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Register a new userGood deal bounds with convex constraints
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Takuji Arai
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We investigate the structure of good deal bounds, which are subintervals of a no-arbitrage pricing bound, for financial market models with convex constraints as an extension of Arai and Fukasawa (2014). The upper and lower bounds of a good deal bound are naturally described by a convex risk measure. We call such a risk measure a good deal valuation; and study its properties. We also discuss superhedging cost and Fundamental Theorem of Asset Pricing for convex constrained markets.
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We shall provide in this paper good deal pricing bounds for contingent claims induced by the shortfall risk with some loss function. Assumptions we impose on loss functions and contingent claims are very mild. We prove that the upper and lower bounds of good deal pricing bounds are expressed by convex risk measures on Orlicz hearts. In addition, we obtain its representation with the minimal penalty function. Moreover, we give a representation, for two simple cases, of good deal bounds and calculate the optimal strategies when a claim is traded at the upper or lower bounds of its good deal pricing bound.
This paper expands the work on distributionally robust newsvendor to incorporate moment constraints. The use of Wasserstein distance as the ambiguity measure is preserved. The infinite dimensional primal problem is formulated; problem of moments duality is invoked to derive the simpler finite dimensional dual problem. An important research question is: How does distributional ambiguity affect the optimal order quantity and the corresponding profits/costs? To investigate this, some theory is developed and a case study in auto sales is performed. We conclude with some comments on directions for further research.
Let $(Phi,Psi)$ be a conjugate pair of Orlicz functions. A set in the Orlicz space $L^Phi$ is said to be order closed if it is closed with respect to dominated convergence of sequences of functions. A well known problem arising from the theory of risk measures in financial mathematics asks whether order closedness of a convex set in $L^Phi$ characterizes closedness with respect to the topology $sigma(L^Phi,L^Psi)$. (See [26, p.3585].) In this paper, we show that for a norm bounded convex set in $L^Phi$, order closedness and $sigma(L^Phi,L^Psi)$-closedness are indeed equivalent. In general, however, coincidence of order closedness and $sigma(L^Phi,L^Psi)$-closedness of convex sets in $L^Phi$ is equivalent to the validity of the Krein-Smulian Theorem for the topology $sigma(L^Phi,L^Psi)$; that is, a convex set is $sigma(L^Phi,L^Psi)$-closed if and only if it is closed with respect to the bounded-$sigma(L^Phi,L^Psi)$ topology. As a result, we show that order closedness and $sigma(L^Phi,L^Psi)$-closedness of convex sets in $L^Phi$ are equivalent if and only if either $Phi$ or $Psi$ satisfies the $Delta_2$-condition. Using this, we prove the surprising result that: emph{If (and only if) $Phi$ and $Psi$ both fail the $Delta_2$-condition, then there exists a coherent risk measure on $L^Phi$ that has the Fatou property but fails the Fenchel-Moreau dual representation with respect to the dual pair $(L^Phi, L^Psi)$}. A similar analysis is carried out for the dual pair of Orlicz hearts $(H^Phi,H^Psi)$.
We consider the problem of finding a model-free upper bound on the price of an American put given the prices of a family of European puts on the same underlying asset. Specifically we assume that the American put must be exercised at either $T_1$ or $T_2$ and that we know the prices of all vanilla European puts with these maturities. In this setting we find a model which is consistent with European put prices and an associated exercise time, for which the price of the American put is maximal. Moreover we derive a cheapest superhedge. The model associated with the highest price of the American put is constructed from the left-curtain martingale transport of Beiglb{o}ck and Juillet.
This paper considers utility indifference valuation of derivatives under model uncertainty and trading constraints, where the utility is formulated as an additive stochastic differential utility of both intertemporal consumption and terminal wealth, and the uncertain prospects are ranked according to a multiple-priors model of Chen and Epstein (2002). The price is determined by two optimal stochastic control problems (mixed with optimal stopping time in the case of American option) of forward-backward stochastic differential equations. By means of backward stochastic differential equation and partial differential equation methods, we show that both bid and ask prices are closely related to the Black-Scholes risk-neutral price with modified dividend rates. The two prices will actually coincide with each other if there is no trading constraint or the model uncertainty disappears. Finally, two applications to European option and American option are discussed.
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