Financial Mathematics for Cryptocurrencies

CHAPTER 11Price Return Models

This chapter develops canonical models for price returns, emphasizing their application to digital assets. We examine model properties, calibration challenges, and implications for risk management and derivative pricing in cryptocurrency markets.

• Present autoregressive models for return dynamics and discuss stationarity requirements.
• Introduce continuous-time price processes and compare their modeling trade-offs.
• Develop stochastic volatility frameworks and address practical calibration issues.

We begin with discrete-time autoregressive models and their application to cryptocurrency returns.

11.1 AR MODELS

Empirical analysis of price returns has evidenced autocorrelation, which is not expected for shorter horizons, as per SF1 in Chapter 10. This has led to the introduction of autoregressive models (AR models) that are used to model price returns [242]. These models are discrete.

Stationarity is an important concept in time series analysis, as it implies the stability of the statistical properties over time. While there are several formal definitions of stationarity, we will refer to weak stationarity¹ in this chapter.

Definition 11.1 (Stationarity [Weak]):

A time series is said to be weakly stationary (WSS) if it satisfies both the following conditions:

• Its mean is constant.
• The covariance between two lagged values is only a function of the lag.

Definition 11.1 leads to Corollary 11.1 on the variance of the time series.

Corollary ...