BFS 2002

Plenary Address

A valuation algorithm for incomplete models

Marek Musiela, Thaleia Zariphopoulou

The classical arbitrage pricing is based upon the concept of risk replication. One makes investment allocations in order to dynamically replicate future liability. Unfortunately, this pricing mechanism breaks down in incomplete models because one cannot offset all risk by taking positions in the market. We analyze the notion of value based upon the concept of optimal investment. The price corresponds to the amount which makes an investor indifferent to the various investment opportunities. The resulting valuation algorithm seems to be intuitively obvious. For each time period, one should condition the total risk at the end on the hedgeable one in order to extract the risk that cannot be hedged. Then, price that risk by certainty equivalent, and in the second step, price the remaining hedgeable risk by arbitrage. This defines the total risk at the beginning of the period and the valuation algorithm may be repeated.