Stochastic Finance, 4th Edition

10Minimizing the hedging error

In this chapter, we present an alternative approach to the problem of hedging in an incomplete market model. Instead of controlling the downside risk, we simply aim at minimizing the quadratic hedging error. We begin with a local version of the minimization problem, which may be viewed as a sequential regression procedure. Its solution involves an orthogonal decomposition of a given contingent claim; this extends a classical decomposition theorem for martingales known as the Kunita–Watanabe decomposition. Often, the value process generated by a locally risk-minimizing strategy can be described as the martingale of conditional expectations of the given contingent claim for a special choice of an equivalent martingale ...