Models of the term structure of interest rates play a central role in the modern theory of pricing bonds and other interest rate claims. Term structure models based on the principle of no-arbitrage, especially those of the Heath-Jarrow-Morton (1992) class, have become very popular recently, both with academics and practitioners. Surprisingly however, although the implied volatility function plays a crucial role in these no-arbitrage term structure models, there is little systematic evidence to guide optimal model specification within this broad class. We study the implied volatility in the Heath-Jarrow-Morton framework using Eurodollar futures options data. We estimate a daily time series of forward rates within the HJM framework such that, by construction, the predicted futures prices from our model exactly match the observed futures prices. Next, we estimate a daily time series of volatility parameters such that the sum of squared errors between futures options prices predicted by the model and observed futures options prices is minimized. We use the six different volatility specifications suggested by Amin and Morton (1994) within the HJM class of models to price interest rate claims. Since the volatilities are the only unobservables, we use these models to infer the volatilities from the market prices of Eurodollar futures options over the 1987-1998 periods. The minimized sum of squared errors in the option prices is used as the measure of accuracy of each specific model. Each model differs from the others in its ability to match the market option prices and the time required for the computation. We compare the performances of the six volatility specifications in the accuracy-versus-computation time tradeoff. We document the systematic biases between the model and market prices as a function of option type, maturity, and moneyness. We also examine alternative numerical implementations of HJM models using the six volatility specifications. In particular, we analyze the impact on accuracy and computation time of using different numbers of time-steps. We also examine the effect of using time-steps of varying lengths within the same estimation procedure, and of ordering the time-steps in different ways. / Ph. D.

「凸性偏誤」（Convexity Bias），非債券的「凸性因子」（Convexity），來自利率非平行變動對債券價格的影響。對利率交換契約而言，有其特殊意義。是指利用一連串到期日連續的期貨契約，作為評價利率交換契約的模型，卻因為在期貨契約到期前，其隱含利率並不等於遠期利率的情況下，採用未經修正過的模型，將錯誤估算交換契約的價格。而此偏誤值因隨著到期日的增加，或利率的波動增高而逐漸擴大，呈曲線特性，故稱之為「凸性偏誤」（Convexity Bias）。 由於完整資料收集不易，本論文的重心就限於探討美國歷史上，從1994年至1996年間，美國聯邦準備理事會（Federal Reserve Board；FED），第一階段利息大幅變動期間，利率的變動對凸性偏誤的影響，並預測之後利率變動時，對利率交換契約價格的影響。旨在以實證資料作完整分析，希望藉此探討凸性偏誤是否也會因利率變動程度的不同，進而對利率交換契約價格產生不同程度的影響。並進一步利用簡單的模型，推算出準確的遠期利率，作為評價利率交換契約的指標。將來若利率發生變動，交換契約的交易雙方，也能因此得到正確的交換契約價格，進行交易或避險，以減低利率風險可能帶來的損失。

經由愈來愈多的實證研究發現，的確在利率的變動過程中，除了包含連續性行為，即遵循”擴散”模式(diffusion process)，亦包含了不連續性行為，也就是有著跳躍(jump)的情形發生。因此顯示出假設利率隨機過程僅為連續性的擴散模型已是不足夠的，跳躍-擴散模型(Jump-diffusion model)顯然會比純粹擴散模型有著更好的解釋能力。而市場模型(LIBOR market model)的提出，則說明了遠期LIBOR利率模型較能描述市場實際的利率型態，並且可方便使用市場資訊，進行模型參數校準。 所以本研究旨在以LIBOR market model 加上跳躍過程，即遠期LIBOR利率的跳躍-擴散模型，分別針對歐洲美元期貨與利率結構型債券中的滾雪球式累息債券建立評價方法。由於所選用動態模型的複雜度，使得封閉解的求出不易，因此在文中，最後是採用蒙地卡羅模擬法，求兩商品的數值解。在後續研究上，本文還挑出了幾個最直接影響商品價值的因素，如殖利率、波動度、跳躍幅度等，進行各種情境下商品價值的敏感度分析，以提供投資人與發行商在考量風險因子所在時的一個參考。