B-spline techniques for volatility modeling

This paper is devoted to the application of B-splines to volatility modeling, specifically the calibration of the leverage function in stochastic local volatility models and the parameterization of an arbitrage-free implied volatility surface calibrated to sparse option data. We use an extension of classical B-splines obtained by including basis functions with infinite support. We first come back to the application of shape-constrained B-splines to the estimation of conditional expectations, not merely from a scatter plot but also from the given marginal distributions. An application is the Monte Carlo calibration of stochastic local volatility models by Markov projection. Then we present a new technique for the calibration of an implied volatility surface to sparse option data. We use a B-spline parameterization of the Radon-Nikodym derivative of the underlying's risk-neutral probability density with respect to a roughly calibrated base model. We show that this method provides smooth arbitrage-free implied volatility surfaces. Finally, we sketch a Galerkin method with B-spline finite elements to the solution of the partial differential equation satisfied by the Radon-Nikodym derivative.Comment: 25 page

Extracting expectations from currency option prices: a comparison of methods

This paper compares the goodness-of-fit and the stability of six methods used to extract risk-neutral probability density functions from currency option prices. We first compare five existing methods commonly employed to recover risk-neutral density functions from option prices. Specifically, we compare the methods introduced by Shimko (1993), Madan and Milne (1994), Malz (1996), Melick and Thomas (1997) and Bliss and Panigirtzoglou (2002). In addition, we propose a new method based on the piecewise cubic Hermite interpolation of the implied volatility function. We use data on 12 emerging market currencies against the US dollar and find that the piecewise cubic Hermite interpolation method is by far the method with the best accuracy in fitting observed option prices. We also find that there is a relative tradeoff between the goodness-of-fit and the stability of the methods. Thus, methods which have a better accuracy in fitting observed option prices appear to be more sensitive to option pricing errors, while the most stable methods have a fairly disappointing fitting. However, for the first two PDF moments as well as the quartiles of the risk-neutral distributions we find that the estimates do not differ significantly across methods. This suggests that there is a large scope for selection between these methods without essentially sacrificing the accuracy of the analysis. Nonetheless, depending on the particular use of these PDFs, some methods may be more suitable than othersRisk-neutral probability density functions, option pricing, exchange rate expectations

Construction of a zero-coupon yield curve for the Nairobi Securities Exchange and its application in pricing derivatives

Thesis submitted in partial fulfillment of the requirements for the degree for PhD in Financial Mathematics at Strathmore UniversityYield curves are used to forecast interest rates for different products when their risk parameters are known, to calibrate no-arbitrage term structure models, and (mostly by investors) to detect whether there is arbitrage opportunity. By yield curve information, investors have opportunity of immunizing/hedging their investment portfolios against financial risks if they have to make an investment with some determined time of maturity. Private sector firms look at yields of different maturities and then choose their borrowing strategy. The differences in yields for long maturity and short maturities are an important indicator for central bank to use in monetary policy process. These differences may show the tightness of the government monetary policy and can be monitored to predict recession in coming years. A lot of research has been done in yield curve modeling and as we will see later in the thesis, most of the models developed had one major shortcoming: non differentiability at the interpolating knot points. The aim of this thesis is to construct a zero coupon yield curve for Nairobi Securities Exchange, and use the risk- free rates to price derivatives, with particular attention given to pricing coffee futures. This study looks into the three methods of constructing yield curves: by use of spline-based models, by interpolation and by using parametric models. We suggest an improvement in the interpolation methods used in the most celebrated spline-based model, monotonicity-preserving interpolation on r(t). We also use operator form of numerical differentiation to estimate the forward rates at the knot points, at which points the spot curve is non-differential. In derivative pricing, dynamical processes (Ito^ processes) are reviewed; and geometric Brownian motion is included, together with its properties and applications. Conventional techniques used in estimation of the drift and volatility parameters such as historical techniques are reviewed and discussed. We also use the Hough Transform, an artificial intelligence method, to detect market patterns and estimate the drift and volatility parameters simultaneously. We look at different ways of calculating derivative prices. For option pricing, we use different methods but apply Bellalahs models in calculation of the Coffee Futures prices because they incorporate an incomplete information parameter

Dynamic Risk Profile of the US Term Structure by Wavelet MRA

A careful examination of interest rate time series from different U.S. Treasury maturities by Wavelet Multiresolution Analysis (MRA) suggests that the first differences of the term structure of interest rate series are periodic or, at least, cyclic, non-stationary, long-term dependent, in particular, anti-persistent. Each nodal time series from a particular maturity has its own uniqueness and accordingly supports the Market Segmentation theory. The findings also imply that affine models are insufficient to describe the dynamics of the interest rate diffusion processes and call for more intensive research that might provide better, most likely fractal or nonlinear, term structure models for each maturity. If this is correct, empirical term structure models may describe chaotic, i.e., diffusion processes with non-unique dynamic equilibria.Wavelet, Interest rates, Hurst exponent, Term structure, Yield curve

