本文針對目前金融市場上已發行的可贖回區間計息連動債券，進行個案評價與分析，希望能讓一般投資人更了解市面上此類型商品的報酬型態，以及潛在的投資風險，並站在發行商的角度，進行商品利潤及避險敏感度的探討。 在模型建構的部分，本文採用Lognormal Forward LIBOR Model ( LFM ) 利率模型，並使用由Longstaff and Schwartz( 2001 )所提出的最小平方蒙地卡羅法，來處理同時具有可贖回與路徑相依特性的商品評價。 此外，關於避險參數部分，由於區間計息商品的報酬型態不具有連續性，若在一般蒙地卡羅法之下直接使用重新模擬的方式來求算，將會造成不準確的結果，Piterbarg（2004b）因而提出Sausage Monte Carlo近似法來加以改善。而經過實證研究，在此方法下所求出的避險參數，將有較佳的穩定性與準確度，本文將其運用至可贖回區間計息連動債券，分析不同方法下的避險參數，並就評價結果提供投資人與發行人建議。

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

This article presents a new approach for calculating sensitivities of European swaptions. The sensitivities are found by applying an adjoint method to a stochastic volatility model, namely the SABR/LIBOR market model. This market model predicts the volatility smile and follows the market fluctuations more accurately than earlier used deterministic volatility market models for complex derivatives. The new adjoint method involves not only sensitivity calculations, it also presents a way of estimating the time discretization error using an a posteriori approach. The error calculation is described in this document but not investigated further. The first step in order to calculate the sensitivities is to calibrate the SABR/LIBOR market model to some market data. In our calculations we used data from June 15 2011 with 6 month intervals between the maturity times. When this calibration is complete all of the parameters in the SABR/LIBOR market model are specified and we can continue with the sensitivity calculations using the new adjoint method. The results from these calculations show that the method is a good choice for estimating sensitivities if we consider a complex financial derivative like the European swaption. The method is quite computational so we recommend that it is only used on a small number of securities with respect to a large number of parameters. The method provides more market-driven price and sensitivity estimations than earlier used methods and can benefit hedging of portfolios.

The main contribution of the diploma thesis is to give a comprehensive picture of the switch from LIBOR to OIS discounting. Prior to the global financial crisis, LIBOR (London Interbank Offered Rate) represented an approximation of the risk-free rate in the valuation of interest rate derivatives. The collapse of Lehman Brothers in 2008 resulted in sharp widening of the LIBOR-OIS spread, an indicator of the interbank market stress. Many derivative practitioners have become concerned about the choice of an appropriate risk-free rate. Traditional valuation approaches using LIBOR discounting have been reviewed. Meanwhile, the OIS (Overnight Indexed Swap) rate has become a better proxy for the risk-free rate, at least for collateralized or centrally cleared transactions. Firstly, the research aims to discover the divergences between LIBOR rates, popular pre-crisis proxies for the riskfree rate, and OIS rates, their post-crisis alternatives. Secondly, it covers the interbank lending market, and analyzes individual LIBOR-OIS spreads for the USD, EUR, GBP and CZK currency. Thirdly, it explores the transition to OIS discounting in connection with an influence on a wide spectrum of interest rate derivatives. Therefore, any potential effects are demonstrated on numerical valuation examples of interest rate swaps in the USD, EUR, and GBP currency. Finally, the diploma thesis addresses a topic of collateral management and clarifies different approaches using LIBOR or OIS rates for collateralized or non-collateralized transactions.

由於近幾年連動式債券的盛行，要如何在眾多的投資商品中找到適合自己的標的，對投資人來說越來越重要。本篇論文選擇目前市面上常見的兩種利率連動債來做評價與分析。一為CMS連結債券，另一為滾雪球型連動債券。 在各個利率模型中，由於BGM Model具備了良好的評價特性，因此成為本論文評價的依據。另外，處理利率動態過程中各個遠期利率相關係數時，本論文採用了Peter Weigel(2004)有關於將市場利率相關係數矩陣降秩的方法，以便增進運算時的效率。最後，考量此兩種連動債券都具備了可贖回的權利，再加上評價時是利用蒙地卡羅來進行模擬，因此採用了最小平方蒙地卡羅來處理可贖回債券的評價。進行一萬次模擬後，可以得出標的債券的價格，之後再對各個因素進行敏感度分析，可以發現影響商品價格的各種原因。 根據評價的結果，可以針對投資人與發行商做不同的策略建議。對投資人而言，在購買此類商品時需注意商品是否具有贖回條款，並且未來的利率走勢是否會大幅影響投資收益。對發行商而言，一般所發行的連動式債券大多較複雜，很難直接在市場上找到可供避險的商品，因此除了利用回權的方式外，用現在市場的的商品來做部分避險也是另一種選擇。

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。

In this thesis, the interest rates derivatives and their valuation based on the future development of interest rates are presented. The Hull-White model focusing on the modeling of the instantaneous spot rates is described in detail. The model is calibrated to the market caplet volatilities and is used to evaluate various interest rates derivatives. The main emphasis is put on the LIBOR market model describing the development of set of forward rates. There are presented and in detail discussed results of the calibration of LMM model on the market swaption volatilities. At the end the two models are compared.