برآورد مدل عامل از ساختار مدت نرخ بهره جهشی: مورد بازار اوراق قرضه دولت تایوان

This paper examines the Ornstein–Uhlenbeck (O–U) process used by Vasicek, J. Financial Econ. 5 (1977) 177, and a jump-diffusion process used by Baz and Das, J. Fixed Income (Jnue, 1996) 78, for the Taiwanese Government Bond (TGB) term structure of interest rates. We first obtain the TGB term structures by applying the B-spline approximation, and then use the estimated interest rates to estimate parameters for the one-factor and two-factor Vasicek and jump-diffusion models. The results show that both the one-factor and two-factor Vasicek and jump-diffusion models are statistically significant, with the two-factor models fitting better. For two-factor models, compared with the second factor, the first factor exhibits characteristics of stronger mean reversion, higher volatility, and more frequent and significant jumps in the case of the jump-diffusion process. This is because the first factor is more associated with short-term interest rates, and the second factor is associated with both short-term and long-term interest rates. The jump-diffusion model, which can incorporate jump risks, provides more insight in explaining the term structure as well as the pricing of interest rate derivatives.

Financial variables such as stock prices, foreign exchange rates, and interest rates are conventionally assumed to follow a diffusion process with continuous time paths when pricing financial assets. Despite their attractive statistical properties and computation convenience, more and more empirical evidence has shown that pure diffusion models are not appropriate for these financial variables. For example, Ball and Torous, 1983, Jarrow and Rosenfeld, 1984, Ball and Torous, 1985a, Ball and Torous, 1985b, Akgiray and Booth, 1986, Lin and Yeh, 1997 and Jorion, 1988 all found evidence indicating the presence of jumps in the stock price process. Akgiray and Booth, 1988, Tucker and Pond, 1988 and Park et al., 1993 studied foreign exchange markets and concluded that the jump-diffusion process is more appropriate for foreign exchange rates. In pricing and hedging with financial derivatives, jump-diffusion models are particularly important, since ignoring jumps in financial prices will cause pricing and hedging risks. The jump-diffusion process is particularly meaningful for interest rates, since the interest rate is an important economic variable, which is, to some extent, controlled by the government as an instrument for its financial policy. Hamilton (1988) investigated US interest rates and found changes in regime for the interest rate process. Das (1994) found movements in interest rates display both continuous and discontinuous jump behavior. Presumably, jumps in interest rates are caused by several market phenomena, such as money market interventions by the Fed, news surprises, shocks in the foreign exchange markets, and so on. Classical term structure of interest rate models, such as the Vasicek (1977) model, the Cox et al., 1985 model, the Brennan and Schwartz (1978) model, and other extended models all assume that processes of state variables (such as the short-term interest rate, the long-term interest rate, or others), which drive interest rate fluctuations, follow various diffusion processes. Their assumptions are inconsistent with the a priori belief and empirical evidence regarding the existence of discontinuous jumps in interest rates. At a cost of additional complexity, Ahn and Thompson (1988) extended the CIR model by adding a jump component to the square root interest rate process. Using a linearization technique, they obtained closed-form approximations for discount bond prices. Similarly, Baz and Das (1996) extended the Vasicek model by adding a jump component to the Ornstein–Uhlenbeck (O–U) interest rate process, and obtained closed-form approximate solutions for bond prices by the same linearization technique. They also showed that the approximate formula is quite accurate. Although theoretical derivations for the jump-diffusion term structure models have been developed, the associated empirical work is quite limited. A formal model of the term structure of interest rates is necessary for the valuation of bonds and various interest rate options. More importantly, parameter values or estimates are required for the implementation of a specific model. To price interest rate options, with closed-form solutions or by numerical methods, one must have values of the parameters in the stochastic processes that determine interest rate dynamics. Hence parameter estimation is a very first step in the application and analysis of interest rate option pricing models. In this study, besides the classical Vasicek (1977) model, we also investigated a jump-diffusion process, which is a mixture of an O–U process with mean-reverting characteristics used by Vasicek (1977) and