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


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.

  • $\begingroup$ Thank you, @Lipton! Does there exist a similar "simple" approach for replication for non European options? What does $c$ function mean? $\endgroup$ – gencho Jan 20 '18 at 18:04
  • $\begingroup$ p means put and c means call. Sorry I don't know about non European options. But it sounds so much harder, because it's basically the valuation function of a dynamic program...not sure how you can easily do that. I'd imagine you can maybe get an inequality instead of equality? $\endgroup$ – Lipton Jan 20 '18 at 18:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.