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Abstract. The latest generation of volatility derivatives goes beyond variance and volatility swaps and probes our ability to price realized variance and sojourn times along bridges for the underlying stock price process. In this paper, we give an operator algebraic treatment of this problem based on Dyson expansions and moment methods and discuss applications to exotic volatility derivatives. The methods are quite flexible and allow for a specification of the underlying process which is semi-parametric or even non-parametric, including state-dependent local volatility, jumps, stochastic volatility and regime switching. We find that volatility derivatives are particularly well suited to be treated with moment methods, whereby one extrapolates the distribution of the relevant path functionals on the basis of a few moments. We consider a number of exotics such as variance knockouts, conditional corridor variance swaps, gamma swaps and variance swaptions and give valuation formulas in detail.

A time-fractional diffusion process defined in a discrete probability setting is studied. Working in continuous time, the infinitesimal generators of random processes are discretized and the diffusion equation generalized by allowing the time derivative to be fractional, i.e. of non-integer order. The properties of the resulting distributions are studied in terms of the MittagLeffler function. We discuss the computation of these distribution functions by deriving new global rational approximations for the Mittag-Leffler function that account for both its initial Taylor series and asymptotic power-law tail behaviours. Furthermore, we derive integral representations for both the continuous and the discrete time-fractional distributions and use these to prove a convergence theorem.MSC 2010 : 26A33, 33E12, 35R11, 60G22

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