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Deviations of price processes from the lognormal behaviour, as postulated by the Black and Scholes model, have been observed both on the stock market and exchanges. This fact along with the empirical observation that the volatility of an asset price is a stocastic variable itself makes the problem of pricing options still open and extremely interesting both theoretically and for practical
applications.
The Black and Scholes option pricing theory has two remarkable features: the hedging strategy eliminates risk entirely, and the option price does not depend on the average return of the underlying.
This has lead to a general framework for derivative pricing, where the absence of arbitrage opportunities leads to the existence of a ‘risk-neutral probability measure’ over which the relevant average should be taken to obtain the price of derivative. However, in most models of stock fluctuations (except for continuous time Brownian motion and binomial)
risk in option trading cannot be eliminated. The natural framework for pricing in this case is the risk minimisation approach, where the optimal trading strategy is determined such that the chosen measure of risk is minimized.
In collaboration with J.P.Bouchaud and D.Sornette, we proposed a method for option pricing and hedging using historical probability distributions. We addressed non-perfect arbitrage, market friction and the presence of ’fat’ tails in incomplete market under transaction costs. An implicit volatility ’smile’ has been predicted. We have given a precise estimate of the residual risk associated with optimal (but imperfect) hedging and of the optimal trading time.
In collaboration we M. Jeanin and D. Samuel we investigate the effect of hedging strategies on the so called
pinning effect, i.e. the tendency of stock's prices to close near the strike price of heavily traded options as the expiration date nears.
We adopt a model introduced by Frey and Stremme (1997) and show that under the original assumptions of the model pinning is driven by two effects: a hedging dependent drift term that pushes the stock price toward the strike price and a hedging dependent volatility term that constrains the stock price near the strike as it approaches it. Finally we show that pinning can be generated by simulating trading in a double auction market. Pinning in the microstructure model is consistent with the Frey and Stremme model when both discrete hedging and stochastic impact are taken into account.