**W**hile chapter 1 highlighted the principles of equity pricing from a rather theoretical point of view, we want to focus now on practical aspects: we will discuss a few commonly used stochastic volatility models and applications to Cliquet pricing; we will also address the pricing of payoffs that depend on the realized variance of an asset. In particular, “variance swaps” have become very liquid instruments and trading volumes are set to grow even further. The respective options on variance are an attractive new class of products on which to work.

In section 1.3.2, we discussed how we can construct martingales that fit a given initial option price surface, the most popular approach being Dupire's implied local volatility. We have already mentioned that in practice, it is rarely possible to obtain a continuum of option prices. Another problem with using an “implied” model is that it does not allow us to control the specific dynamics of the resulting actual stock price process. In this sense, we want to stress that a model that fits very well to some market does not at all guarantee that it produces acceptable prices: for example, consider a stock for which only forwards are traded, but no options. Then a “perfectly fitting” model would be given by a deterministic stock price process.^{1} In this case it is obvious that this “model” cannot be correct if we want to price options on the stock. This argument can be carried over to volatility ...

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