An Introduction to Financial Markets

Chapter Fourteen Option Pricing: Incomplete Markets

In Chapter 13, we have introduced the essential mathematical machinery required to price options. We have done so under the somewhat self-contradictory assumption of market completeness. The practical implication of market completeness is that we price derivatives under the condition that they are of no use, as they can be perfectly replicated, and that any risk can be hedged away. Needless to say, markets are not complete, as a consequence of the following reasons:

• The derivative may be written on a non-traded asset, which cannot be included in a hedging/replication portfolio. This is the case, e.g., for interest rate derivatives.
• The BSM model relies on geometric Brownian motion (GBM), which features continuous sample paths, a gentle tail behavior with moderate kurtosis, as well as a deterministic volatility. Actually, volatility is an unobservable risk factor, which may be hard to hedge.
• Diffusion models with stochastic volatility may yield heavy-tailed distributions that are more compatible with observed behavior than GBM. A more radical approach is to rule out continuity and introduce jumps. Jumps introduce a non-predictable component that may disrupt replication approaches.
• Transaction costs preclude continuous-time hedging, resulting in hedging errors and residual risk.

From a theoretical viewpoint, market incompleteness implies the existence of multiple probability measures compatible with no-arbitrage. From a practical ...