In this paper, principal component analysis (PCA) is applied to three different parametrization of interest rates: zero rates, yield curve, and forward rates. This comparative study is complementary to Akahori, Aoki, and Nagata [1] where they claimed that, under the noarbitrage principle, yield curve cannot be a random walk. Conversely the forward curve could be a random walk. In our result of PCA, however, we observed that of the general beliefs. Our empirical results on the number of factors for the zero rates and the yield curve align with the general beliefs. This is a puzzle.

This article addresses the problem of approximating the price of options on discrete and continuous arithmetic averages of the underlying, i.e., discretely and continuously monitored Asian options, in local volatility models. A "path-integral"-type expression for option prices is obtained using a Brownian bridge representation for the transition density between consecutive sampling times and a Laplace asymptotic formula. In the limit where the sampling time window approaches zero, the option price is found to be approximated by a constrained variational problem on paths in time-price space. We refer to the optimizing path as the most-likely path (MLP). An approximation for the implied normal volatility follows accordingly. The small-time asymptotics and the existence of the MLP are also rigorously recovered using large deviation theory.

Principal component analysis (PCA) is a general method to analyse the factors of the term structure of interest rates. There are usually two or three factors. However, it is shown by Liu that when we apply PCA to forward rates, not spot rates, we need more factors to explain 95% of variability. In order to verify the robustness of this result, we introduce another method based on Fourier series, which is proposed by Malliavin and Mancino. The results reconfirm the observation of Liu with different data sets. In particular, the Fourier series method gives us similar results to PCA.

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In this paper, we perform an estimation of the volatility of the forward rate using the Fourier method proposed by Malliavin and Mancino [4], together with the classical Principal Component Analysis. We have observed similar anomaly with the one in Liu [2]. We also give a remark on the practical use of the Malliavin-Mancino method.