隨著財務工程學的發展，結構型商品的架構日趨複雜，連結標的也更加多元化，可依投資人對市場未來的預期，設計出不同的商品型態，滿足投資人財富管理的需求或企業理財的規劃。但因為一般投資人不容易了解結構型商品複雜的設計，可能發生投資報酬不符預期或忽略商品潛在風險的情況。 本論文以建華銀行「美金組合式商品連動債券」與「固定期限交換利率利差連動債券」為例，進行評價與避險分析，以互換選擇權推導極小值選擇權的評價方法推導次小值選擇權的封閉解，並與蒙地卡羅模擬結果相較；利用對數常態遠期LIBOR利率模型評價連結固定期限交換利率的商品。最後進行投資與避險策略分析。希望能增進投資人對商品風險與報酬的認識，和提供金融機構未來設計相關類型商品時，對於評價與避險之理論基礎和方法的一個參考。

本論文第一章將 LIBOR 市場模型加入股價動態，並求出其風險中立過程下的動態模型，並利用此模型評價股籌交換契約。第二章將 LIBOR 市場模型擴展成兩國的市場模型，加入兩國股價動態，並求出風險中立過程下的動態模型，並利用此模型評價跨國股籌交換契約。本論文第二部份說明如何實際使用此模型，並使用蒙地卡羅模擬檢驗此評價模型的正確性。 / This thesis includes two main chapters. Chapter 2 is entiled as "Equity Swaps in a LIBOR Market Model" and Chapter 3 is entitled as "Cross-Currency Equity Swaps in a LIBOR Market in a Model". The conclusions of this thesis are made in Chapter 4. In Chapter 2, we extends the BGM (Brace, Gatarek and Musiela (1997))interest rate model (the LIBOR market model) by incorporating the stock price dynamics under the martingale measure. As compared with traditional interest rate models, the extended BGM model is easy to calibrate the model parameters and appropriate for pricing equity swaps. The general framework for pricing equity swaps is proposed and applied to the pricing of floating-for-equity swaps with either constant or variable notional principals. The calibration procedure and the practical implementation are also discussed. In Chapter 3, under the arbitrage-free framework of HJM, we simultaneously extends the BGM model (the LIBOR market model) from a single-currency economy to a cross-currency case and incorporates the stock price dynamics under the martingale measure. The resulting model is very general for pricing almost every kind of (cross-currency) equity swaps traded in OTC markets. The calibration procedure and the hedging strategies are also provided in this paper for practical operation. The pricing formulas of the equity swaps with either a constant or a variable notional principal and with hedged or un-hedged exchange rate risk are derived and discussed as examples.

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.

目前中國大陸金融市場伴隨著經濟體日趨成熟與影響力的情形之下，已逐漸開放，結構型理財產品多元化發展，使中國大陸的衍生性商品市場越趨活躍，相繼陸續開放承作各類型的金融商品，金融市場更加完備。不論是發行商或是投資人，皆應更深入了解結構型理財產品，才有助於中國大陸衍生性商品市場的拓展。 本文針對中國大陸金融市場上已發行的股票掛鈎型與利率掛鈎型的結構型理財產品進行個案評價與風險分析，第一個評價個案為「追蹤能源類股股權掛鈎結構型理財產品」，以封閉解作為評價基礎，並採用蒙地卡羅法模擬本商品的提前到期機率；第二個評價個案為「每日計息雙區間可贖回結構型理財產品」，採用BGM模型進行評價，並利用最小平方蒙地卡羅模擬法，考慮發行商提前贖回條款並計算每一期的配息；分別對兩個個案進行評價，針對評價結果分析發行商的利潤與避險策略，並給予投資人投資與避險的建議。

大陸的金融市場近年開放快速，推出的產品也漸趨多樣化，主要在市場上較為活躍的是股權衍生性商品與利率衍生性商品。在2007下半年，爆發次級房貸風暴，美國採取降息的手段來挽救經濟，美國的經濟仍屬疲弱，對於全球，乃至於大陸都有一定程度的影響。此時，對一般投資人而言，投資若能同時具有保本的功能，將具有相當的吸引力。 本文主要是針對大陸金融市場已發行的衍生性商品做為評價與分析，能夠讓一般的投資人知道目前大陸的金融產品發展的情況。並且分析商品適合的投資人，以及所面對的風險；對發行者來說，則探討其獲利表現與發行策略的分析。 分析的產品為興業銀行發行的掛鉤紅籌股的結構債與東亞銀行發行的每日區間計息債券以3個月期LIBOR標的。分別以蒙地卡羅模擬法與LIBOR Market Model(也稱為BGM模型)進行分析，探討發行者的利潤與何種投資人適合購買。

本文分別評價了中國大陸地區發行之利率結構型商品及股權結構型商品，並針對其風險及條款設計進行分析。文中所選的利率結構型商品為「每日計息利率連動理財產品」，在對數常態遠期LIBOR模型的假設下，我們先利用市場報價校準參數化之波動度及相關係數函數，再使用最小平方法蒙地卡羅模擬利率路徑，以處理此商品的提前贖回條件；為了產生非標準期間之遠期LIBOR利率，在模擬過程中加入了Brigo和Mercurio(2006)提到的漂移項插補法。另一個股權結構型商品為「人民幣A股多資產連結理財產品」，由於此商品連結標的多達五個，本文中使用風險中立下股價的動態過程，以及蒙地卡羅模擬來求算其合理價格。此外，針對這兩個商品所需要注意的風險，本文皆提出了建議。

隨著低利率時代的來臨，投資人不能再從定存或證券中獲取高報酬率，在另一方面，許多的結構型商品相繼出現，如高收益票券、投資型定存、投資型保單等，打著高收益的稱號來吸引市場上的投資人購買。但是許多投資人持有負面的見解，認為此種商品並非無風險，甚至時常出現血本無歸的情形，究竟投資人如何在眾多商品中選擇出最有利的商品？另外，近年來金融業的商品朝向國際化與多元化發展，但是國內銀行及券商能夠承作金融商品創新及設計有限，不僅無法滿足國內投資人，對於證券商與銀行業來說也有不利的影響，因此健全結構型商品的發展才能使得金融業，證券商與一般投資人三贏的局面。 / 本文分別評價了ING銀行發行之利率結構型商品及元大證券之股權結構型商品，並針對其風險及條款設計進行分析。文中所選的利率結構型商品為「ING五年期目標贖回連動債券」，在對數常態遠期LIBOR模型的假設下，我們先利用市場報價校準參數化之波動度及相關係數函數，再使用最小平方法蒙地卡羅模擬利率路徑，以處理此商品的提前贖回條件。另一個股權結構型商品為「「絕對保富」結構型商品」，由於此商品連結標的多達三個，本文中使用風險中立下股價的動態過程，以及蒙地卡羅模擬來求算其合理價格。此外，針對這兩個商品所需要注意的風險，本文皆提出了建議。