Entropy martingale optimal transport and nonlinear pricing–hedging duality

The objective of this paper is to develop a duality between a novel entropy martingale optimal transport (EMOT) problem and an associated optimisation problem. In EMOT, we follow the approach taken in the entropy optimal transport (EOT) problem developed in Liero et al. (Invent. Math. 211:969–1117, 2018), but we add the constraint, typical of martingale optimal transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures. In the associated problem, the objective functional, related via Fenchel conjugacy to the entropic term in EMOT, is no longer linear as in (martingale) optimal transport. This leads to a novel optimisation problem which also has a clear financial interpretation as a nonlinear subhedging problem. Our theory allows us to establish a nonlinear robust pricing–hedging duality which also covers a wide range of known robust results. We also focus on Wasserstein-induced penalisations and study how the duality is affected by variations in the penalty terms, with a special focus on the convergence of EMOT to the extreme case of MOT.

     

Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem

In this paper, we study a class of quadratic backward stochastic differential equations (BSDEs), which arises naturally in the utility maximization problem with portfolio constraints. We first establish the existence and uniqueness of solutions for such BSDEs and then give applications to the utility maximization problem. Three cases of utility functions, the exponential, power, and logarithmic ones, are discussed. This study is part of my PhD thesis supervised by Professor Ying Hu and defended at the University of Rennes 1 (in France) in October 2007.     

Changes of numeraire for pricing futures, forwards, and options

A change of numeraire argument is used to derive a general optionparity, or equivalence, result relating American call and putprices, and to obtain new expressions for futures and forwardprices. The general parity result unifies and extends a numberof existing results. The new futures and forward pricing formulasare often simpler to compute in multifactor models than existingalternatives. We also extend previous work by deriving a generalformula relating exchange options to ordinary call options.A number of applications to diffusion models, including stochasticvolatility, stochastic interest rate, and stochastic dividendrate models, and jump-diffusion models are examined.     

Option pricing and the martingale restriction

In the absence of frictions, the value of the under-lying assetimplied by option prices must equal its actual market value.With frictions, however, this requirement need not hold. UsingS&P 100 index options data, I find that the implied costof the index is significantly higher in the options market thanin the stock market, and is directly related to measures oftransaction costs and liquidity. I show that the Black-Scholesmodel has strong bid-ask spread, trading volume, and open interestbiases. Option pricing models that relax the martingale restrictionperform significantly better.     

Testing the martingale hypothesis for gross returns

We propose an alternative ratio statistic for measuring predictability of stock prices. Our statistic is based on actual returns rather than logarithmic returns and is therefore better suited to capturing price predictability. It captures not only linear dependence in the same way as the variance ratio statistics of Lo and MacKinlay (1988) but also some nonlinear dependencies. We derive the asymptotic distribution of the statistics under the null hypothesis that simple gross returns are unpredictable after a constant mean adjustment. This represents a test of the weak form of the Efficient Market Hypothesis. We also consider the multivariate extension, in particular, we derive the restrictions implied by the EMH on multiperiod portfolio gross returns. We apply our methodology to test the gross return predictability of various financial series.     

Testing the martingale restriction for option implied densities

This paper contributes to the forecasting literature by presenting a new evaluation method for density and probability estimates. This procedure is particularly well suited for analyzing time series of forecasts implied from option prices, although the results are very general and can be applied also outside this framework. A small scale simulation study documents that valid and accurate inference can be drawn for option implied densities with the proposed method. The new testing procedure is demonstrated in an empirical application on density estimates implied from $/£ currency options.     

An explicit martingale version of the one-dimensional Brenier theorem

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge–Kantorovich mass transport problem was introduced in (Beiglböck et al. in Finance Stoch. 17:477–501, 2013; Galichon et al. in Ann. Appl. Probab. 24:312–336, 2014). Further, by suitable adaptation of the notion of cyclical monotonicity, Beiglböck and Juillet (Ann. Probab. 44:42–106, 2016) obtained an extension of the one-dimensional Brenier theorem to the present martingale version. In this paper, we complement the previous work by extending the so-called Spence–Mirrlees condition to the case of martingale optimal transport. Under some technical conditions on the starting and the target measures, we provide an explicit characterization of the corresponding optimal martingale transference plans both for the lower and upper bounds. These explicit extremal probability measures coincide with the unique left- and right-monotone martingale transference plans introduced in (Beiglböck and Juillet in Ann. Probab. 44:42–106, 2016). Our approach relies on the (weak) duality result stated in (Beiglböck et al. in Finance Stoch. 17:477–501, 2013), and provides as a by-product an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.     

