No Arbitrage Condition for Positive Diffusion Price Processes

<p style='text-indent:20px;'>This paper provides invariance theorems that facilitate testing for the existence of an asset price bubble in a market where the price evolves as a Markov diffusion process. The test involves only the properties of the price process' quadratic variation under the statistical probability. It does not require an estimate of either the equivalent local martingale measure or the asset's drift. To augment its use, a new family of stochastic volatility price processes is also provided where the processes' strict local martingale behavior can be characterized.</p>

Within the setup of continuous-time semimartingale financial markets, we show that a multiprior Gilboa–Schmeidler minimax expected utility maximizer forms a portfolio consisting only of the riskless asset if and only if among the investor’s priors there exists a probability measure under which all admissible wealth processes are supermartingales. Furthermore, we show that under a certain attainability condition (which is always valid in finite or complete markets) this is also equivalent to the existence of an equivalent (local) martingale measure among the investor’s priors. As an example, we generalize a no betting result due to Dow and Werlang.

The Levy term structure model due to Eberlein and Raible is extended to non-homogeneous driving processes. The classes of equivalent martingale and local martingale measures for various filtrations are characterized. It turns out that in a number of standard situations the martingale measure is unique.

Portfolio Optimization Under Convex Incentive Schemes

Let M(X) be a family of all equivalent local martingale measures for some locally bounded d-dimensional process X, and V be a positive process. Main result of the paper (Theorem 2.1) states that the process V is a supermartingale whatever Q in M(X), if and only if this process admits the following decomposition: V_t = V_0 + \int_0^t H_s dX_s - C_t, t>= 0, where H is an integrand for X, and C is an adapted increasing process. We call such a representation the optional because, in contrast to Doob-Meyer decomposition, it generally exists only with an adapted (optional) process C. We apply this decomposition to the problem of hedging European and American style contingent claims in a setting of incomplete security markets.

Asset Price Bubbles in Incomplete Markets

A general theory of stochastic integral in the abstract topological measurable space is established. The martingale measure is defined as a random set function having some martingale property. All square integrable martingale measures constitute a Hilbert space M 2. For each μ∈M 2, a real valued measure 〈μ〉 on the predictable σ-algebra ℘ is constructed. The stochastic integral of a random function \(\mathfrak{h} \in L^2 \left( {\left\langle \mu \right\rangle } \right)\) with respect to μ is defined and investigated by means of Riesz's theorem and the theory of projections. The stochastic integral operator I μis an isometry from L 2(〈μ〉) to a stable subspace of M 2, its inverse is defined as a random Radon-Nikodym derivative. Some basic formulas in stochastic calculus are obtained. The results are extended to the cases of local martingale and semimartingale measures as well.

In a one price economy, the Fundamental Theorem of Asset Pricing (FTAP) establishes that no-arbitrage is equivalent to the existence of an equivalent martingale measure. Such an equivalent measure can be derived as the normal unit vector of the hyperplane that separates the attainable gain subspace and the convex cone representing arbitrage opportunities. However, in two-price financial models (where there is a bid–ask price spread), the set of attainable gains is not a subspace anymore. We use convex optimization, and the conic property of this region to characterize the “no-arbitrage” principle in financial models with the bid–ask price spread present. This characterization will lead us to the generation of a set of price factor random variables. Under such a set, we can find the lower and upper bounds (supper-hedging and sub-hedging bounds) for the price of any future cash flow. We will show that for any given cash flow, for which the price is outside the above range, we can build a trading strategy that provides one with an arbitrage opportunity. We will generalize this structure to any two-price finite-period financial model.

We now return to general continuous-time financial market models in the setting of §6.1, i.e. there are d+1 primary traded assets whose price processes are given by stochastic processes S 0,... , S d , which are assumed to be adapted, right-continuous with left-limits (RCLL) and strictly positive semimartin-gales on a filtered probability space (Ω, F ℙ, F) (as usual F = (F t ) t≤T ). We assume that the market is free of arbitrage, in the sense that there exist equivalent martingale measures, but it contains non-attainable contingent claims, i.e. there are cash flows which cannot be replicated by self-financing trading strategies. In view of Theorem 6.1.5 this means that we do not have a unique equivalent martingale measures. We try to answer the obvious questions in this setting: how should we price the non-attainable contingent claims, i.e. which of the possible equivalent martingale measures should we pick for our valuation formula based on expectation, and, how can we construct hedging strategies for the nonattainable contingent claims to ‘minimise the risk? We try to answer these two questions in the general setting and then consider a prominent example of an incomplete market, a market with stochastic volatility, in more detail.KeywordsTrading StrategyStochastic VolatilityContingent ClaimMartingale MeasureIncomplete MarketThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A Visual Classification of Local Martingales

Development of the snow- and ice-accounting routine (SIAR)

The paper studies connections between risk aversion and martingale measures in a discrete‐time incomplete financial market. An investor is considered whose attitude toward risk is specified in terms of the index b of constant proportional risk aversion. Then dynamic portfolios are admissible if the terminal wealth is positive. It is assumed that the return (risk) processes are bounded. Sufficient (and nearly necessary) conditions are given for the existence of an optimal dynamic portfolio which chooses portfolios from the interior of the set of admissible portfolios. This property leads to an equivalent martingale measure defined through the optimal dynamic portfolio and the index 0 < b≤ 1. Moreover, the option pricing formula of Davis is given by this martingale measure. In the case of b= 1; that is, in the case of the log‐utility, the optimal dynamic portfolio defines the numéraire portfolio.

This paper provides a thorough survey of the European option pricing, with new trends in the risk measurement, under exponential Lévy models. We develop all steps of pricing from equivalent martingale measures construction to numerical valuation of the option price under these measures. We then construct an algorithm, based on Rockafellar and Uryasev representation and fast Fourier transform, to compute Risk indicators, like the VaR and the CVaR of derivatives. The results are illustrated with an example of each exponential Lévy class. The main contribution of this paper is to build a comprehensive study from the theoretical point of view to practical numerical illustration and to give a complete characterization of the studied equivalent martingale measures by discussing their similarity and their applicability in practice. Furthermore, this work proposes applications to the Fourier inversion technique in risk measurement.

We give a sufficient condition to identify the q-optimal signed and the q-optimal absolutely continuous martingale measures in exponential Levy models. As a consequence, we find that in the one-dimensional case, the q-optimal equivalent martingale measures may exist only if the tails for upward jumps are extraordinarily light. Moreover, we derive the convergence of q-optimal signed, resp. absolutely continuous, martingale measures to the minimal entropy martingale measure as q approaches one. Finally, some implications for portfolio optimization are discussed.

In this paper, we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete, there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for convex payoffs, the option price is increasing in the jump-risk parameter. We apply this result to deduce general inequalities, comparing the prices of contingent claims under various martingale measures, which have been proposed in the literature as candidate pricing measures. Our proofs are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities.