Incompleteness of markets driven by a mixed diffusion

The Risk Averse Investor's Equilibrium Equity Premium in a Semi Martingale Market with Arbitrary Jumps

“…In this paper, we use the semi martingale approach to determine equilibrium equity premium in a production economy with jumps as opposed to option pricing. The problem of deriving ordering results for option prices has been adressed in several papers [ (Karoui & Shreve, 1998), (Hobson, 1998), (Bellamy, 2000), (Henderson, 2002), (Hendersonn & Hobson, 2003), (Hendersonnn & Kluge, 2003), (Moller, 2003), (Eberlein & Jacod, 1997), (Frey & Sin, 1999), (Jakubenas, 2002), (Gushchin & Mordecki, 2002)]. The results for models with nontrivial pricing intervals and the corresponding comparison results are less complete.…”

“…Comparison results for diffusion processes are discussed in (Karoui & Shreve, 1998) and nontrivial bounds for stochastic volatility models are given in (Frey & Sin, 1999). (Bellamy, 2000) (see also (Henderson & Hobson, 2003)) prove that the price of a European call for a diffusion with jumps is bounded below by the corresponding Black-Scholes price and above by the trivial upper price [see also (Bergman & Wiener, 1996) and (Hobson, 1998) for alternative proofs]. An important generalization of the technique introduced in (Karoui & Shreve, 1998) and (Bellamy, 2000) has been established by (Gushchin & Mordecki, 2002) who derive a general comparison result for one-dimensional semimartingales to some Markov process w.r.t convex ordering of terminal values.…”

Bounds for perpetual American option prices in a jump diffusion model

“…To prove the lower bound, note that the corresponding result for European options holds (see Theorem 4.1 of Bellamy and Jeanblanc (2000) or Theorem 5.1 of Ekström and Tysk (2005)). By approximating the American option with a sequence of Bermudan options, the bound carries over to our setting.…”

Asset Pricing under Uncertainty

“…As an example, Kramkov [23] shows that the super-hedging cost of a call corresponds to the value of the option under the least favorable martingale measure. Likewise, Bellamy and Jeanblanc [3] prove that in a jump-diffusion model the price range for a call option corresponds to the interval given by the no-arbitrage conditions that is the super-hedging strategy of a call option is to buy the underlying. It seems therefore that super-hedging is not an effective methodology since the option buyer will prefer to buy the underlying rather than pay the same price to have an option.…”