In this thesis we use techniques from white noise analysis to study solutions of semilinear stochastic differential equations in a Hilbert space H: {dX[subscript]t = (AX[subscript]t + F(t,X[subscript]t)) dt + ơ(t,X[subscript]t) δB[subscript]t, t∈ (0,T], X[subscript]0 = ξ, where A is a generator of either a C[subscript]0-semigroup or an n-times integrated semigroup, and B is a cylindrical Wiener process. We then consider applications to forward interest rate models, such as in the Heath-Jarrow-Morton framework. We also reformulate a phenomenological model of the forward rate. / Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Science, 2009

In this thesis, we propose a new and simple approach of extending the single-factor Heston stochastic volatility model to a more flexible one in solving option pricing problems.  In this approach, the volatility process for the underlying asset dynamics depends on the time to maturity of the option. As this idea is inspired by the Heath-Jarrow-Morton framework which models the evolution of the full dynamics of forward rate curves for various maturities, we name this approach as the HJM-type stochastic volatility (HJM-SV)  model. We conduct an empirical analysis by calibrating this model to real-market option data for underlying assets including an equity  (ABB stock) and a market index (EURO STOXX 50), for two separated time spans from Jan 2017 to Dec 2017 (before the COVID-19 pandemic) and from Nov 2019 to Nov 2020 (after the start of COVID-19 pandemic). We investigate the optimal way of dividing the set of option maturities into three classes, namely, the short-maturity, middle-maturity, and long-maturity classes. We calibrate our HJM-SV model to the data in the following way, for each class a single-factor Heston stochastic volatility model is calibrated to the corresponding market data. We address the question that how well the new HJM-SV model captures the feature of implied volatility surface given by the market data.

In this dissertation, we investigate the spread option pricing problem with stochastic interest rate. First, we will review the basic concept and theories of stochastic calculus, give an introduction of spread options and provide some examples of spread options in different markets. We will also review the market efficiency theory, arbitrage and assumptions that are commonly used in mathematical finance. In Chapter 3, we will review existing spread pricing models and term-structure models such as Vasicek Mode, and the Heath-Jarrow-Morton framework. In Chapter 4, we will use the martingale approach to derive a partial differential equation for the price of the spread option with stochastic interest rate. In Chapter 5, we will study the spread option numerically. We will conclude this dissertation with ideas for future research.

No / We show that the unified HJM-based approach of constructing Gaussian dynamic term structure models developed by Li, Ye, and Yu (2016) nests most existing GDTSMs as special cases. We also discuss issues of interest rate derivatives pricing under this approach and using integration to construct Markov representations of HJM models.

有鑑於附有最低保證給付投資型保險期末現金流量與選擇權如出一轍，是以應用平賭訂價理論(The Martingale Pricing Method)嵌入HJM利率模型，對隨機利率下附有最低保證給付投資型保險進行評價。並對繳費方式與利率型態兩議題所構成四種類型附有最低保證給付投資型保險作實地數據模擬與評價，以及敏感度分析。 研究結果可以歸納為四點結論。 (1) 單就附有最低保證給付投資型保險簡化版（忽略期中死亡理賠與期滿生存機率）而言： 可視為是最低保證給付折現與以之為履約價的買權組合。因此，當影響因子僅與買權有相關性時，附有最低保證給付投資型保險與理論買權的敏感度分析結果，如出一轍。連動標的期初價格與波動度變動於附有最低保證給付投資型保險影響便是實證。 (2) 延續上點論述衍生： 當影響因子同時對買權與附有最低保證給付折現具有相關性時，由於買權佔整個保險價值比重過低，是以主要影響力皆來自附有最低保證給付的變動。附有最低保證給付與固定利率折現因子變動對於保險價值影響，即反應此結果。 (3) 分別就繳費方式不同下，投保年齡與投保期限變動對於附有最低保證給付投資保險的影響而言： 躉繳型繳費方式下，由第二點結論可得，投保期限越長保費越低，是以當投保年齡越大，期中死亡率提高，且期間短的保費較高的情況下，投保年齡變動對於附有最低保證給付投資型保險影響為正向；分期繳型繳費方式下，由於條款設定不同，無法與躉繳型一概而論，反映在投保期間越長保單價值與保費皆增加，但若是比較其增加的幅度（二階條件小於零）逐漸減少，倒是與躉繳型投資保險投保期間與保費關係意思相同，只是呈現方式不同。分期繳型投資型保險保單價值與投保年齡關係，從投保期限與保費關係以及高年齡層死亡率較高，可以得知，隨著投保年齡的增加，分期繳型投資保險中因為死亡理賠的現金流量產生機會提高，而此部分期間短保單價值較低，是以投保年齡與保單價值呈現反比關係，但是保單價值平準化後的保費，源於平準因子每期存活率因投保年齡增加而減少，造成投保年齡越高，保費也越高。 (4) 就性別而言： 躉繳型附有最低保證給付投資保險，由於女性相較於男性死亡率較低，容易取得期間較長的期滿保證金，而此部分價值較低，是以女生保費較男生便宜；分期繳型附有最低保證給付投資保險，則是相反的表現，由於此部分價值較高，是以女性的保險價值高於男性，同時因女性平準因子中的存活率也比男性高，是以每期所要繳交的保費也比男性低廉。 (5) 就利率型態而言： 隨機利率下躉繳型投資型保險與固定利率下躉繳型投資保險相較，便宜許多，主要是因為利率型態為隨機，且期初利率期間結構打破水平狀態的假設，真實反應正常期初利率期間結構(Normal Interest Rate Term Structure)，是以評價出的保費較固定利率型態下的保費低廉，甚至於分期繳型附有最低保證給付投資保險，在隨機利率下，隨著投保期限增加，保費反而下降。

