Risk-Neutral Valuation pp 289-326 | Cite as

# Incomplete Markets

## Abstract

We now return to general continuous-time financial market models in the setting of §6.1, i.e. there are *d* + 1 primary traded assets whose price processes are given by stochastic processes *S* _{0},..., *S* _{ d }, which are assumed to be adapted, right-continuous with left-limits (RCLL) and strictly positive semi-martingales on a filtered probability space (*Ω, F, ℙ, F*) (as usual *F* = (*F* _{ t })_{ t≤T }). We assume that the market is free of arbitrage, in the sense that there exist equivalent martingale measures, but it contains non-attainable contingent claims, i.e. there are cash flows that cannot be replicated by self-financing trading strategies. In view of Theorem 6.1.5 this means that we do not have a unique equivalent martingale measure. We try to answer the obvious questions in this setting: how should we price the non-attainable contingent claims, i.e. which of the possible equivalent martingale measures should we pick for our valuation formula based on expectation, and, how can we construct hedging strategies for the non-attainable contingent claims to ‘minimize the risk? We try to answer these two questions in the general setting and then consider a prominent example of an incomplete market, a market with stochastic volatility, in more detail.

## Keywords

Trading Strategy Stochastic Volatility Price Process Contingent Claim Martingale Measure## Preview

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