Bounds on Stock Price probability distributions in Local-Stochastic Volatility models

Details Bounds on Stock Price probability distributions in Local-Stochastic Volatility models Bounds on Stock Price probability distributions in Local-Stochastic Volatility models

2010-06-16

arxiv.org/abs/1006.3337v1

Economy

Quantitative Finance

Pricing of Securities

Scientific paper

We show that in a large class of stochastic volatility models with additional skew-functions (local-stochastic volatility models) the tails of the cumulative distribution of the log-returns behave as exp(-c|y|), where c is a positive constant depending on time and on model parameters. We obtain this estimate proving a stronger result: using some estimates for the probability that Ito processes remain around a deterministic curve from Bally et al. '09, we lower bound the probability that the couple (X,V) remains around a two-dimensional curve up to a given maturity, X being the log-return process and V its instantaneous variance. Then we find the optimal curve leading to the bounds on the terminal cdf. The method we rely on does not require inversion of characteristic functions but works for general coefficients of the underlying SDE (in particular, no affine structure is needed). Even though the involved constants are less sharp than the ones derived for stochastic volatility models with a particular structure, our lower bounds entail moment explosion, thus implying that Black-Scholes implied volatility always displays wings in the considered class of models. In a second part of this paper, using Malliavin calculus techniques, we show that an analogous estimate holds for the density of the log-returns as well.

Affiliated with

Also associated with

No associations

Rating

Bounds on Stock Price probability distributions in Local-Stochastic Volatility models does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Bounds on Stock Price probability distributions in Local-Stochastic Volatility models, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Bounds on Stock Price probability distributions in Local-Stochastic Volatility models will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-550892