Interest rates forecasting: between Hull and White and the {CIR}{#}. How to make a single factor model work

In this work we present our findings of the so‐called CIR#, which is a modified version of the Cox, Ingersoll & Ross (CIR) model, turned into a forecasting tool for any term structure. The main feature of the CIR# model is its ability to cope with negative interest rates, cluster volatility and jumps. By considering a dataset composed of money market interest rates during turmoil and calmer periods, we show how the CIR# performs in terms of directionality of rates and forecasting error. Comparison is carried out with a revamped version of the CIR model (denoted CIRadj), the Hull and White model and the EWMA which is often adopted whenever no structure in data is assumed. Testing and validation is performed on both historical and had hoc data with different metrics and clustering criteria to confirm the analysis

Dynamic hybrid pricing formulation for equity warrants

Equity warrants are instruments issued by a company that give the stockholder the privilege of buying a stock at a certain strike price within a particular timeframe. Motivated by empirical studies, the Black-Scholes option pricing model is not suitable to price a warrant since both assumptions of constant volatility and constant interest rates in the model are incompatible. This study proposed the Heston-Cox-Ingersoll- Ross (Heston-CIR) hybrid model to identify the effects of stochastic volatility and stochastic interest rates in pricing equity warrants. The study constructed new analytical pricing formulas for equity warrants by using Cauchy transformation and partial differential equation approaches. The local optimization method is employed to obtain the estimated parameter values by calibrating the Heston-CIR model. The effectiveness of the proposed model is investigated through the empirical study using the data from Bursa Malaysia. The proposed model shows significant improvement on the computation time in estimating nine model parameters, ranging from 38.12 to 62.62 seconds compared to the existing models. Moreover, the empirical study suggested that the proposed model is accurate when compared to the real market over five years period. This model also produced smallest pricing errors among the existing models. The finding also suggested equity warrants in moneyness opportunity, 88.75% of the warrants are profitable. In conclusion, the proposed model performs the best in identifying the effects of stochastic volatility and stochastic interest rates in pricing equity warrants

Modeling of volatility-linked financial products

This thesis is the collation of four papers, adapted from their original versions as to form here four distinct chapters. In the first chapter we illustrate and solve the pricing problem of a target volatility option (TVO) using three different methodologies. In the second chapter we study the pricing PDE for a general contingent claim involving an asset and its realized volatility, and then solve it for a variety of actual models and payoffs. The third chapter introduces a class of time-changed stochastic processes based on which a martingale asset price evolution can be devised. Pricing equations for volatility-linked derivatives are also obtained in this framework. In the final chapter we analyze one specific model of this class; we conclude that it does show high flexibility in explaining the forward volatility skew dynamics and that it can capture certain interesting stylized facts

Mortality linked derivatives and their pricing

This thesis addresses the absence of explicit pricing formulae and the complexity of proposed models (incomplete markets framework) in the area of mortality risk management requiring the application of advanced techniques from the realm of Financial Mathematics and Actuarial Science. In fact, this is a multi-essay dissertation contributing in the direction of designing and pricing mortality-linked derivatives and offering the state of art solutions to manage longevity risk. The first essay investigates the valuation of Catastrophic Mortality Bonds and, in particular, the case of the Swiss Re Mortality Bond 2003 as a primary example of this class of assets. This bond was the first Catastrophic Mortality Bond to be launched in the market and encapsulates the behaviour of a well-defined mortality index to generate payoffs for bondholders. Pricing this type of bond is a challenging task and no closed form solution exists in the literature. In my approach, we adapt the payoff of such a bond in terms of the payoff of an Asian put option and present a new methodology to derive model-independent bounds for catastrophic mortality bonds by exploiting the theory of comonotonicity. While managing catastrophic mortality risk is an upheaval task for insurers and re-insurers, the insurance industry is facing an even bigger challenge - the challenge of coping up with increased life expectancy. The recent years have witnessed unprecedented changes in mortality rate. As a result academicians and practitioners have started treating mortality in a stochastic manner. Moreover, the assumption of independence between mortality and interest rate has now been replaced by the observation that there is indeed a correlation between the two rates. Therefore, my second essay studies valuation of Guaranteed Annuity Options (GAOs) under the most generalized modeling framework where both interest rate and mortality risk are stochastic and correlated. Pricing these types of options in the correlated environment is an arduous task and a closed form solution is non-existent. In my approach, I employ the use of doubly stochastic stopping times to incorporate the randomness about the time of death and employ a suitable change of measure to facilitate the valuation of survival benefit, there by adapting the payoff of the GAO in terms of the payoff of a basket call option. I then derive general price bounds for GAOs by employing the theory of comonotonicity and the Rogers-Shi (Rogers and Shi, 1995) approach. Moreover, I suggest some `model-robust' tight bounds based on the moment generating function (m.g.f.) and characteristic function (c.f.) under the affine set up. The strength of these bounds is their computational speed which makes them indispensable for annuity providers who rely heavily on Monte Carlo simulations to calculate the fair market value of Guaranteed Annuity Options. In fact, sans Monte Carlo, the academic literature does not offer any solution for the pricing of the GAOs. I illustrate the performance of the bounds for a variety of affine processes governing the evolution of mortality and the interest rate by comparing them with the benchmark Monte Carlo estimates. Through my work, I have been able to express the payoffs of two well known modern mortality products in terms of payoffs of financial derivatives, there by filling the gaps in the literature and offering state of art techniques for pricing of these sophisticated instruments

Time series forecasting with the CIR# model: from hectic markets sentiments to regular seasonal tourism

This research aims to propose the so-called CIR#, which takes its cue from the well- known Cox-Ingersoll-Ross (CIR) model originally devised for pricing, as a general econometric model. To this end, we present the results on two very different time series such as Polish interest rates (subject to market sentiments) and seasonal tourism (subject to pandemic lock-down measures). For interest rates, as reference models, we consider an improved version of the CIR model (denoted CIRadj), the Hull and White model, the exponentially weighted moving average (EWMA) which is often adopted whenever no structure is assumed in the data and a popular machine learning model such as the short-term memory network (LSTM). For tourism, as a benchmark, we consider seasonal autoregressive integrated moving average (SARIMA) complemented by the generalized autoregressive conditional heteroskedasticity (GARCH) for modelling the variance, the classic Holt-Winters model and the aforementioned LSTM. Results support the claim that the CIR# performs better than the other models in all considered cases being able to deal with erratic behaviour in data

Some extensions of the Black-Scholes and Cox-Ingersoll-Ross models

In this thesis we will study some financial problems concerning the option pricing in complete and incomplete markets and the bond pricing in the short-term interest rates framework. We start from well known models in pricing options or zero-coupon bonds, as the Black-Scholes model and the Cox-Ingersoll-Ross model and study some their generalizations. In particular, in the first part of the thesis, we study a generalized Black-Scholes equation to derive explicit or approximate solutions of an option pricing problem in incomplete market where the incompleteness is generated by the presence of a non-traded asset. Our aim is to give a closed form representation of the indifference price by using the analytic tool of (C0) semigroup theory. The second part of the thesis deals with the problem of forecasting future interest rates from observed financial market data. We propose a new numerical methodology for the CIR framework, which we call the CIR# model, that well fits the term structure of short interest rates as observed in a real market