Nonparametric Finance

Chapter 16Quadratic and Local Quadratic Hedging

Quadratic hedging was introduced in Sections 13.1.3 and 15.1. In quadratic hedging we find the best approximation of the option in the sense of the mean-squared error. Quadratic hedging is related to the idea of statistical arbitrage: The fair price is defined as such price that makes the probability of gains and losses small for the writer of the option.

Quadratic hedging makes it possible to price and hedge options in a completely nonparametric way. In quadratic hedging we can derive prices and hedging coefficients without any modeling assumptions, making only some rather weak assumptions about square integrability and about a bounded mean–variance trade-off. There are many ways to implement quadratic hedging nonparametrically. We use kernel estimation in our implementation.

Let be the discounted payoff of an European option. For example, for a call option . In quadratic hedging the mean-squared hedging error

is minimized among strategies and among the initial investment . The terminal value of the gains process is defined by

where ...