Retrieving risk neutral moments and expected quadratic variation from option prices

Abstract

This paper derives exact formulas for retrieving risk neutral moments of future payoffs of any order from generic European-style option prices. It also provides an exact formula for retrieving the expected quadratic variation of the stock market implied by European option prices, which nowadays is used as an estimate of the implied volatility, and a formula approximating the jump component of this measure of variation. To implement the above formulas to discrete sets of option prices, the paper suggests a numerical procedure and provides upper bounds of its approximation errors. The performance of this procedure is evaluated through a simulation and an empirical exercise. Both of these exercises clearly indicate that the suggested numerical procedure can provide accurate estimates of the risk neutral moments, over different horizons ahead. These can be in turn employed to obtain accurate estimates of risk neutral densities and calculate option prices, efficiently, in a model-free manner. The paper also shows that, in contrast to the prevailing view, ignoring the jump component of the underlying asset can lead to seriously biased estimates of the new volatility index suggested by the Chicago Board Options Exchange.

The Aggregation Property and its Applications to Realised Higher Moments

“…[2013] investigate the convergence of the discretely-monitored swap rate to its continuouslymonitored counterpart and derive bounds on ε that get tighter as the monitoring frequency increases; generalise these results and provide conditions for signing ε; Hobson and Klimmek [2012] derive model-free bounds for ε; Broadie and Jain [2008] derive fair-value swap rates for discretely-monitored variance swaps under various stochastic volatility diffusion and jump models, claiming that for most realistic contract specifications ε is smaller than the error due to violation of assumption (b); extend their analysis to include a much wider variety of processes by considering the asymptotic expansion of ε. Finally, Rompolis and Tzavalis [2017] derive bounds for the jump error ι and demonstrate, via simulations and an empirical study, that price jumps induce a systematic negative bias which is particularly apparent when there are large downward jumps.…”

Model-Free Discretisation-Invariant Swap Contracts

“…[2013] investigate the convergence of the discretely-monitored swap rate to its continuouslymonitored counterpart and derive bounds on δ that get tighter as the monitoring frequency increases; generalise these results and provide conditions for signing δ; Hobson and Klimmek [2012] derive model-free bounds for δ; Broadie and Jain [2008] derive fair-value swap rates for discretely-monitored variance swaps under various stochastic volatility diffusion and jump models, claiming that for most realistic contract specifications δ is smaller than the error due to violation of assumption (b); extend their analysis to include a much wider variety of processes by considering the asymptotic expansion of δ. Finally, Rompolis and Tzavalis [2013] derive bounds for the jump error ι and demonstrate, via simulations and an empirical study, that price jumps induce a systematic negative bias which is particularly apparent when there are large downward jumps. Neuberger [2012] finds a way to avoid the errors arising from assumptions (a) and (b): by discarding the conventional definition of realised variance and using instead the log variance pay-off function λ (x) := 2 e x − 1 − x where x denotes the log return.…”

A general property for time aggregation

“…(1) the general properties that must be satisfied by any aggregating function of any multivariate 3 Assuming (hypothetically) that monitoring of the RV is continuous and that there are no jumps in the forward price, a unique fair-value variance swap rate is derived from market prices of traded options using the Carr and Madan [2001] replication integral. However, in the real world neither of these assumptions hold and many papers investigate the discretisation error: [Broadie and Jain, 2008, Carr and Wu, 2009, Carr and Lee, 2009, Davis et al, 2014, Hobson and Klimmek, 2012, Jarrow et al, 2013; and jump error [Ait-Sahalia, 2004, Rompolis andTzavalis, 2017] and many others. The combination of these errors lead to large deviations between fair-value and market swap rates.…”