Higher-Order Volatility

Higher-Order Volatility: Time Series

“…Here we recall some basic definitions and results from Carey (2005Carey ( , 2006. Let X t ( ) be a positive-valued adapted stochastic process on a filtered probability space !, F , F t ( ),Q ( ) , t !…”

“…j,t = lim "t!0 # j,t denote j -th order instantaneous volatility. In Carey (2005) a simple thought experiment shows how to interpret ! j,t and its limit !…”

“…X t ( ) derivation. This formula and a related one were originally obtained in Carey (2005Carey ( , 2006 under the assumption that the finite-period volatilities are deterministic for very small time horizons !t . While not as clean as the assumption we make above, it perhaps carries more intuition.…”

“…This choice is motivated both by the analytical simplification that results, and by the empirical behaviour of the marked-implied ! which, in the equity index setting at least, tends to decrease as time to expiration increases (see Carey (2005)). The option pricing formula then simplifies to:…”

“…A general formula giving j -th order volatility in terms of the Merton jumpdiffusion parameters can be found in Carey (2005). With this formula, j -th order volatility is readily computed from the jump-diffusion parameters.…”