Exercise: In-out parity for barrier options

For any barrier $B$ and strike $K$:

where KO is the knock-out and KI is the knock-in (down or up, matching).

Tasks

1. Derive in-out parity from a portfolio argument: hold both KI and KO. What's the combined payoff at expiry?

2. Numerically verify for a down-and-out + down-and-in call: $S_0 = 100, K = 100, B = 80, T = 1, r = 0.05, \sigma = 0.2$. Use the FDM solver from the lesson.

3. Application. A market maker quotes the down-and-out call at \9.20$and the down-and-in call at$$1.50$. The vanilla call BS price is$$10.45$. Is there an arbitrage? Compute the net trade.
4. Generalisation. Does in-out parity hold for American barrier options? Why or why not?