Shifting martingale measures and the birth of a bubble as a submartingale

Biagini, Francesca; Föllmer, Hans; Nedelcu, Sorin

doi:10.1007/s00780-013-0221-8

Cited by 47 publications

(77 citation statements)

References 25 publications

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“…Instead we have an absence of arbitrage piecewise on sequential time intervals. See for example [1,15].…”

Section: Traded Optionsmentioning

confidence: 99%

“…We consider a European call with pay-off function g satisfying (1). By (4) we have Π(g) Q ≤ S 0 = 1 for all Q ∈ Q LM G , and Π(g) Q ≥ g(S 0 ) = g(1) if further Q ∈ Q M G , and these are strict inequalities unless g is trivial, or affine on the range R. Note that no lower bound except Π(g) Q ≥ 0 is known when Q ∈ Q LM G \Q M G ).…”

Section: The Range Of European Call Option Pricesmentioning

confidence: 99%

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Options Prices in Incomplete Markets

Jacod

Protter

Crépey³

et al. 2017

ESAIM: ProcS

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Abstract. In this paper we consider the valuation of an option with time to expiration T and pay-off function g which is a convex function (as is a European call option), and constant interest rate r = 0, for a variety of underlying price process models constructed from two independent Poisson processes, and an independent Brownian motion. This gives rise to incomplete market models with an infinite number of risk neutral measures. The collection of risk neutral measures gives rise to different prices, which comprise intervals that we calculate. The intervals can vary dramatically depending on the model parameters.

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“…Instead we have an absence of arbitrage piecewise on sequential time intervals. See for example [1,15].…”

Section: Traded Optionsmentioning

confidence: 99%

Section: The Range Of European Call Option Pricesmentioning

confidence: 99%

Options Prices in Incomplete Markets

Jacod

Protter

Crépey³

et al. 2017

ESAIM: ProcS

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show abstract

“…Note, however, that this modelling approach is different from the related literature on financial bubbles (see e.g. Cox and Hobson [3], Jarrow et al [6,7], Heston et al [5], Biagini et al [2]), in the sense that we are more concerned with the effects of a burst of the bubble and not so much on its formation and price impacts. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 95%

Worst-Case Optimal Investment with a Random Number of Crashes

2013

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We study a portfolio optimization problem in a market which is under the threat of crashes. At random times, the investor receives a warning that a crash in the risky asset might occur. We construct a strategy which renders the investor indifferent about an immediate crash of maximum size and no crash at all. We then verify that this strategy outperforms every other trading strategy using a direct comparison approach. We conclude with numerical examples and calculating the costs of hedging against crashes.

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“…A regime switch from an initial martingale measure P * 0 ∈ P * UI , which does not exhibit a bubble, to another martingale measure P * 1 ∈ P * NUI would thus describe the sudden appearance of a bubble as in [66]. But the flow in the space P * could also move slowly from P * 0 to P * 1 as in [7], and this would induce the slow birth of a bubble as a submartingale.…”

Section: P Versus P *mentioning

confidence: 99%

Probabilistic Aspects of Finance

Föllmer

Schied

2012

SSRN Journal

Self Cite

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In the past decades, advanced probabilistic methods have had significant impact on the field of finance, both in academia and in the financial industry. Conversely, financial questions have stimulated new research directions in probability. In this survey paper, we review some of these developments and point to some areas that might deserve further investigation. We start by reviewing the basics of arbitrage pricing theory, with special emphasis on incomplete markets and on the different roles played by the "real-world" probability measure and its equivalent martingale measures. We then focus on the issue of model ambiguity, also called Knightian uncertainty. We present two case studies in which it is possible to deal with Knightian uncertainty in mathematical terms. The first case study concerns the hedging of derivatives, such as variance swaps, in a strictly pathwise sense. The second one deals with capital requirements and preferences specified by convex and coherent risk measures. In the final two sections we discuss mathematical issues arising from the dramatic increase of algorithmic trading in modern financial markets. Why does Brownian motion appear in the financial context? Here is a first rough argument. At each fixed time, the price of a stock could be seen as a temporary equilibrium resulting from a large number of decisions to buy or sell, made in a random and more or less independent manner: Many coins are thrown successively, and so Brownian motion should arise as a manifestation of the central limit theorem. This is the "Coin-Tossing View of Finance", as it is called by J. Cassidy in How Markets Fail [17]. This rough argument can be refined by using microeconomic assumptions on the behavior of agents and on the ways they generate a random demand, and then the

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Shifting martingale measures and the birth of a bubble as a submartingale

Cited by 47 publications

References 25 publications

Options Prices in Incomplete Markets

Options Prices in Incomplete Markets

Worst-Case Optimal Investment with a Random Number of Crashes

Probabilistic Aspects of Finance

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