[FAM-news] This week FAM Seminars

In a continuous arbitrage free market without a risk-free bond the relationship between the minimal martingale measure Q, the resulting short rate for a locally risk-free bond priced according to the chosen measure Q, and the implied instantaneous Sharpe-ratio is considered using a dynamic CAPM approach. We define locally efficient portfolios and investigate their relevance for maximizing terminal utility in an incomplete market. For a totally unhedgeable price for instantaneous risk, isoelastic utility of terminal wealth can be maximized using a portfolio consisting of the locally risk-free bond and a locally efficient fund only. More general, optimal self-financing hedging strategies can be described using (Forward-) Backward-SDEs. We derive the relationship between the optimal portfolio, the optimal martingale measure in the dual problem and the optimal value function of the problem. In a markovian market model we find a non-linear PDE for the value function. From the solution we can construct under additional assumptions the optimal portfolio and the solution of the dual problem. Furthermore, we find the intertemporal price for risk relative to the locally risk-free bond to equal the standard deviation of the variance optimal martingale measure. In a market with zero bonds, the absolute intertemporal price for risk is related to the discounted variance optimal martingale measure and the zero bond prices.