次級房貸風暴造成全球金融海嘯，投資人對金融衍生性商品一度避之唯恐不及。在各方韃伐之下，卻忽略衍生性商品作為風險管理的工具及促進市場效率及完整性的重要性，未來在金融市場著重風險控管的趨勢下，衍生性商品仍會扮演不可或缺的角色。 本論文針對市場上交易量較大的利率衍生性金融商品及匯率衍生性金融商品，進行個案的評價與分析，提供投資人或發行者一個明確易懂的評價分析方式，能使市場上衍生性商品的交易更具效率性。 本論文以瑞士銀行所發行的「上下限固定期限交換利率利差連動債券」及「數據百慕達式匯率連動債券」為例，分別以LIBOR Market Model和最小平方蒙地卡羅法做為評價方式。依據評價結果分析發行商的避險策略與投資人所面對的投資風險。

Student Number : 0003819T - MSc dissertation - School of Computational and Applied Mathematics - Faculty of Science / The over-the-counter (OTC) interest rate derivative market is large and rapidly developing. In March 2005, the Bank for International Settlements published its “Triennial Central Bank Survey” which examined the derivative market activity in 2004 (http://www.bis.org/publ/rpfx05.htm). The reported total gross market value of OTC derivatives stood at $6.4 trillion at the end of June 2004. The gross market value of interest rate derivatives comprised a massive 71.7% of the total, followed by foreign exchange derivatives (17.5%) and equity derivatives (5%). Further, the daily turnover in interest rate option trading increased from 5.9% (of the total daily turnover in the interest rate derivative market) in April 2001 to 16.7% in April 2004. This growth and success of the interest rate derivative market has resulted in the introduction of exotic interest rate products and the ongoing search for accurate and efficient pricing and hedging techniques for them. Interest rate caps and (European) swaptions form the largest and the most liquid part of the interest rate option market. These vanilla instruments depend only on the level of the yield curve. The market standard for pricing them is the Black (1976) model. Caps and swaptions are typically used by traders of interest rate derivatives to gamma and vega hedge complex products. Thus an important feature of an interest rate model is not only its ability to recover an arbitrary input yield curve, but also an ability to calibrate to the implied at-the-money cap and swaption volatilities. The LIBOR market model developed out of the market’s need to price and hedge exotic interest rate derivatives consistently with the Black (1976) caplet formula. The focus of this dissertation is this popular class of interest rate models. The fundamental traded assets in an interest rate model are zero-coupon bonds. The evolution of their values, assuming that the underlying movements are continuous, is driven by a finite number of Brownian motions. The traditional approach to modelling the term structure of interest rates is to postulate the evolution of the instantaneous short or forward rates. Contrastingly, in the LIBOR market model, the discrete forward rates are modelled directly. The additional assumption imposed is that the volatility function of the discrete forward rates is a deterministic function of time. In Chapter 2 we provide a brief overview of the history of interest rate modelling which led to the LIBOR market model. The general theory of derivative pricing is presented, followed by a exposition and derivation of the stochastic differential equations governing the forward LIBOR rates. The LIBOR market model framework only truly becomes a model once the volatility functions of the discrete forward rates are specified. The information provided by the yield curve, the cap and the swaption markets does not imply a unique form for these functions. In Chapter 3, we examine various specifications of the LIBOR market model. Once the model is specified, it is calibrated to the above mentioned market data. An advantage of the LIBOR market model is the ability to calibrate to a large set of liquid market instruments while generating a realistic evolution of the forward rate volatility structure (Piterbarg 2004). We examine some of the practical problems that arise when calibrating the market model and present an example calibration in the UK market. The necessity, in general, of pricing derivatives in the LIBOR market model using Monte Carlo simulation is explained in Chapter 4. Both the Monte Carlo and quasi-Monte Carlo simulation approaches are presented, together with an examination of the various discretizations of the forward rate stochastic differential equations. The chapter concludes with some numerical results comparing the performance of Monte Carlo estimates with quasi-Monte Carlo estimates and the performance of the discretization approaches. In the final chapter we discuss numerical techniques based on Monte Carlo simulation for pricing American derivatives. We present the primal and dual American option pricing problem formulations, followed by an overview of the two main numerical techniques for pricing American options using Monte Carlo simulation. Callable LIBOR exotics is a name given to a class of interest rate derivatives that have early exercise provisions (Bermudan style) to exercise into various underlying interest rate products. A popular approach for valuing these instruments in the LIBOR market model is to estimate the continuation value of the option using parametric regression and, subsequently, to estimate the option value using backward induction. This approach relies on the choice of relevant, i.e. problem specific predictor variables and also on the functional form of the regression function. It is certainly not a “black-box” type of approach. Instead of choosing the relevant predictor variables, we present the sliced inverse regression technique. Sliced inverse regression is a statistical technique that aims to capture the main features of the data with a few low-dimensional projections. In particular, we use the sliced inverse regression technique to identify the low-dimensional projections of the forward LIBOR rates and then we estimate the continuation value of the option using nonparametric regression techniques. The results for a Bermudan swaption in a two-factor LIBOR market model are compared to those in Andersen (2000).

