# Hedge cost Exotic payoff

For an exotic payoff with no analytical formula, the standard is to use MC simulation or solve a PDE. On the other hand the price should be the future hedge costs with the hedge ratios implied by the model. How can we reconcile both approaches? I imagine one is implied in the other even-though it is indirect, i just can't see how.

Thanks,

• I'm not sure I fully understand your question, but the idea of the price as the seller's hedging costs is 'built into' the PDE. Even if the black Scholes pde didn't have an analytical form, it still would be the price that makes the P&L of the seller's hedging portfolio $0$. The hedging ratios/delta strategies aren't always going to be in the simple form but in complete markets there's always some sort of self financing strategy that'll hedge the short position in the option. So even if the exotic has a PDE that doesn't have closed form solutions, the PDE was constructed so that's it's fair. – Slade May 13 at 18:02
• +1. And the link between the PDE solution and MC lies in the Feynman-Kac formula. – Quantuple May 14 at 6:51
• If your question is whether one can get hedge ratios from an MC or PDE pricer, the answer is yes. Simply reprice with a different spot to get delta, etc. – Ivan May 14 at 19:01