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Register a new userShapes of implied volatility with positive mass at zero
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We study the shapes of the implied volatility when the underlying distribution has an atom at zero and analyse the impact of a mass at zero on at-the-money implied volatility and the overall level of the smile. We further show that the behaviour at small strikes is uniquely determined by the mass of the atom up to high asymptotic order, under mild assumptions on the remaining distribution on the positive real line. We investigate the structural difference with the no-mass-at-zero case, showing how one can--theoretically--distinguish between mass at the origin and a heavy-left-tailed distribution. We numerically test our model-free results in stochastic models with absorption at the boundary, such as the CEV process, and in jump-to-default models. Note that while Lees moment formula tells that implied variance is at most asymptotically linear in log-strike, other celebrated results for exact smile asymptotics such as Benaim and Friz (09) or Gulisashvili (10) do not apply in this setting--essentially due to the breakdown of Put-Call duality.
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This paper presents how to apply the stochastic collocation technique to assets that can not move below a boundary. It shows that the polynomial collocation towards a lognormal distribution does not work well. Then, the potentials issues of the related collocated local volatility model (CLV) are explored. Finally, a simple analytical expression for the Dupire local volatility derived from the option prices modelled by stochastic collocation is given.
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We derive a backward and forward nonlinear PDEs that govern the implied volatility of a contingent claim whenever the latter is well-defined. This would include at least any contingent claim written on a positive stock price whose payoff at a possibly random time is convex. We also discuss suitable initial and boundary conditions for those PDEs. Finally, we demonstrate how to solve them numerically by using an iterative finite-difference approach.
In informationally efficient financial markets, option prices and this implied volatility should immediately be adjusted to new information that arrives along with a jump in underlyings return, whereas gradual changes in implied volatility would indicate market inefficiency. Using minute-by-minute data on S&P 500 index options, we provide evidence regarding delayed and gradual movements in implied volatility after the arrival of return jumps. These movements are directed and persistent, especially in the case of negative return jumps. Our results are significant when the implied volatilities are extracted from at-the-money options and out-of-the-money puts, while the implied volatility obtained from out-of-the-money calls converges to its new level immediately rather than gradually. Thus, our analysis reveals that the implied volatility smile is adjusted to jumps in underlyings return asymmetrically. Finally, it would be possible to have statistical arbitrage in zero-transaction-cost option markets, but under actual option price spreads, our results do not imply abnormal option returns.
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