The South African interest rate market has mainly been focused on vanilla interest rate products and hence can be seen as underdeveloped in this regard when compared, for instance, to the associated equity market. Market participants subscribe this aspect to a lack of demand and sophistication of investors within the market. This is, however, expected to change given the influx of international banks into the South African market over the past couple of years. The current market methodology, for the pricing of vanilla interest rate options in the South African market, is the standard Black model with some mechanism to incorporate interest rate smiles. This mechanism is typically in the form of the SABR model. The most signi cant drawback of this approach is the fact that it models each forward rate in isolation. Hence, there is no way to incorporate the joint dynamics between different forward rates and consequently cannot be used for the pricing of exotic interest rate options. In anticipation of these new market developments, we explore the possibility of calibrating the LIBOR market model to the South African market. This dissertation follows a bottom up approach and hence considers all aspects associated with such an implementation. The work mainly focuses on the calibration to at-the-money interest rate options. A possible extension to the SABR model, while remaining within the LMM framework, is considered in the final chapter. Copyright / Dissertation (MSc)--University of Pretoria, 2012. / Mathematics and Applied Mathematics / unrestricted

Instantaneous rate models, although theoretically satisfying, are less so in practice. Instantaneous rates are not observable and calibra- tion to market data is complicated. Hence, the need for a market model where one models LIBOR rates seems imperative. In this modeling process, we aim at regaining the Black-76 formula[7] for pricing caps and °oors since these are the ones used in the market. To regain the Black-76 formula we have to model the LIBOR rates as log-normal processes. The whole construction method means calibration by using market data for caps, °oors and swaptions is straightforward. Brace, Gatarek and Musiela[8] and, Miltersen, Sandmann and Sondermann[25] showed that it is possible to con- struct an arbitrage-free interest rate model in which the LIBOR rates follow a log-normal process leading to Black-type pricing for- mulae for caps and °oors. The key to their approach is to start directly with modeling observed market rates, LIBOR rates in this case, instead of instantaneous spot rates or forward rates. There- after, the market models, which are consistent and arbitrage-free[6], [22], [8], can be used to price more exotic instruments. This model is known as the LIBOR Market Model. In a similar fashion, Jamshidian[22] (1998) showed how to con- struct an arbitrage-free interest rate model that yields Black-type pricing formulae for a certain set of swaptions. In this particular case, one starts with modeling forward swap rates as log-normal processes. This model is known as the Swap Market Model. Some of the advantages of market models as compared to other traditional models are that market models imply pricing formulae for caplets, °oorlets or swaptions that correspond to market practice. Consequently, calibration of such models is relatively simple[8]. The plan of this work is as follows. Firstly, we present an em- pirical analysis of the standard risk-neutral valuation approach, the forward risk-adjusted valuation approach, and elaborate the pro- cess of computing the forward risk-adjusted measure. Secondly, we present the formulation of the LIBOR and Swap market models based on a ¯nite number of bond prices[6], [8]. The technique used will enable us to formulate and name a new model for the South African market, the SAFEX-JIBAR model. In [5], a new approach for the estimation of the volatility of the instantaneous short interest rate was proposed. A relationship between observed LIBOR rates and certain unobserved instantaneous forward rates was established. Since data are observed discretely in time, the stochastic dynamics for these rates were determined un- der the corresponding risk-neutral measure and a ¯ltering estimation algorithm for the time-discretised interest rate dynamics was pro- posed. Thirdly, the SAFEX-JIBAR market model is formulated based on the assumption that the forward JIBAR rates follow a log-normal process. Formulae of the Black-type are deduced and applied to the pricing of a Rand Merchant Bank cap/°oor. In addition, the corre- sponding formulae for the Greeks are deduced. The JIBAR is then compared to other well known models by numerical results. Lastly, we perform some computational analysis in the following manner. We generate bond and caplet prices using Hull's [19] stan- dard market model and calibrate the LIBOR model to the cap curve, i.e determine the implied volatilities ¾i's which can then be used to assess the volatility most appropriate for pricing the instrument under consideration. Having done that, we calibrate the Ho-Lee model to the bond curve obtained by our standard market model. We numerically compute caplet prices using the Black-76 formula for caplets and compare these prices to the ones obtained using the standard market model. Finally we compute and compare swaption prices obtained by our standard market model and by the LIBOR model. / Economics / D.Phil. (Operations Research)

