{'fraction_non_alphanumeric': 0.019784172661870502, 'fraction_numerical': 0.0, 'mean_word_length': 4.856140350877193, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Stochastic Volatility : Time Scales and Perturbations

Our goal is to address the following fundamental question in pricing and hedging derivatives. How traded call options, quoted in terms of implied volatilities, can be used to price and hedge more complicated contracts. One can approach this difficult problem in two different ways: modeling the evolution of the underlying or modeling the evolution of the implied volatility surface. In both cases one requires that the model is free of arbitrage. Modeling the underlying usually involves the specification of a multi-factor Markovian model under the risk-neutral pricing measure. The calibration to the observed implied volatilities of the parameters of that model, including the market prices of risk, is a challenging task because of the complex relation between call option prices and model parameters (through a pricing partial differential equation for instance). A main problem with this approach is to find the “right model” which will produce a stable parameter estimation. We like to think of this problem as the “(t, T,K)” problem: for a given present time t and a fixed maturity T , it is usually easy with low dimensional models to fit the skew with respect to strikes K. Getting a good fit of the term structure of implied volatility, that is when a range of observed maturities are taken into account, is a much harder problem which can be handled with a sufficient number of parameters and eventually including jumps in the model. The main problem remains: the stability with respect to t of these calibrated parameters. However this is an highly desirable quality if one wants to use the model to