A note on P- vs. Q-expected loss portfolio constraints

Abstract

We consider portfolio optimization problems with expected loss constraints un-der the physical measure P and the risk neutral measure Q, respectively. Using Merton’s portfolio as a benchmark portfolio, the optimal terminal wealth of the Q-risk constraint problem can be easily replicated with the standard delta hedg-ing strategy. Motivated by this, we consider the Q-strategy fulfilling the P-risk constraint and compare its solution with the true optimal solution of theP-riskconstraint problem. We show the existence and uniqueness of the optimal solution to theQ-strategy fulfilling theP-risk constraint, and provide a tractable evalua-tion method. The Q-strategy fulfilling the P-risk constraint is not only easier toimplement with standard forwards and puts on a benchmark portfolio than the P-risk constraint problem, but also easier to solve than either of theQ- or P-riskconstraint problem. The numerical test shows that the difference of the values ofthe two strategies (the Q-strategy fulfilling the P-risk constraint and the optimal strategy solving the P-risk constraint problem) is reasonably small.