This thesis consists of three papers in the area of interest rate derivatives modelling. The pricing and hedging of (exotic) interest rate derivatives is one of the most demanding and complex problems in option pricing theory and is of great practical importance in the market. Models used in production at various banks can broadly be divided in three groups: 1- or 2-factor instantaneous short/forward rate models (such as Hull &amp; White (1990) or Cheyette (1996)), LIBOR/swap market models (introduced by Brace, Gatarek &amp; Musiela (1997), Miltersen, Sandmann &amp; Sondermannn (1997) and Jamshidian (1997)) and the one or two-dimensional Markov-functional models of Hunt, Kennedy &amp; Pelsser (2000)). In brief and general terms the main characters of the above mentioned three modelling frameworks can be summarised as follows. Short/forward rate models are by nature computationally efficient (implementations may be done using PDE or lattice methods) but less flexible in terms of fitting of implied volatility smiles and correlations between various rates. Calibration is hence typically performed in a ‘local’ (product by product based) sense. LIBOR market models on the other hand may be calibrated in a ‘global’ sense (i.e. fitting close to everything implying that one calibration may in principle be used for all products) but are of high dimension and an accurate implementation has to be done using the Monte Carlo method. Finally, Markov-functional models can be viewed as designed to combine the computational efficiency of short/forward rate models with flexible calibration properties. The defining property of a Markov-functional model is that each rate and discount factor at all times can be written as functionals of some (preferably computationally simple) Markovian driving process. While this is a property of most commonly used interest rate models Hunt et al. (2000) introduced a technique to numerically determine a set of functional forms consistent with market prices of vanilla options across strikes and expiries. The term a ‘Markov-functional model’ is typically referring to this type of model as opposed to the more general meaning, a terminology that is adopted also in this thesis. Although Markov-functional models are indeed a popular choice in practice there are a few outstanding points on the practitioners’ wish list. From a conceptual point of view there is still work to be done in order to fully understand the implications of various modelling choices and how to efficiently calibrate and use the model. Part of the reason for this is that while the properties of the short/forward rate and the LIBOR market models may be understood from their defining SDEs this is less clear for a Markov-functional model. To aid the understanding of the Markov-functional model Bennett &amp; Kennedy (2005) compares one-dimensional LIBOR and swap Markov-functional models with the one-factor separable LIBOR and swap market models and concludes that the models are similar distributionally across a wide range of viable market conditions. Although this provides good intuition there is still more work to be done in order to fully understand the implications of various modelling choices, in particular in a two or higher dimensional setting. The first two papers in this thesis treat extensions of the standard Markov-functional model to be able to use a higher dimensional driving process. This allows a more general understanding of the Markov-functional modelling framework and enables comparisons with multi-factor LIBOR market models. From a practical point of view it provides more powerful modelling of correlations among rates and hence a better examination and control of some types of exotic products. Another desire among practitioners is to develop an efficient way of using a process of stochastic volatility type as a driver in a Markov-functional model. A stochastic volatility Markov-functional model has the virtue of both being able to fit current market prices across strikes and to provide better control over the future evolution of rates and volatilities, something which is important both for pricing of certain products and for risk management. Although there are some technical challenges to be solved in order to develop an efficient stochastic volatility Markov-functional model there are also many (more practical) considerations to take into account when choosing which type of driver to use. To shed light on this the third paper in the thesis performs a data driven study in order to motivate and develop a suitable two-dimensional stochastic volatility process for the level of interest rates. While the main part of the paper is general and not directly linked to any complete interest rate model for exotic derivatives, particular care is taken to examine and equip the process with properties that will aid use as a driver for a stochastic volatility Markov-functional model. / <p>Diss. Stockholm :  Stockholm School of Economics, 2011. Introduction together with 3 papers</p>

