In Gatheral's book on stochastic volatility, he writes the price of an option as $$\int_K^\infty dS \phi (S - K)$$
where $\phi$ is a density.
Where does this come from?
I have multiple questions:
- Why does he write $dS$ before the integrand?
- Why does he use the word "option" and not "call option", since that's what it seems to be?
- Where does the formula come from? I get that in the usual black scholes setting, the price is $\int_K^\infty (S - K) dQ$ where $Q$ is the risk-neutral measure. Does this result also hold in general stochastic volatility setting, and if so, how do we get from $\int_K^\infty (S - K) dQ$ to $\int_K^\infty (S-K) \phi dQ$? Is he saying $\phi$ is the density of .. $S$ with respect to the measure $dS$??