This paper extends the affine class of term structure models to describe the joint dynamics of exchange rates and interest rates. In particular, the issue of how to reconcile the low volatility of interest rates with the high volatility of exchange rates is addressed. The incomplete market approach of introducing exchange rate volatility that is orthogonal to both interest rates and the pricing kernels is shown to be infeasible in the affine setting. Models in which excess exchange rate volatility is orthogonal to interest rates but not orthogonal to the pricing kernels are proposed and validated via Kalman filter estimation of maximal 5-factor models for 6 country pairs.

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According to past research utilizing Bitcoin and other cryptocurrencies, Bitcoin has been shown to lead most other cryptocurrencies in terms of price movements. However, existing studies tend to focus on the direction of the lead-lag relationship instead of the duration of the lead-lag time. Furthermore, they are handicapped by the reliance on low-frequency data such as daily prices. This paper showcases the measurement of the lead-lag duration between cryptocurrencies using ultra-high-frequency tick-by-tick data, via the pair of Bitcoin and Cardano. Tick-by-tick data bring unique challenges in terms of methodology. The vast majority of time series econometrics methods are designed for use with data collected at regularly spaced time intervals, such as every hour, every day, etc. Tick-by-tick data, on the other hand, are not synchronized in any way and do not arrive at consistently spaced time intervals. Consequently, an asynchronous data integration methodology is utilized to estimate the Bitcoin price lead over Cardano price for each month beginning in January 2019 and continuing through May 2021. The length of the lead time ranges from 16 seconds to 118 seconds, with an average of around 57 seconds. Throughout the study period, the lengths of the lead time manifest a general trend of decline, which is shown to be statistically significant via non-parametric tests. Testing of seasonal patterns turns out to be not significant. The methodology and the findings of this paper have implications for both academics and practitioners, for example, when studying and implementing statistical arbitrage with cryptocurrencies.

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