You said:
if the index's return is negative then the note's total payoff will be 1000 x (1 - R)
When the return is positive, then the payoff is 1000 + 1000 * 2.5 * max{R - 0.1, 0}.
Hence, your payout function should be as follows:
v(T)= Indicator {Index(T) < 1000, Index(T); 1000 < Index(T) < 1100, 1000; Index(T) > 1100, 1000 + 2.5*(Index(T)-1100) } and indeed you can re-write as
V(T) = Index(T) - max(Index(T)-1000,0) + 2.5*max(Index(T)-1100,0)
A) If the underlying process that drives the Index price does not correspond with the assumed process by Black Scholes then you cannot value this note via Black Scholes. But you can run a monte carlo simulation based approach to value such note. Here are couple steps to get you started
You essentially need to to decide on the type of model you need to apply. For that you need to know which model best describes the price dynamics of the underlying index. Be mindful in case there are any correlating brownian motions or correlating processes which would make it a little more exciting to deal with.
Next, you would need to simulate different index price paths to evolve the index price.
Then, you derive the final payoff, using the index value at each price path at the time that coincides with the expiration of your note.
You then properly discount each payoff.
At the end you average the discounted payoffs to get to your expected discounted future value of the note which is the price anyone would want to trade at if he/she believed that your model and variable inputs properly described the evolution of the index.
B) If the underlying processes correspond then you can simply value the first portion of this note via "no-arbitrage argument" and treat the other payoff components as separate call options.