a compound Poisson jump process, for interest rates. Closed-form approximate solutions for discount bond prices were derived by Baz and Das (1996). The approximate model is essentially a one-factor term structure model. It has the disadvantage that all bond returns are perfectly correlated, and it may not be adequate to characterize the term structure of interest rates and its changing shape over time. However, the model at least can incorporate jump risks into the term structure model, making the model more complete relative to the pure diffusion Vasicek model. In addition, similar to the simple diffusion Vasicek model, the short-term interest rate can move to negative values under the extended jump-diffusion Vasicek model. Realizing the drawback of one-factor models, in this study, we extended both the Vasicek (1977) model and the Baz and Das (1996) one-factor jump-diffusion model to a multi-factor model. We then developed a methodology for estimating both the one-factor and two-factor Vasicek and jump-diffusion term structure of interest rate models and completed an empirical study for Taiwanese Government Bond (TGB) interest rates. The state variables (such as the instantaneous short-term interest rate, and other factors) that drive the term structure of interest rates dynamics were not observable, and the observed bond prices are functions of the state variables. Thus, we can use the change of variable technique to obtain the likelihood function in terms of the observed bond prices, in order to conduct a maximum likelihood estimate. The estimation procedure of this study is similar to Chen and Scott, 1993 and Pearson and Sun, 1994. In the empirical study, for analytical purposes, we first need to obtain the TGB term structure of interest rates (that is the zero-coupon bond yield curve). However, the term structure is not directly observable because most of the TGBs are not zero-coupon bonds. Thus the coupon bond price, which may contain coupon effects, can not provide a good substitute for calculating the term structure of interest rates. Fortunately, a coupon bond is nothing but a portfolio of zero-coupon bonds with maturity consistent with coupon dates. We, thus, can use a curve fitting methodology, called the B-spline approximation suggested by Shea (1984) and successfully generalized and adopted by Steeley, 1991 and Lin and Paxson, 1993, to fit the TGB term structure of interest rates. We use the prices of 45 TGBs to obtain the term structure from January 6, 1996 to August 29, 1998. We then use the estimated weekly interest rates on the 30-day, 180-day, 5- and the 10-year zero-coupon TGBs to estimate parameters in the one-factor and two-factor Vasicek and jump-diffusion models. Thus, our methodology is a two-stage approach for investigating factor models of the term structure of interest rates. In the first stage, the B-spline approximation is applied to the cross-sectional bond prices to obtain the weekly term structure of interest rates. We then use the longitudinal term structure data obtained in the first stage to estimate the term structure models. This approach is different from that of De Munnik and Schotman, 1994, Sercu and Wu, 1997 and Ferguson and Raymar, 1998, who used cross-sectional data to estimate the term structure models. Although the cross-sectional approach is quite straightforward, it has several limitations. First, the cross-sectional approach cannot easily be applied to multi-factor term structure models since it will be difficult to identify all parameters in its non-linear least squares estimation. Second the cross-sectional approach is inconsistent with inter-temporal equilibrium of the term structure models, in which parameters are assumed to be constant over time. Although one can pool the cross-sectional data and time-series data as done by De Munnik and Schotman (1994), to overcome this problem, problems will still arise since there will be too many parameters to be estimated. With our two-staged methodology, we can investigate the term structure dynamics more freely and flexibly. The first stage is purely statistical and is aimed at fitting the market term structure adequately. We then take the obtained term structures as exogenous to examine the economic meaning of the term structure models. The major disadvantage of the two-stage approach is that it is difficult to assess the estimation error, since efforts can be introduced in either of the two stages. The empirical results show that both the one-factor and two-factor Vasicek and jump-diffusion models are statistically significant, with the two-factor models fitting better. This is as expected, since one-factor models do not fit the versatile term structure of interest rates very well. For both the two-factor Vasicek and jump-diffusion models, compared with the second factor, the first factor exhibits characteristics of stronger mean-reversion, higher