Momentum and reversion in risk neutral martingale probabilities

A time homogeneous, purely discontinuous, parsimonous Markov martingale model is proposed for the risk neutral dynamics of equity forward prices. Transition probabilities are in the variance gamma class with spot dependent parameters. Markov chain approximations give access to option prices. The model is estimated on option prices across strike and maturity for five days at a time. Properties of the estimated processes are described via an analysis of return quantiles, momentum functions that measure the response of tail probabilities to such moves. Momentum and reversion are also addressed via the construction of reverse conditional expectations. Term structures for the moments of marginal distributions support a decay in skewness and excess kurtosis with maturity at rates slower than those implied by Lévy processes. Out of sample performance is additionally reported. It is observed that risk neutral dynamics by and large reflect the presence of momentum in numerous probabilities. However, there is some reversion in the upper quantiles of risk neutral return distributions.     

A class of stochastic volatility models and the q-optimal martingale measure

This paper proposes an approach under which the q-optimal martingale measure, for the case where continuous processes describe the evolution of the asset price and its stochastic volatility, exists for all finite time horizons. More precisely, it is assumed that while the ‘mean–variance trade-off process’ is uniformly bounded, the volatility and asset are imperfectly correlated. As a result, under some regularity conditions for the parameters of the corresponding Cauchy problem, one obtains that the qth moment of the corresponding Radon–Nikodym derivative does not explode in finite time.     

A note on the existence of unique equivalent martingale measures in a Markovian setting

Simple sufficient conditions for the existence of a unique equivalent martingale measure are provided. Furthermore, these conditions give us a handle on situations where an equivalent martingale measure cannot exist. The existence of a unique equivalent martingale measure is of relevance to problems in mathematical finance. Two examples of models for which the question of existence was unresolved are studied. By means of our results existence of a unique equivalent measure up to an explosion time is proved.     

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

In an incomplete financial market model, we study a flow in the space of equivalent martingale measures and the corresponding shifting perception of the fundamental value of a given asset. This allows us to capture the birth of a perceived bubble and to describe it as an initial submartingale which then turns into a supermartingale before it falls back to its initial value zero.     

Robust pricing and hedging under trading restrictions and the emergence of local martingale models

We pursue the robust approach to pricing and hedging in which no probability measure is fixed, but call or put options with different maturities and strikes can be traded initially at their market prices. We allow the inclusion of robust modelling assumptions by specifying a set of feasible paths on which (super)hedging arguments are required to work. In a discrete-time setup with no short selling, we characterise absence of arbitrage and show that if call options are traded, then the usual pricing–hedging duality is preserved. In contrast, if only put options are traded, a duality gap may appear. Embedding the results into a continuous-time framework, we show that the duality gap may be interpreted as a financial bubble and link it to strict local martingales. This provides an intrinsic justification of strict local martingales as models for financial bubbles arising from a combination of trading restrictions and current market prices.     

ARCH and GARCH models vs. martingale volatility of finance market returns

ARCH and GARCH models assume either i.i.d. or ‘white noise’ as is usual in regression analysis, while also assuming memory in a conditional mean square fluctuation with stationary increments. We will show that ARCH/GARCH is inconsistent with uncorrelated increments, violating the i.i.d. and ‘white’ assumptions, and violating finance data and the efficient market hypothesis as well.     

如何应对外资银行竞争问题初探

按照我国跨入WTO大门时为金融市场对外开放制订的时间表，中国银行业从2006年12月11日起对外资全面开放，对所有外资银行实现“国民待遇”。我们叫喊多年“狼来了”之后，外国的“银狼”真的来了。中外资银行之间面对面的较量已经拉开帷幕。面对外资银行的激烈竞争，中国银行业需要做些什么？     

The dynamics of the free-rider problem in takeovers

We explore the dynamics of a takeover bid. In contrast to precedingmodels, if the initial takeover bid is unsuccessful a raideris allowed to make a new tender offer in order to try and securethe remaining shares. Numerical analysis shows that the raider'stender offer rises over time as be accumulates more shares.The anticipation of a higher tender offer in the future makesshareholders more inclined to hold their shares and forces theraider to offer a higher premium than is predicted by statictheories. As the time between tender offers goes to zero, weshow analytically that the expected profit from engaging ina takeover goes to zero.