Es sei ein Heath-Jarrow-Morton Zinsstrukturmodell df(t,T) = alpha(t,T)dt + sigma(t,T)dX_t gegeben, angetrieben von einem mehrdimensionalen Semimartingal X. Das Ziel dieser Arbeit besteht darin, die Existenz endlich dimensionaler Realisierungen für solche Modelle zu untersuchen, wobei wir als treibende Prozesse die Klasse der Grigelionis Prozesse wählen, die insbesondere Levy Prozesse enthält. Zur Bearbeitung der Fragestellung werden zwei veschiedene Ansätze verfolgt. Wir dehnen die Ideen aus der Differenzialgeometrie von Björk und Svensson (2001) auf die vorliegende Situation aus und zeigen, dass das in der zitierten Arbeit bewiesene Kriterium für die Existenz endlich dimensionaler Realisierungen in unserem Fall als notwendiges Kriterium dienlich ist. Dieses Resultat wird auf konkrete Volatilitätsstrukturen angewandt. Im Kontext von sogenannten Benchmark Realisierungen, die eine natürliche Verallgemeinerung von Short Rate Realisierungen darstellen, leiten wir Integro-Differenzialgleichungen her, die für die Untersuchung der Existenz endlich dimensionaler Realisierungen hilfreich sind. Als Verallgemeinerung eines Resultats von Jeffrey (1995) beweisen wir außerdem, dass Zinsstrukturmodelle, die eine generische Benchmark Realisierung besitzen, notwendigerweise eine singuläre Hessesche Matrix haben. Beide Ansätze zeigen, dass neue Phänomene auftreten, sobald der treibende Prozess X Sprünge macht. Es gibt dann auf einmal nur noch sehr wenige Zinsstrukturmodelle, die endlich dimensionale Realisierungen zulassen, was ein beträchtlicher Unterschied zu solchen Modellen ist, die von einer Brownschen Bewegung angetrieben werden. Aus diesem Grund zeigen wir, dass für die in der Literatur oft behandelten Modelle mit deterministischer Richtungsvolatilität eine Folge von endlich dimensionalen Systemen existiert, die gegen das Zinsmodell konvergieren. / Let f(t,T) be a term structure model of Heath-Jarrow-Morton type df(t,T) = alpha(t,T)dt + sigma(t,T)dX_t, driven by a multidimensional semimartingale X. Our objective is to study the existence of finite dimensional realizations for equations of this kind. Choosing the class of Grigelionis processes (including in particular Levy processes) as driving processes, we approach this problem from two different directions. Extending the ideas from differential geometry in Björk and Svensson (2001), we show that the criterion for the existence of finite dimensional realizations, proven in the aforementioned paper, still serves as a necessary condition in our setup. This result is applied to concrete volatility structures. In the context of benchmark realizations, which are a natural generalization of short rate realizations, we derive integro-differential equations, suitable for the analysis of the realization problem. Generalizing Jeffrey (1995), we also prove a result stating that forward rate models, which generically possess a benchmark realization, must have a singular Hessian matrix. Both approaches reveal that, with regard to the results known for driving Wiener processes, new phenomena emerge, as soon as the driving process X has jumps. In particular, the occurrence of jumps severely limits the range of models that admit finite dimensional realizations. For this reason we prove, for the often considered case of deterministic direction volatility structures, the existence of finite dimensional systems converging to the forward rate model.

