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Mathematical Finance's Blog

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Analytical approximations of local‐Heston volatility model and error analysis

November 28, 2017 Comments (0)

Abstract
This paper studies the expansion of an option price (with bounded Lipschitz payoff) in a stochastic volatility model including a local volatility component. The stochastic volatility is a square root process, which is widely used for modeling the behavior of the variance process (Heston model). The local volatility part is of general form, requiring only appropriate growth and boundedness assumptions. We rigorously establish tight error estimates of our expansions, using Malliavin...

On the market viability under proportional transaction costs

November 28, 2017 Comments (0)

Abstract
This paper studies the market viability with proportional transaction costs. Instead of requiring the existence of strictly consistent price systems as in the literature, we show that strictly consistent local martingale systems (SCLMS) can successfully serve as the dual elements such that the market viability can be verified. We introduce two weaker notions of no arbitrage conditions on market models named no unbounded profit with bounded risk (NUPBR) and no local arbitrage with...

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November 28, 2017 Comments (0)


On American VIX options under the generalized 3/2 and 1/2 models

May 4, 2017 Comments (0)

Abstract
In this paper, we extend the 3/2 model for VIX studied by Goard and Mazur and introduce the generalized 3/2 and 1/2 classes of volatility processes. Under these models, we study the pricing of European and American VIX options, and for the latter, we obtain an early exercise premium representation using a free‐boundary approach and local time‐space calculus. The optimal exercise boundary for the volatility is obtained as the unique solution to an integral equation of Volterra type. We...

Small‐cost asymptotics for long‐term growth rates in incomplete markets

May 4, 2017 Comments (0)

Abstract
This paper provides a rigorous asymptotic analysis of long‐term growth rates under both proportional and Morton–Pliska transaction costs. We consider a general incomplete financial market with an unspanned Markov factor process that includes the Heston stochastic volatility model and the Kim–Omberg stochastic excess return model as special cases. Using a dynamic programming approach, we determine the leading‐order expansions of long‐term growth rates and explicitly construct strategies...

On the C‐property and w∗‐representations of risk measures

May 2, 2017 Comments (0)

Abstract
We identify a large class of Orlicz spaces for which the topology fails the C‐property introduced by Biagini and Frittelli. We also establish a variant of the C‐property and use it to prove a ‐representation theorem for proper convex increasing functionals, satisfying a suitable version of Delbaen's Fatou property, on Orlicz spaces with . Our results apply, in particular, to risk measures on all Orlicz spaces other than .

Optimal cash holdings under heterogeneous beliefs

April 18, 2017 Comments (0)

Abstract
This paper explores a one‐period model for a firm that finances its operations through debt provided by heterogeneous creditors. Creditors differ in their beliefs about the firm's investment outcomes. We show the existence of Stackelberg equilibria in which the firm holds cash reserves in order to provide incentives for pessimistic creditors to invest in the firm. We find interest rates and cash holdings to be complementary tools for increasing debt capacity. In markets with a high...

A note on the long rate in factor models of the term structure

April 18, 2017 Comments (0)

Abstract
In this paper, we consider factor models of the term structure based on a Brownian filtration. We show that the existence of a nondeterministic long rate in a factor model of the term structure implies, as a consequence of the Dybvig–Ingersoll–Ross theorem, that the model has an equivalent representation in which one of the state variables is nondecreasing. For two‐dimensional factor models, we prove moreover that if the long rate is nondeterministic, the yield curve flattens out, and...

Conic martingales from stochastic integrals

April 18, 2017 Comments (0)

Abstract
In this paper, we introduce the concept of conic martingales. This class refers to stochastic processes that have the martingale property but that evolve within given (possibly time‐dependent) boundaries. We first review some results about the martingale property of solution to driftless stochastic differential equations. We then provide a simple way to construct and handle such processes. Specific attention is paid to martingales in [0, 1]. One of these martingales proves to be...