Asset price bubbles: Invariance theorems

Using the Ray-Knight theorem we give conditions for anonnegative diffusion without drift to reach zero or not. These results also givenecessary and sufficient conditions for such a diffusion process to be a martingale (and notjust a local martinagle). We apply these results in order to give necessary and sufficientconditions for nonnegative diffusion to have equivalent local martingale measures.

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.

Asset Price Bubbles in Incomplete Markets

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.

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.

The degree of variation of trading prices with respect to time is volatility-measured by the standard deviation of returns. We present the estimation of stochastic volatility from the stochastic differential equation for evenly spaced data. We indicate that, the price process is driven by a semi-martingale and the data are evenly spaced. The results of Malliavin and Mancino [1] are extended by adding a compensated poisson jump that uses a quadratic variation to calculate volatility. The volatility is computed from a daily data without assuming its functional form. Our result is well suited for financial market applications and in particular the analysis of high frequency data for the computation of volatility.

The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics but also in insurance, finance, and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the Itō and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Itō integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral, and its Itō integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Lévy alpha -stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably, these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, which generalizes the standard diffusion equation, solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE and check it by Monte Carlo.

In this paper, we employ deep learning techniques to detect financial asset bubbles by using observed call option prices. The proposed algorithm is widely applicable and model‐independent. We test the accuracy of our methodology in numerical experiments within a wide range of models and apply it to market data of tech stocks in order to assess if asset price bubbles are present. Under a given condition on the pricing of call options under asset price bubbles, we are able to provide a theoretical foundation of our approach for positive and continuous stochastic asset price processes. When such a condition is not satisfied, we focus on local volatility models. To this purpose, we give a new necessary and sufficient condition for a process with time‐dependent local volatility function to be a strict local martingale.

The sustainability of stock price fluctuations indicated by many empirical studies hardly reconciles with the existing models in standard financial theories. This paper proposes a recursive dynamic asset pricing model based on the comprehensive impact of the sentiment investor, the information trader and the noise trader. The dynamic process of the asset price is characterized and a numerical simulation of the model is provided. The model captures the features of the actual stock price that are consistent with the empirical evidence on the sustainability of stock price fluctuations. It also offers a partial explanation for other financial anomalies, for example, asset price’s overreaction, asset bubble and the financial crisis. The major finding is that investor sentiment is the key factor to understand the sustainability of stock price fluctuations.

The relationship between quadratic variation for compound renewal processes and M-Wright functions is discussed. The convergence of quadratic variation is investigated both as a random variable (for given t) and as a stochastic process.

We consider any continuous hermitian kernel M ( Δ , Δ ′ ) M(\Delta ,\Delta ’) on P × P \mathcal {P} \times \mathcal {P} where P \mathcal {P} is the prering of intervals of [0,1]. Conditions on M are given to find an interval covariance function K ( Δ , Δ ′ ) K(\Delta ,\Delta ’) so that K ( Δ , Δ ′ ) = M ( Δ , Δ ′ ) K(\Delta ,\Delta ’) = M(\Delta ,\Delta ’) for all nonoverlapping Δ \Delta and Δ ′ \Delta ’ in P \mathcal {P} . The problem is solved by first treating finite hermitian matrices A and finding a positive definite matrix B so that b i j = a i j , i ≠ j {b_{ij}} = {a_{ij}},i \ne j , so that tr B is minimized. Using natural correspondence between interval covariance functions and stochastic processes, a decomposition theorem is derived for stochastic processes of bounded quadratic variation into an orthogonal process and a process having minimal quadratic variation.

In the years following the publication of Black and Scholes (J Political Econ, 81(3), 637–654, 1973), numerous alternative models have been proposed for pricing and hedging equity derivatives. Prominent examples include stochastic volatility models, jump-diffusion models, and models based on Levy processes. These all have their own shortcomings, and evidence suggests that none is up to the task of satisfactorily pricing and hedging extremely long-dated claims. Since they all fall within the ambit of risk-neutral valuation, it is natural to speculate that the deficiencies of these models are (at least in part) attributable to the constraints imposed by the risk-neutral approach itself. To investigate this idea, we present a simple two-parameter model for a diversified equity accumulation index. Although our model does not admit an equivalent risk-neutral probability measure, it nevertheless fulfils a minimal no-arbitrage condition for an economically viable financial market. Furthermore, we demonstrate that contingent claims can be priced and hedged, without the need for an equivalent change of probability measure. Convenient formulae for the prices and hedge ratios of a number of standard European claims are derived, and a series of hedge experiments for extremely long-dated claims on the S&P 500 total return index are conducted. Our model serves also as a convenient medium for illustrating and clarifying several points on asset price bubbles and the economics of arbitrage.

Under proportional transaction costs, a price process is said to have a consistent price system, if there is a semimartingale with an equivalent martingale measure that evolves within the bid-ask spread. We show that a continuous, multi-asset price process has a consistent price system, under arbitrarily small proportional transaction costs, if it satisfies a natural multi-dimensional generalization of the stickiness condition introduced by Guasoni (2006).

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Market Imperfections, Equilibrium and Arbitrage