Today the topic was pricing …
Today the topic was pricing of non-attainable contingent claims X (incomplete markets). Although there is no unique price for X in this case, it is possible to identify an interval [V-,V+] in which each value represents a fair (arbitrage) price. Here, V- is the supremum over Q-expected discounted payoffs Y/B1 with Y≤X being attainable and V+ is the infimum over Q-expected discounted payoffs Y/B1 with Y≥X being attainable. I showed how to reformulate these two problems as standard linear programming problems, which can be used to compute V-,V+ in concrete examples. Moreover, using duality, I showed that V- (V+) equals the infimum (supremum) of the Q-expected discounted payoff X/B1 with Q varying over the set of risk-neutral probability measures. This result was finally illustrated by an example.
Exercises (for Tuesday next week): I.12, 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 1.19.