Submitted by Daniel Benabou (dbenabas@gmail.com) on 2018-09-24T14:05:05Z No. of bitstreams: 1 Daniel Benabou - Final.pdf: 1509370 bytes, checksum: b5f974a956000f0474b19f3aa92f13db (MD5) / Approved for entry into archive by Joana Martorini (joana.martorini@fgv.br) on 2018-09-24T15:01:39Z (GMT) No. of bitstreams: 1 Daniel Benabou - Final.pdf: 1509370 bytes, checksum: b5f974a956000f0474b19f3aa92f13db (MD5) / Approved for entry into archive by Isabele Garcia (isabele.garcia@fgv.br) on 2018-09-24T20:11:14Z (GMT) No. of bitstreams: 1 Daniel Benabou - Final.pdf: 1509370 bytes, checksum: b5f974a956000f0474b19f3aa92f13db (MD5) / Made available in DSpace on 2018-09-24T20:11:14Z (GMT). No. of bitstreams: 1 Daniel Benabou - Final.pdf: 1509370 bytes, checksum: b5f974a956000f0474b19f3aa92f13db (MD5) Previous issue date: 2018-08-24 / Com o objetivo de obter a estrutura de curvas futuras de swaps de energia, este trabalho foca na implementação numérica do modelo de Heath, Jarrow e Morton (1992) utilizando somente as informações dos contratos de swaps negociados no Sistema Elétrico Brasileiro, através do modelo discreto do HJM conhecido como Modelo de Brace, Garatek e Musiela (1997), também referido como Modelo de Mercado. A estrutura de volatilidade foi obtida de forma não-paramétrica através de curvas suaves e de vértices sintéticos obtidos por interpolação dos dados de venda de uma comercializadora tratados através do método de Análise de Componentes Principais (PCA). Os dados analisados foram contratos firmados entre o início de 2013 e o primeiro quadrimestre de 2015. / For the purpose of obtaining the structure of future swap energy curves in the Brazilian market, this paper focuses on the numerical implementation of the Heath, Jarrow and Morton model (1992) using market information regarding the swap contracts traded in the Brazilian energy system, with its multi-factor discrete form, the Brace, Garatek and Musiela (1997) model, also known as Market Model. The volatility structure is obtained with smooth curves and synthetic vertices, obtained thru swap contracts negotiated by a Brazilian energy trading company. Also, the volatility structures where analyzed with the Principal Components Analysis (PCA). The analyzed data where swap contracts stablished between the beginning of 2013 until the first quarter of 2015.

Submitted by Nelson Nojima (nelson.nojima@gmail.com) on 2013-09-09T22:05:53Z No. of bitstreams: 1 Dissert-NelsonNojima_v27a.pdf: 752789 bytes, checksum: 162111784a7b3f201faadac1a7a9417f (MD5) / Approved for entry into archive by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br) on 2013-09-10T12:39:18Z (GMT) No. of bitstreams: 1 Dissert-NelsonNojima_v27a.pdf: 752789 bytes, checksum: 162111784a7b3f201faadac1a7a9417f (MD5) / Made available in DSpace on 2013-09-10T12:56:54Z (GMT). No. of bitstreams: 1 Dissert-NelsonNojima_v27a.pdf: 752789 bytes, checksum: 162111784a7b3f201faadac1a7a9417f (MD5) Previous issue date: 2013-08-23 / For the purpose of pricing interest rate derivatives in the Brazilian market, this work focuses on the numerical implementation of the Heath, Jarrow and Morton model (1992) in its multi-factor discrete form, which allows for great flexibility on the estimation of the forward rate curve under a volatility term structure based on orthogonal factors (PCA), thus facilitating the Monte Carlo simulation of its dynamics as a consequence of the independence of these factors. The volatility term structure is built in a non-parametric way based on synthetic buckets obtained via interpolation of historical data of BM&FBOVESPA DI Futures ranging from Jan/2nd/2003 to Dec/28th/2012. The Brace and Musiela (1994) adaptation of the HJM Model was adopted in this study. The following derivatives are priced: DI Futures, options on the IDI index, and options on DI Futures. / Com o objetivo de precificar derivativos de taxas de juros no mercado brasileiro, este trabalho foca na implementação do modelo de Heath, Jarrow e Morton (1992) em sua forma discreta e multifatorial através de uma abordagem numérica, e, que possibilita uma grande flexibilidade na estimativa da taxa forward sob uma estrutura de volatilidade baseada em fatores ortogonais, facilitando assim a simulação de sua evolução por Monte Carlo, como conseqüência da independência destes fatores. A estrutura de volatilidade foi construída de maneira a ser totalmente não paramétrica baseada em vértices sintéticos que foram obtidos por interpolação dos dados históricos de cotações do DI Futuro negociado na BM&FBOVESPA, sendo o período analisado entre 02/01/2003 a 28/12/2012. Para possibilitar esta abordagem foi introduzida uma modificação no modelo HJM desenvolvida por Brace e Musiela (1994).

