How to recreate a payoff of exotic options with vanillas? ( using density function)

Question

How to replicate exotic option (maybe hybrid) using vanillas?

I have heard once that the idea is to find a derivative wrt the strike of the option value, so that to get a CDF of some distribution, and by having second derivative wrt the strike to get a PDF which in turn somehow helps...

1. Can somebody give an idea/reference to the approach I vagually described above?
2. Are there other standard approaches?

P.S. I do not know any book telling about it.. But I found a post here, by nicolas, that probably tells the approach I am asking about.

Thank you

Answer 1

Replication theorem:

For any twice-continuously differentiable $f(x)$, the value of a European option with payoff $f(\cdot)$ and expiry $T$ is the weighted integral of call and put options with weights equal to the second derivative of $f(\cdot)$:

$E(f(S(T)) = f(K^*) + f'(K)(S(0)-K^*) + \int_{-\infty}^{K^*}p(0, S(0); T, K)f''(K) dK + \int_{K^*}^{\infty}c(0, S(0); T, K)f''(K) dK$

for any $K^*$.

You can find it in Interest Rate Model by Andersen and Piterbarg.