Advances in non linear models for time series: methods and applications to economic and financial data

When linear models fail to explain the dynamic behavior of economic and financial time series, the researcher has to turn his attention to the nonlinear world. This work has been devoted to develop novel methodological proposals that may be useful in explaining the evolution over time of economic indicators and financial instruments. In Chapter 2, the well established Markov regime switching framework is extended letting the transition probabilities vary over time according to an observation-driven updating mechanism. An extensive simulation study shows the ability of our new model to track several dynamic patterns in transition probabilities. In the illustration to U.S. Industrial Production growth rate, we show that the model can capture the dynamic features of regime transition probabilities for means and variances. In Chapter 3, we adapt the new methodology in order to model the electricity spot prices. The Markov regime switching has been extensively used in literature to deal with the spikes that affect the evolution over time of this commodity prices. The non-homogenous occurrence of jumps may be successfully explained by an hidden Markov chain with time-varying transition probabilities that can be also influenced by exogenous variables. The information related to forecasted reserve margin and forecast demand can be easily included in our proposal to improve the model fit as well as to describe the occurrence of spikes. In Chapter 4, we propose a novel semi-nonparametric model to describe accurately the volatility of financial returns. The finite sample properties are investigated under both correct and incorrect model specification. The latter case suggests that our model is able to recover the functional form of volatility as long as the sample size increases. In empirical relevant settings, features like the asymmetric effect of negative and positive current shocks on future volatility, known as leverage effect, as well as the role played by the market condition in influencing the volatility evolution might be captured by our proposal

Stochastic Mortality Modelling and Management of Longevity Risk with Pricing and Reserving Applications to Annuity Products

Over the past decades, the life insurance sector has been faced with a number of challenges that emerged as a result of the growing longevity and stagnating birth rates for highly developed societies. In the field of actuarial application one could therefore ask for the implications of (long-term) mortality trends and (short-term) population fluctuation on an insurer's pricing and reserving of pension contracts, particularly in interaction with uncertainty in the capital markets. The first part of the thesis focuses on the mathematical description and projection of the mortality of homogeneous populations or insurance cohorts. Besides a survey of the most important representatives, a comprehensive analysis and comparison of stochastic and deterministic mortality forecasting models is carried out. In the second part a full stochastic model approach for two typical old-age provision products is set up and applied in terms of a management of longevity risk. On the one hand, a deferred conventional life annuity is analysed with regard to the combined effects of stochastic mortality and interest rates on different premium principles and risk capital allocation. On the other hand, a modern unit-linked annuity insurance, namely a deferred variable annuity contract, with an additional guaranteed minimum death benefit during the deferment period and a minimum income benefit at retirement is discussed. Mathematically, the guarantees represent options on the greater of the net asset value and a predetermined insurance benefit. For this reason, the existence and uniqueness of an extra fair percentage guarantee charge is proven. Furthermore, a sensitivity analysis of the fair charge and risk neutral option prices concerning different model parameters is considered and several profitability and risk measures are determined

Forecasting and Risk Management Techniques for Electricity Markets

This book focuses on the recent development of forecasting and risk management techniques for electricity markets. In addition, we discuss research on new trading platforms and environments using blockchain-based peer-to-peer (P2P) markets and computer agents. The book consists of two parts. The first part is entitled “Forecasting and Risk Management Techniques” and contains five chapters related to weather and electricity derivatives, and load and price forecasting for supporting electricity trading. The second part is entitled “Peer-to-Peer (P2P) Electricity Trading System and Strategy” and contains the following five chapters related to the feasibility and enhancement of P2P energy trading from various aspects

Dissecting the Yield Curve: The International Evidence

We develop a term structure model that decomposes nominal yields into the sum of an expectation, term premium, and convexity term and in turn of their real and inflation counterparts. The model explicitly captures the interrelation between yield-only and macroeconomic factors while allowing for aggregate stochastic volatility. We extract the components from the nominal and real yield curve of the United States, the Euro Area, the United Kingdom, and Japan. We find that short-rate expectations have steadily declined over the last two decades and account for the bulk of yield dynamics. Term premia increase with maturity but explain a smaller fraction of yield forecast error variance than previously documented. With regard to yield comovement, the United States generates the strongest spillovers at the long end of the yield curve, whereas the Japanese market is the top importer of shocks