volatility, and more frequent and significant jumps in the case of the jump-diffusion process. This is because the first factor is more associated with short-term interest rates. The second factor is associated with both short- and long-term interest rates, which is as a general factor driving the whole term structure dynamics. There is not a great difference in fitting power of the two-factor Vasicek model and the jump-diffusion model, but the jump-diffusion model, which can incorporate jump risks, provide more insight in explaining the term structure as well as the pricing of interest rate derivatives. Following the internationalization and liberalization of financial markets, the Taiwanese capital market has become one of the most important markets in the Asia–Pacific area. Although the TGB market is small and not liquid compared with other developed bond markets, it is increasingly important and worthy of study. Moreover, since the assumption of an appropriate stochastic process for the interest rate and the estimation of its associated parameters is of critical importance when pricing and hedging with the term structure of interest rates and interest rate derivatives, the results and the methodology for estimating parameters in the jump-diffusion process have important implications for the area of financial engineering. The rest of this paper is organized as follows. Section 2 specifies the Vasicek and the jump-diffusion term structure of interest rate models, and extends the one-factor model to a multi-factor model. Section 3 presents the empirical methodology used in this study. Section 4 describes the B-spline approximation methodology for fitting the term structure of interest rates. Section 5 specifies the data, and analyzes the results of parameters estimation and term structure fitting. Section 6 is the summary of the study.

In this paper, we investigate the O–U process used by Vasicek (1977) and a jump-diffusion process, which is a mixture of an O–U process and a compound Poisson jump process, used by Baz and Das (1996), for the term structure of interest rates. We develop a methodology for estimating both the one-factor and two-factor Vasicek and jump-diffusion term structure models, and complete an empirical study for the TGB interest rates. Since the state variables that drive the term structure of interest rates dynamics are not observable, we use the change of variable technique to obtain the likelihood function in terms of the observed bond prices, and conduct a maximum likelihood estimate. We use a two-stage approach to investigate the term structure models. The first stage involves application of the B-spline approximation to cross-sectional bond prices to obtain weekly term structures. This is a purely statistical procedure with the goal of fitting the market term structure adequately. We then use the longitudinal term structure data obtained in the first stage as exogenous to estimate the term structure models. The major disadvantage of the two-stage approach is that it is difficult to assess the estimation error, since errors can arise in either of the two stages. The research sample contained weekly prices of 45 TGBs from January 6, 1996 to August 29, 1998. Having obtained the term structure by applying the B-spline approximation, we then use the estimated weekly interest rates on the 30-day, 180-day, 5- and the 10-year zero coupon TGBs to estimate parameters for the one-factor and two-factor Vasicek and jump-diffusion models. The results show that both the one-factor and two-factor Vasicek and jump-diffusion model are statistically significant, with the two-factor models fitting better. This is as expected, since one-factor models do not fit the versatile term structure of interest rates very well. For both the two-factor Vasicek and jump-diffusion models, compared with the second factor, the first factor exhibits characteristics of stronger mean reversion, higher volatility, and more frequent and significant jumps in the case of the jump-diffusion process. The first factor is more often associated with shorter-term interest rates, and the second factor is associated with both short-term and longer-term interest rates. There is not a great difference in the fitting power of the two-factor Vasicek model and the jump-diffusion model, but the jump-diffusion model, which can incorporate jump risks, provides more insight in explaining the term structure as well as the pricing of interest rate derivatives. Since the assumption of an appropriate stochastic process for the interest rate and the estimation of its associated parameters are of critical importance when pricing and hedging with the term structure of interest rates and interest rate derivatives, the results and the methodology for estimating parameters in the jump-diffusion process have important implications in the area of financial engineering.