Instantaneous rate models, although theoretically satisfying, are less so in practice. Instantaneous rates are not observable and calibra- tion to market data is complicated. Hence, the need for a market model where one models LIBOR rates seems imperative. In this modeling process, we aim at regaining the Black-76 formula[7] for pricing caps and °oors since these are the ones used in the market. To regain the Black-76 formula we have to model the LIBOR rates as log-normal processes. The whole construction method means calibration by using market data for caps, °oors and swaptions is straightforward. Brace, Gatarek and Musiela[8] and, Miltersen, Sandmann and Sondermann[25] showed that it is possible to con- struct an arbitrage-free interest rate model in which the LIBOR rates follow a log-normal process leading to Black-type pricing for- mulae for caps and °oors. The key to their approach is to start directly with modeling observed market rates, LIBOR rates in this case, instead of instantaneous spot rates or forward rates. There- after, the market models, which are consistent and arbitrage-free[6], [22], [8], can be used to price more exotic instruments. This model is known as the LIBOR Market Model. In a similar fashion, Jamshidian[22] (1998) showed how to con- struct an arbitrage-free interest rate model that yields Black-type pricing formulae for a certain set of swaptions. In this particular case, one starts with modeling forward swap rates as log-normal processes. This model is known as the Swap Market Model. Some of the advantages of market models as compared to other traditional models are that market models imply pricing formulae for caplets, °oorlets or swaptions that correspond to market practice. Consequently, calibration of such models is relatively simple[8]. The plan of this work is as follows. Firstly, we present an em- pirical analysis of the standard risk-neutral valuation approach, the forward risk-adjusted valuation approach, and elaborate the pro- cess of computing the forward risk-adjusted measure. Secondly, we present the formulation of the LIBOR and Swap market models based on a ¯nite number of bond prices[6], [8]. The technique used will enable us to formulate and name a new model for the South African market, the SAFEX-JIBAR model. In [5], a new approach for the estimation of the volatility of the instantaneous short interest rate was proposed. A relationship between observed LIBOR rates and certain unobserved instantaneous forward rates was established. Since data are observed discretely in time, the stochastic dynamics for these rates were determined un- der the corresponding risk-neutral measure and a ¯ltering estimation algorithm for the time-discretised interest rate dynamics was pro- posed. Thirdly, the SAFEX-JIBAR market model is formulated based on the assumption that the forward JIBAR rates follow a log-normal process. Formulae of the Black-type are deduced and applied to the pricing of a Rand Merchant Bank cap/°oor. In addition, the corre- sponding formulae for the Greeks are deduced. The JIBAR is then compared to other well known models by numerical results. Lastly, we perform some computational analysis in the following manner. We generate bond and caplet prices using Hull's [19] stan- dard market model and calibrate the LIBOR model to the cap curve, i.e determine the implied volatilities ¾i's which can then be used to assess the volatility most appropriate for pricing the instrument under consideration. Having done that, we calibrate the Ho-Lee model to the bond curve obtained by our standard market model. We numerically compute caplet prices using the Black-76 formula for caplets and compare these prices to the ones obtained using the standard market model. Finally we compute and compare swaption prices obtained by our standard market model and by the LIBOR model. / Economics / D.Phil. (Operations Research)

