# What does $\int dS \phi (S - K)$ mean in Gatheral's book?

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:

1. Why does he write $dS$ before the integrand?
2. Why does he use the word "option" and not "call option", since that's what it seems to be?
3. 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 $dS$??

1. It's a mere a question of notation and you can consider that $$\int f(x) dx \equiv \int dx f(x)$$
3. This comes from the definition of the risk-neutral measure $\Bbb{Q}$, as the measure under which the prices of self-financing portfolios are martingales when expressed under the money market numéraire $B_t$. Assuming zero discount rates, $B_t = 1, \forall t\geq 0$ and this becomes \begin{align} C_0 &= B_0 \Bbb{E}_0^\Bbb{Q}[B_T^{-1}C_T] \\ &= \Bbb{E}_0^\Bbb{Q}[ C_T] \\ &= \Bbb{E}_0^\Bbb{Q}[ (S_T - K)^+ ] \\ &=\int_{-\infty}^{+\infty}(S-K)^+ q(S) dS \\ &=\int_{K}^{\infty} (S-K) q(S) dS \\ &\equiv \int_{K}^{\infty} dS q(S) (S-K) \end{align} It's just that he denoted the risk-neutral pdf $q(S)$ by $\phi$.