Thesis (MSc)--Stellenbosch University, 2015. / ENGLISH ABSTRACT : The standard-Heath–Jarrow–Morton (HJM) framework is well-known for its application to pricing and hedging interest rate derivatives. This study implemented the extended HJM framework introduced by Eberlein and Raible (1999), in which a Brownian motion (BM) is replaced by a wide class of processes with jumps. In particular, the HJM driven by the generalised hyperbolic processes was studied. This approach was motivated by empirical evidence proving that models driven by a Brownian motion have several shortcomings, such as inability to incorporate jumps and leptokurticity into the price dynamics. Non-homogeneous Lévy processes and the change of measure techniques necessary for simplification and derivation of pricing formulae were also investigated. For robustness in numerical valuation, several transform methods were investigated and compared in terms of speed and accuracy. The models were calibrated to liquid South African data (ATM) interest rate caps using two methods of optimisation, namely the simulated annealing and secant-Levenberg–Marquardt methods. Two numerical valuation approaches had been implemented in this study, the COS method and the fractional fast Fourier transform (FrFT), and were compared to the existing methods in the context. Our numerical results showed that these two methods are quite efficient and very competitive. We have chose the COS method for calibration due to its rapidly speed and we have suggested a suitable approach for truncating the integration range to address the problems it has with short-maturity options. Our calibration results provided a nearly perfect fit, such that it was difficult to decide which model has a better fit to the current market state. Finally, all the implementations were done in MATLAB and the codes included in appendices. / AFRIKAANSE OPSOMMING : Die standaard-Heath–Jarrow–Morton-raamwerk (kortom die HJM-raamwerk) is daarvoor bekend dat dit op die prysbepaling en verskansing van afgeleide finansiële instrumente vir rentekoerse toegepas kan word. Hierdie studie het die uitgebreide HJM-raamwerk geïmplementeer wat deur Eberlein en Raible (1999) bekendgestel is en waarin ’n Brown-beweging deur ’n breë klas prosesse met spronge vervang word. In die besonder is die HJM wat deur veralgemeende hiperboliese prosesse gedryf word ondersoek. Hierdie benadering is gemotiveer deur empiriese bewyse dat modelle wat deur ’n Brown-beweging gedryf word verskeie tekortkominge het, soos die onvermoë om spronge en leptokurtose in prysdinamika te inkorporeer. Nie-homogene Lévy-prosesse en die maatveranderingstegnieke wat vir die vereenvoudiging en afleiding van prysbepalingsformules nodig is, is ook ondersoek. Vir robuustheid in numeriese waardasie is verskeie transformmetodes ondersoek en ten opsigte van spoed en akkuraatheid vergelyk. Die modelle is vir likiede Suid-Afrikaanse data vir boperke van rentekoerse sonder intrinsieke waarde gekalibreer deur twee optimiseringsmetodes te gebruik, naamlik die gesimuleerde uitgloeimetode en die sekans-Levenberg–Marquardt-metode. Twee benaderings tot numeriese waardasie is in hierdie studie gebruik, naamlik die kosinusmetode en die fraksionele vinnige Fourier-transform, en met bestaande metodes in die konteks vergelyk. Die numeriese resultate het getoon dat hierdie twee metodes redelik doeltreffend en uiters mededingend is. Ons het op grond van die motiveringspoed van die kosinus-metode daardie metode vir kalibrering gekies en ’n geskikte benadering tot die trunkering van die integrasiereeks voorgestel ten einde die probleem ten opsigte van opsies met kort uitkeringstermyne op te los. Die kalibreringsresultate het ’n byna perfekte passing gelewer, sodat dit moeilik was om te besluit watter model die huidige marksituasie die beste pas. Ten slotte is alle implementerings in MATLAB gedoen en die kodes in bylaes ingesluit.

During the last decade several Japanese life insurance companies failed mainly due to interest losses. In fact, interest rate risk dominates mortality risk for a portfolio of business in force. When the interest rates are modelled as random variables, the yields on bonds are the sum of expected short spot rates and a risk premium for random bond prices. However, in our study, we assume a risk-neutral environment, i.e. zero risk premiums. As tools to deal with stochastic interest rates, various interest rate term structure models are considered. The Vasicek model, the Heath-Jarrow-Morton (hereafter “HJM”) approach and Cairns’ model are explained in detail. The history and nature of the very low interest rate environment in Japan is described in line with the monetary policy framework of the central bank. An unusual interest rate movement in the very low interest rate environment is identified. A modified HJM approach and Cairns’ model are chosen in our study. Cairns’ model is used to graduate the initial yield curve. The HJM approach with a specific volatility function and modified to deal with very low interest rates is used for simulating subsequent developments of the initial yield curve. After the introduction of various concepts needed to investigate a life insurance policy portfolio, we prepare for simulation by collecting information and by fitting parameters to market observations. The Yen swap curve is chosen as a base yield curve. The simulation results show how much initial capital and/or margins are needed in order to avoid the ruin of a portfolio.