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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 $dS$??
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  1. It's a mere a question of notation and you can consider that $$ \int f(x) dx \equiv \int dx f(x) $$
  2. It is indeed implied that the option under scrutiny is a (European) call option
  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 $\phi$.
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  • $\begingroup$ Yes but 1 is terrible form and is not suggestive of an author that values clarity of notation. $\endgroup$ – James Spencer-Lavan Jul 25 '18 at 21:31
  • $\begingroup$ Not a huge find of this notation either. $\endgroup$ – Quantuple Jul 25 '18 at 21:33
  • $\begingroup$ It is not traditional notation, but it is nice in that right after you see the integral sign it tells you what the variable of integration is (without having to move your eyes to the right). So you can read it as "integral with respect to dS of bla bla". $\endgroup$ – Alex C Jul 26 '18 at 9:00

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