Mathematical models for financial asset prices which include, for example, stochastic volatility or jumps are incomplete in that derivative securities are generally not replicable by trading in the underlying. In earlier work (2004) the first author provided a geometric condition under which trading in the underlying and a finite number of vanilla options completes the market. We complement this result in several ways. First, we show that the geometric condition is not necessary and a weaker, necessary and sufficient, condition is presented. While this condition is generally not directly verifiable, we show that it simplifies to matrix non-degeneracy in a single point when the pricing functions are real analytic functions. In particular, any stochastic volatility model is then completed with an arbitrary European type option. Further, we show that adding path-dependent options such as a variance swap to the set of primary assets, instead of plain vanilla options, also completes the market.
1. Introduction. It is well known that the Black-Scholes financial market model, consisting of a log-normal asset price diffusion and a non-random money market account, is complete: every contingent claim is replicated by a portfolio formed by dynamic trading in the two assets. Ultimately this result rests on the martingale representation property of Brownian motion. As soon as we attempt to correct the empirical deficiencies of the asset model by including, say, stochastic volatility, completeness is lost if we continue to regard the original two assets as the only tradables: there are no longer enough assets to 'span the market'. However there are traded options markets for many assets such as single stocks or stock indices, so it is a natural question to ask whether the market becomes complete when these are included. An early result in this direction was provided by Romano and Touzi [17] who showed that a single call option completes the market when there is stochastic volatility driven by one extra Brownian motion (under some additional assumptions; see Section 5 below). But providing a general theory has proved surprisingly problematic. There are two main approaches, succinctly labelled 'martingale models' and 'market models' by Schweizer and Wissel [19]. In the former-which is the approach taken in Davis [4] and in this paper-one starts with a stochastic basis (Ω, F , (F t ) T ∈R+ , P). P is a risk-neutral measure, so all discounted asset prices are P-martingales which can be constructed by conditional expectation: the price process for an asset that has the integrable F T -measurable value H at some final time T is S H t = E[e -r(T -t) H| F t ] for t < T , where r is the riskless interest rate. The distinction between an 'underlying asset' and a 'contingent claim' largely disappears in this approach. A specific model is obtained by specifying some process whose natural filtration is (F t ), for example a diffusion process as in Section 2 below. In a 'market model' one specifies directly the price processes of all traded assets, be they underlying assets or derivatives. For the latter, say a call option with strike K and exercise time T on an asset S t , it is generally more convenient to model a proxy such as the implied volatility σt which is related in a one-to-one way to the price process A t of the call by A t = BS(S t , K, r, σt , Tt), where BS(• • • ) is the Black-Scholes formula. This is the approach pursued by Schönbucher [18] and, in different variants, in recent papers by Schweizer and Wissel [19] and Jacod and Protter [10]. This is not the place to debate the merits of these approaches; suffice it to say that the problem with martingale models is that the modelling of asset volatilities is too indirect, while the problem with market models is the extremely awkward set of conditions required for absence of arbitrage.
The paper is organised as follows. We first describe our market, i.e. we model the factor process spanning the filtration and write assets prices as conditional expectations. Then in Section 3 we give a necessary and sufficient condition for completeness of our market. Section 4 explores the case when the coefficients of the SDE solved by the factor process and option payoffs are such that the prices are real analytic function of the time and the factor process. We show that the completeness question reduces from non-degeneracy of a certain matrix in the whole domain to its non-degeneracy in a single point. The result is then applied in Section 5 to show completeness of stochastic volatility models. Section 6 explores the use of path dependent derivatives, in particular variance swaps, in place of European type options and Section 7 concludes.
- Market model. Consider a market in which investors can trade in d risky assets A 1 , . . . , A d together with a riskless money market account paying interest at a constant rate r ≥ 0. We assume there is no arbitrage in the market and we want to investigate market completeness on [0, T ]. We therefore assume existence of an equivalent martingale measure and we chose to work under this measure, which we denote P. The market is spanned by some factor process. More precisely, market factors are modeled with a d-dimensional diffusion process (ξ t ) t≥0 , solution to an SDE:
where w t is a d-dimensional Brownian motion on (Ω, F , P), w.r.t. its natural filtration, and where we assume that (A1) σ(t, x)σ(t, x) T is strictly positive definite for a.e. (t, x) ∈ (0, T ) × R d , (2.1) has a unique strong solution.
The assumption of ellipticity above seems natural and corresponds to full factor representation. The assumption that the state space of ξ is the whole of R d is a simplification which allows us to expose the main ideas without superficial technicalities. In general one could take a general open connected set D ⊂ R d as the state space. Behaviour of ξ at the boundaries would then imply the appropriate boundary conditions for PDE for
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