Naoto Kunitomo

The asymptotic expansion approach to the valuation of interest rate contingent claims

We propose a new methodology for the valuation problem of financial contingent claims when the underlying asset prices follow a general class of continuous Ito processes. Our method can be applied to a wide range of valuation problems including complicated contingent claims associated with the term structure of interest rates. We illustrate our method by giving two examples: the valuation problems of swaptions and average (Asian) options for interest rates. Our method gives some explicit formulas for solutions, which are sufficiently numerically accurate for practical purposes in most cases. The continuous stochastic processes for spot interest rates and forward interest rates are not necessarily Markovian nor diffusion processes in the usual sense; nevertheless our approach can be rigorously justified by the Malliavin–Watanabe Calculus in stochastic analysis.

Effects Of Stochastic Interest Rates And Volatility On Contingent Claims

We investigate the effects of the stochastic interest rates and the volatility f the underlying asset price on the contingent claim prices including futures and options prices. The futures price can be decomposed into the forward price and the additional terms and the options price can be decomposed into the Black-Scholes formula and several additional terms via the asymptotic expansion approach in the small disturbance asymptotics developed by Kunitomo and Takahashi(1995,1998,2001), which is based on Malliavin-Watanabe Calculus in stochastic analysis. We illustrate our new formulae and their numerical accuracy by using some modi ed CIR type processes for the short term interest rates and stochastic volatility.

Stationary and Non-stationary Simultaneous Switching Autoregressive Models with an Application to Financial Time Series

A common observation among economists on many economic time-series, including major financial time-series, is the asymmetrical movement between the downward phase and the upward phase of their sample paths. Since this feature of time irreversibility cannot be described by the Gaussian ARMA, ARIMA or ARCH time-series models, we propose stationary and non-stationary simultaneous switching autoregressive (SSAR) models, which are nonlinear switching time-series models. We discuss some properties of these time-series models and the estimation method for their unknown parameters. The asymmetrical conditional heteroscedasticity can be easily incorporated into the SSAR models. We also report a simple empirical result on Nikkei 225 Spot and Futures indices by using a non-stationary SSAR model. n n n nJEL Classification Numbers: C22, C32.

Estimation of Asymmetrical Volatility for Asset Prices: The Simultaneous Switching ARIMA Approach

The asymmetrical movement between the downward and upward phases of the sample paths of many financial time series has been commonly noted by economists. Since this feature cannot be described by the Autoregressive Integrated Moving-average (ARIMA) model and the Autoregressive Conditional Heteroskedastic (ARCH) model, we introduce a class of the Simultaneous Switching Autoregressive Integrated Moving-Average (SSARIMA) model with ARCH disturbances. The asymmetrical volatility function of financial time series with daily effects can easily be estimated by this modelling. We also report a simple empirical result on stock price daily indices of the Nikkei-225 and SP-500.