This thesis focuses on the non-arbitrage (fair) pricing of interest rate derivatives, in particular caplets and swaptions using the LIBOR market model (LMM) developed by Brace, Gatarek, and Musiela (1997) and Swap market model (SMM) developed Jamshidan (1997), respectively. Today, in most financial markets, interest rate derivatives are priced using the renowned Black-Scholes formula developed by Black and Scholes (1973). We present new pricing models for caplets and swaptions, which can be implemented in the financial market other than the Black-Scholes model. We theoretically construct these "new market models" and then test their practical aspects. We show that the dynamics of the LMM imply a pricing formula for caplets that has the same structure as the Black-Scholes pricing formula for a caplet that is used by market practitioners. For the SMM we also theoretically construct an arbitrage-free interest rate model that implies a pricing formula for swaptions that has the same structure as the Black-Scholes pricing formula for swaptions. We empirically compare the pricing performance of the LMM against the Black-Scholes for pricing caplets using Monte Carlo methods.

Title: Stochastic interest rates modeling Author: Jakub Černý Abstract: This present work studies different stochastic models of interest rates. Theoretical part of this work describes short-rate models, HJM fra- mework and LIBOR Market model. It focuses in detail on widely known short-rate models, i.e. Vašíček, Hull-White and Ho-Lee model, and on LI- BOR Market model. This part ends by valuation of interest rate options and model calibration to real data. Analytical part of the work analyses valuation of real non-standard interest rate derivative using different models. Part of this derivative valuation is comparison among models in terms of general valuation and also in terms of capturing the dynamics of interest rates. The aim of this work is to describe different stochastic models of interest rates and mainly to compare them with each other.

傳統上描述利率期間結構，不外乎藉由瞬間短期利率的隨機過程（如：Hull and White模型），或瞬間遠期利率的隨機過程（如：HJM模型）。應用這些方式理論上雖然可行，但是市場上並無法觀察得知這些瞬間利率。 Brace-Gatarek-Musiela利率模型（簡稱BGM模型）是將HJM模型間斷化，直接推導市場上可觀察得到之LIBOR利率的隨機過程，用它來描述市場利率期間結構，並利用數學的技巧，推導出符合對數常態的型式，方便使用Black公式來求解，且同時考慮LIBOR利率之波動程度，透過與市場資料的校準，符合市場上的利率期間結構及利率波動結構，有助於利率衍生性商品的訂價與避險。 由於市場上有愈來愈多的利率衍生性商品，不是由單純的cap、swaption來組成，例如：路徑相依選擇權、美式選擇權、回顧型選擇權…等，這些新奇選擇權要求出評價公式很難，所以通常使用數值方法來評價。常用的數值方法有蒙地卡羅模擬法及樹狀圖評價法，由於使用蒙地卡羅模擬法處理起來較耗時，而且評價美式選擇權比較麻煩，而樹狀圖評價法較省時，且應用較廣。因此，本文除了詳細推導BGM利率模型，並建構出BGM利率模型下的利率樹，來對這些新奇選擇權做評價。 最後做一實證分析，以市場上的所發行的利率連動債券為例，對於匯豐銀行美元護本109利率連動債券的設計、評價、損益分析及其相關議題做詳盡的探討。

在中國金融市場逐步開放，結構型理財商品的發行與需求日益增加的情形下，本文以目前市場上已發行的利率連結商品與匯率連結商品為個案，進行評價與分析。在利率連結商品方面，以連結6個月美元LIBOR利率的「美元12個月期逆浮動利率連動債」為例，採用Brace, Gaterek and Musiela（1997）提出的LIBOR市場模型（又稱BGM模型），由市場觀察到的即期LIBOR利率與交換利率，求出遠期利率的起始值，並利用市場上利率上限選擇權（CAP）報價，校準遠期利率波動度結構，而遠期利率間的相關係數矩陣則以歷史資料來估計，然後以蒙地卡羅模擬法進行商品評價；在匯率連結商品方面，以連結日元兌澳元、英鎊、歐元匯率的「美元三個月期組合匯率理財專案」為例，採用Garman and Kohlhagen（1983）外匯選擇權的匯率動態過程，利用歷史資料求出各匯率變動率波動度以及各匯率間的相關係數矩陣，然後以蒙地卡羅模擬法進行商品評價。此外亦針對兩種商品的敏感性與避險參數作分析，最後分別由發行商與投資人的觀點，探討其發行與投資該商品的策略與風險所在。