We study the problem of super-replication for game options under proportional transaction costs. We consider a multidimensional continuous time model, in which the discounted stock price process ...satisfies the conditional full support property. We show that the super-replication price is the cheapest cost of a trivial super-replication strategy. This result is an extension of previous papers (see Statist. Decisions 27 (2009) 357-369 and Ann. Appl. Probab. 18 (2008) 491-520) which considered only European options. In these papers the authors showed that with the presence of proportional transaction costs the super-replication price of a European option is given in terms of the concave envelope of the payoff function. In the present work we prove that for game options the super-replication price is given by a game variant analog of the standard concave envelope term. The treatment of game options is more complicated and requires additional tools. We combine the theory of consistent price systems together with the theory of extended weak convergence which was developed in Weak convergence of stochastic processes for processes viewed in the Strasbourg manner (1981) Preprint. The second theory is essential in dealing with hedging which involves stopping times, like in the case of game options.
We consider a financial market with one riskless and one risky asset. The super-replication theorem states that there is no duality gap in the problem of super-replicating a contingent claim under ...transaction costs and the associated dual problem. We give two versions of this theorem. The first theorem relates a numéraire-based admissibility condition in the primal problem to the notion of a local martingale in the dual problem. The second theorem relates a numéraire-free admissibility condition in the primal problem to the notion of a uniformly integrable martingale in the dual problem.
In this paper we investigate the possible values of basket options. Instead of postulating a model and pricing the basket option using that model, we consider the set of all models which are ...consistent with the observed prices of vanilla options, and, within this class, find the model for which the price of the basket option is largest. This price is an upper bound on the prices of the basket option which are consistent with no-arbitrage. In the absence of additional assumptions it is the lowest upper bound on the price of the basket option. Associated with the bound is a simple super-replicating strategy involving trading in the individual calls.
We prove a general version of the super-replication theorem, which applies to Kabanov's model of foreign exchange markets under proportional transaction costs. The market is described by a ...matrix-valued cadlag bid-ask process evolving in continuous time. We propose a new definition of admissible portfolio processes as predictable (not necessarily right- or left- continuous) processes of finite variation related to the bid-ask process by economically meaningful relations. Under the assumption of existence of a strictly consistent price system (SCPS), we prove a closedness property for the set of attainable vector-valued contingent claims. We then obtain the super-replication theorem as a consequence of that property, thus generalizing to possibly discontinuous bid-ask processes analogous results obtained by Kabanov (Financ. Stoch. 3, 237-248, 1999), Kabanov and Last (Math. Financ. 12, 63-70, 2002) and Kabanov and Stricker (Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann, pp 125-136, 2002). Rasonyi's counter-example (Lecture Notes in Mathematics 1832, 394-398, 2003) served as an important motivation for our approach. PUBLICATION ABSTRACT
We derive in closed form distribution-free bounds and optimal hedging strategies for spread options. Upper bounds are obtained when the spread option's joint distribution is constrained by the prices ...of traded options with all available strikes of a given maturity. The difference between the upper bound and the market price is a useful new measure of codependence, which we refer to as the market implied antimonotonicity gap.
Let
Z
t,
z
ν
be a
R
d+1
-valued mixed diffusion process controlled by
ν with initial condition
Z
t,
z
ν
(
t)=
z. In this paper, we characterize the set of initial conditions such that
Z
t,
z
ν
can be ...driven above a given stochastic target at time
T by proving that the corresponding value function is a discontinuous viscosity solution of a variational partial differential equation. As applications of our main result, we study two examples: a problem of optimal insurance under self-protection and a problem of option hedging under jumping stochastic volatility where the underlying stock pays a random dividend at a fixed date.
We study the situation of an agent who can trade on a financial market and can also transform some assets into others by means of a production system, in order to price and hedge derivatives on ...produced goods. This framework is motivated by the case of an electricity producer who wants to hedge a position on the electricity spot price and can trade commodities which are inputs for his system. This extends the essential results of Bouchard and Nguyen (Math Finance,
2011
) to continuous time markets. We introduce the generic concept of
conditional sure profit
along the idea of the
no sure profit
condition of Rásonyi (Optimality and risk-modern trends in mathematical finance: the Kabanov Fetschrift,
2009
). The condition allows one to provide a closedness property for the set of super-hedgeable claims in a very general financial setting. Using standard separation arguments, we then deduce a dual characterization of the latter and provide an application to power futures pricing.
The classical Black–Scholes hedging strategy of a European contingent claim may require rapid changes in the replicating portfolio. One approach to avoid this is to impose a priori bounds on the ...variations of the allowed trading strategies, called gamma constraints. Under such a restriction, it is in general no longer possible to replicate a European contingent claim, and super-replication is a commonly used alternative. This paper characterizes the infimum of the initial capitals that allow an investor to super-replicate the contingent claim by carefully choosing an investment strategy obeying a gamma constraint. This infimum is shown to be the unique viscosity solution of a nonstandard partial differential equation. Due to the lower gamma bound, the “intuitive” partial differential equation is not parabolic and the actual equation satisfied by the infimum is the parabolic majorant of this equation. The derivation of the viscosity property is based on new results on the small time behavior of double stochastic integrals.
La stratégie de couverture classique d'une option européenne, dictée par le modèle de Black et Scholes, peut conduire à des rebalancements rapides du portefeuille répliquant. Afin d'éviter de telles situations indésirables, nous introduisons des contraintes spécifiques sur le portefeuille, appelées contraintes gamma. Il n'est alors pas possible en général de répliquer parfaitement l'option européene. Par conséquent, la surréplication est alors une alternative fréquemment utilisée. Dans ce papier, on caractérise l'infimum des capitaux initiaux qui permet à un investiseur de surrépliquer l'actif contingent en choisissant soigneusement une stratégie de portefeuile satisfaisant à une contrainte gamma. Nous montrons que cet infimum est l'unique solution de viscosité d'une équation aux dérivées partielles non standard. A cause de la borne inférieure sur la contrainte gamma, l'équation aux dérivées partielles « intuitive » n'est pas parabolique, et l'équation effectivemet satisfaite par l'infimum est le mojorant parabolique de l'équation « intuitive ». L'obtention de la propriété de viscosité s'appuie sur des résultats nouveaux portant sur des intégrales stochastiques doubles.
PDF
