Pricing and hedging with globally and instantaneously vanishing risk

Summary

Using a backward stochastic differential equation (BSDE) approach in a Brownian motion setting, we determine in an incomplete market an initial price Y₀ for a non-attainable claim ξ ∈ L^p, 1 < p < ∞, that takes the hedging risk into account. Y₀ is chosen to be the best price such that the minimal replication error of the optimal hedging strategy with initial capital Y₀ has vanishing risk, the risk being measured with a certain coherent risk measure. This risk measure allows an interpretation as measuring total risk by accumulating instantaneously measured risks. Intertemporal consistency properties of best prices and uniqueness of optimal hedging strategies with vanishing risk are shown. The result can be interpreted as a robust predictable martingale representation property of Brownian motion w.r.t. the intersection of the scenario set defining the risk measure and the set of equivalent martingale measures for the assets traded in the market.