Extreme Returns in Asset Prices


Throughout this chapter, we assume that speculative prices s t like those pertaining to stocks, foreign currencies, futures etc. are evaluated at discrete times t = 0, 1, 2, . . ., where the periods can be days or weeks. Thus, if s 0 is the price of an investment at time t = 0, then the return—the difference of prices taken relatively to the initial price—at time T is (s Ts 0)/s 0. Our primary interest concerns daily returns under discrete compounding (arithmetic returns)
$$ \tilde r_t = \frac{{s_t - s_{t - 1} }} {{s_{t - 1} }}$$
or the daily returns under continuous compounding (log-returns)
$$ r_t = \log (s_t ) - \log (s_{t - 1} ).$$
These quantities are close to each other if the ratio s t/s t−1 is close to 1. We will focus on the latter concept. Log-returns are also called geometric returns in the financial literature.


Asset Price Stochastic Volatility GARCH Model Tail Index Return Series 
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