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Incompleteness of markets driven by a mixed diffusion. (English) Zbl 0951.91028

It is well known that in an incomplete market there are several equivalent martingale measures and that perfect hedging is not always possible. In this situation, one way to price options is to choose a particular equivalent martingale measure, for example the Föllmer-Schweizer minimal probability, the canonical martingale measure which minimizes the relative entropy with respect to the original probability measure, or the equivalent martingale measure associated with a utility function. In all these cases, the price is a viable price, i.e., it does not induce any arbitrage opportunities; however, there is no consensus on the choice of this martingale measure.
Another approach is to determine the range of prices compatible with no arbitrage or the minimal super-replication strategy. However, when the dynamics of the stock price are driven by a Wiener process or in a general semi-martingale framework, these two approaches are closely related: the supremum of the possible prices is equal to the minimum initial value of an admissible self-financing strategy that super-replicates the contingent claim.
When the incompleteness arises from stochastic volatility and/or portfolio constraints, the minimal price needed to super-replicate a given contingent claim. E. Eberlein and J. Jacod [Finance Stoch. 1, 131-140 (1997; Zbl 0889.90020)] showed the absence of non-trivial bounds on European claim prices in a model where prices are driven by a purely discontinuous Lévy process with unbounded jumps.
This paper addresses the problem of the range of viable prices for mixed diffusion dynamics. An incomplete market driven by a pair of Wiener and Poisson processes is considered. The upper bound is proved to be a trivial one: for example, the minimal strategy to hedge a European call is a long position in the underlying asset. On the contrary, this paper shows that the lower bound is a nontrivial one, but the corresponding Black-Scholes function evaluated at the current stock price.

MSC:

91B28 Finance etc. (MSC2000)
60J75 Jump processes (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B24 Microeconomic theory (price theory and economic markets)
60J60 Diffusion processes

Citations:

Zbl 0889.90020
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