Option pricing and hedging with minimum local expected shortfall

We propose a versatile Monte-Carlo method for pricing and hedging options when the market is incomplete, for an arbitrary risk criterion (chosen here to be the expected shortfall), for a large class of stochastic processes, and in the presence of transaction costs. We illustrate the method on plain vanilla options when the price returns follow a Student-t distribution. We show that in the presence of fat-tails, our strategy allows to significantly reduce extreme risks, and generically leads to low Gamma hedging. Similarly, the inclusion of transaction costs reduces the Gamma of the optimal strategy.Comment: 23 pages, 7 figures, 8 table

Processus multifractals en finance et valorisation d'options par minimisation de risques extrêmes

PALAISEAU-Polytechnique (914772301) / SudocSudocFranceF

The skewed multifractal random walk with applications to option smiles

We generalize the construction of the multifractal random walk (MRW) due to Bacry et al (Bacry E, Delour J and Muzy J-F 2001 Modelling financial time series using multifractal random walks Physica A 299 84) to take into account the asymmetric character of financial returns. We show how one can include in this class of models the observed correlation between past returns and future volatilities, in such a way that the scale invariance properties of the MRW are preserved. We compute the leading behaviour of q-moments of the process, which behave as power laws of the time lag with an exponent ζq=p-2p(p-1)λ2 for even q=2p, as in the symmetric MRW, and as ζq=p + 1-2p2λ2-α (q=2p + 1), where λ and α are parameters. We show that this extended model reproduces the 'HARCH' effect or 'causal cascade' reported by some authors. We illustrate the usefulness of this 'skewed' MRW by computing the resulting shape of the volatility smiles generated by such a process, which we compare with approximate cumulant expansion formulae for the implied volatility. A large variety of smile surfaces can be reproduced.