is to find the most essential components and parameters in the SDE model and to adjust them so that the model is capable of giving similar results as the SSA. The simulations are carried out using the distributed computing resources (GRID) provided by Techila Technologies Ltd. Conference: Neuroinformatics 2008, Stockholm, Sweden, 7 Sep - 9 Sep, 2008. Presentation Type: Poster Presentation Topic: Computational Neuroscience Citation: Intosalmi J, Manninen T, Ruohonen K and Linne M (2008). Stochastic modeling of neuronal signaling. Front. Neuroinform. Conference Abstract: Neuroinformatics 2008. doi: 10.3389/conf.neuro.11.2008.01.023 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Jul 2008; Published Online: 28 Jul 2008. * Correspondence: Jukka Intosalmi, Tampere University of Technology, Tampere, Finland, jukka.intosalmi@tut.fi Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Scholar Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne PubMed Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

We consider the valuation of collateralized derivative contracts such as bond option or Caplet contracts. We allow for posting different collaterals such as securities or cash for the derivatives and its hedges. The pricing is based on modeling the joint evolution of collateral rate and the spread between collaterals. The Hull–White models are applied to collateral rate and spread to generate the closed pricing formula for zero coupon bond option. We also derive the pricing formula for Caplet under the Libor Market model and SABR model framework.

The uncertainty of geometric imperfections in a series of nominally equal I-beams leads to a variability of corresponding buckling loads. Its analysis requires a stochastic imperfection model, which can be derived either by the simple variation of the critical eigenmode with a scalar random variable, or with the help of the more advanced theory of random fields. The present paper first provides a concise review of the two different modeling approaches, covering theoretical background, assumptions and calibration, and illustrates their integration into commercial finite element software to conduct stochastic buckling analyses with the Monte–Carlo method. The stochastic buckling behavior of an example beam is then simulated with both stochastic models, calibrated from corresponding imperfection measurements. The simulation results show that for different load cases, the response statistics of the buckling load obtained with the eigenmode-based and the random field-based models agree very well. A comparison of our simulation results with corresponding Eurocode 3 limit loads indicates that the design standard is very conservative for compression dominated load cases.

In this paper, using a geometric method introduced in (Henry-Labordere Large Deviations and Asymptotic Methods in Finance (2015) [12]) and initiated by (Avellaneda et al. Risk Mag. (2002) [4]), we derive an asymptotic swaption implied volatility at the first-order for a general stochastic volatility Libor Market Model. This formula is useful to quickly calibrate a model to a full swaption matrix. We apply this formula to a specific model where the forward rates are assumed to follow a multi-dimensional CEV process correlated to a SABR process. For a caplet, this model degenerates to the classical SABR model and our asymptotic swaption implied volatility reduces naturally to the Hagan-al formula (Hagan et al. Willmott Mag. 88–108 (2002) [11]). The geometry underlying this model is the hyperbolic manifold \({\mathbb H}^{n+1}\) with n the number of Libor forward rates.

This paper uses an extensive set of market data of forward swap rates and swaptions covering 3 July 2002 to 21 May 2009 to identify a two-dimensional stochastic volatility process for the level of rates. The process is identified step by step by increasing the requirement of the model and introducing appropriate adjustments. The first part of the paper investigates the smile dynamics of forward swap rates at their setting dates. Comparing the SABR (with different ) and Heston stochastic volatility models informs us what different specifications of the driving SDEs have to offer in terms of reflecting the dynamics of the smile across dates. The outcome of the analysis is that a normal SABR model () satisfactorily passes all tests and seems to provide a good match to the market. In contrast, we find that the Heston model does not. The next step is to seek a model of the forward swap rates (in their own swaption measure) based on only two Brownian motions that enables a specification with common parameters. It turns out that this can be done by extending the SABR model with a time-dependent volatility function and a mean-reverting volatility process. The performance of the extended (SABR with mean-reversion) model is analysed over several historical dates and is shown to be a stable and flexible choice that allows for good calibration across expiries and strikes. Finally, a time-homogeneous candidate stochastic volatility process that can be used as a driver for all swap rates is identified. This candidate process may in future work be used as a building block for a separable stochastic volatility LIBOR market model or a stochastic volatility Markov-functional model.

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

From classical mechanics to unitary propagation without quantization

Reuse of wastewater to irrigate food crops is being practiced in many parts of the world and is becoming more commonplace as the competition for, and stresses on, freshwater resources intensify. But there are risks associated with wastewater irrigation, including the possibility of transmission of pathogens causing infectious disease, to both workers in the field and to consumers buying and eating produce irrigated with wastewater. To manage these risks appropriately we need objective and quantitative estimates of them. This is typically achieved through one of two modelling approaches: deterministic or stochastic. Each parameter in a deterministic model is represented by a single value, whereas in stochastic models probability functions are used. Stochastic models are theoretically superior because they account for variability and uncertainty, but they are computationally demanding and not readily accessible to water resource and public health managers. We constructed models to estimate risk of enteric virus infection arising from the consumption of wastewater-irrigated horticultural crops (broccoli, cucumber and lettuce), and compared the resultant levels of risk between the deterministic and stochastic approaches. Several scenarios were tested for each crop, accounting for different concentrations of enteric viruses and different lengths of environmental exposure (i.e. the time between the last irrigation event and harvest, when the viruses are liable to decay or inactivation). In most situations modelled the two approaches yielded similar estimates of risk (within 1 order-of-magnitude). The two methods diverged most markedly, up to around 2 orders-of-magnitude, when there was large uncertainty associated with the estimate of virus concentration and the exposure period was short (1 day). Therefore, in some circumstances deterministic modelling may offer water resource managers a pragmatic alternative to stochastic modelling, but its usefulness as a surrogate will depend upon the level of uncertainty in the model parameters.