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.     

Esscher transforms and the minimal entropy martingale measure for exponential Lévy models

In this paper we offer a systematic survey and comparison of the Esscher martingale transform for linear processes, the Esscher martingale transform for exponential processes, and the minimal entropy martingale measure for exponential Lévy models, and present some new results in order to give a complete characterization of those classes of measures. We illustrate the results with several concrete examples in detail.     

Approximating functionals of local martingales under lack of uniqueness of the Black–Scholes PDE solution

When the underlying stock price is a strict local martingale process under an equivalent local martingale measure, the Black–Scholes PDE associated with a European option may have multiple solutions. In this paper, we study an approximation for the smallest hedging price of such an European option. Our results show that a class of rebate barrier options can be used for this approximation. Among them, a specific rebate option is also provided with a continuous rebate function, which corresponds to the unique classical solution of the associated parabolic PDE. Such a construction makes existing numerical PDE techniques applicable for its computation. An asymptotic convergence rate is also studied when the knock-out barrier moves to infinity under suitable conditions.     

GARCH options via local risk minimization

We apply a quadratic hedging scheme developed by Föllmer, Schweizer, and Sondermann to European contingent products whose underlying asset is modeled using a GARCH process and show that local risk-minimizing strategies with respect to the physical measure do exist, even though an associated minimal martingale measure is only available in the presence of bounded innovations. More importantly, since those local risk-minimizing strategies are in general convoluted and difficult to evaluate, we introduce Girsanov-like risk-neutral measures for the log-prices that yield more tractable and useful results. Regarding this subject, we focus on GARCH time series models with Gaussian innovations and we provide specific sufficient conditions concerning the finiteness of the kurtosis, under which those martingale measures are appropriate in the context of quadratic hedging. When this equivalent martingale measure is adapted to the price representation we are able to recover the classical pricing formulas of Duan and Heston and Nandi, as well as hedging schemes that improve the performance of those proposed in the literature.     

Optional decomposition and Lagrange multipliers

Let be the set of equivalent martingale measures for a given process , and let be a process which is a local supermartingale with respect to any measure in . The optional decomposition theorem for states that there exists a predictable integrand such that the difference is a decreasing process. In this paper we give a new proof which uses techniques from stochastic calculus rather than functional analysis, and which removes any boundedness assumption.     

On the hedging of options on exploding exchange rates

We study a novel pricing operator for complete, local martingale models. The new pricing operator guarantees put-call parity to hold for model prices and the value of a forward contract to match the buy-and-hold strategy, even if the underlying follows strict local martingale dynamics. More precisely, we discuss a change of numéraire (change of currency) technique when the underlying is only a local martingale, modelling for example an exchange rate. The new pricing operator assigns prices to contingent claims according to the minimal cost for superreplication strategies that succeed with probability one for both currencies as numéraire. Within this context, we interpret the lack of the martingale property of an exchange rate as a reflection of the possibility that the numéraire currency may devalue completely against the asset currency (hyperinflation).     

No arbitrage of the first kind and local martingale numéraires

A supermartingale deflator (resp. local martingale deflator) multiplicatively transforms nonnegative wealth processes into supermartingales (resp. local martingales). A supermartingale numéraire (resp. local martingale numéraire) is a wealth process whose reciprocal is a supermartingale deflator (resp. local martingale deflator). It has been established in previous works that absence of arbitrage of the first kind (\(\mbox{NA}_{1}\)) is equivalent to the existence of the (unique) supermartingale numéraire, and further equivalent to the existence of a strictly positive local martingale deflator; however, under \(\mbox{NA}_{1}\), a local martingale numéraire may fail to exist. In this work, we establish that under \(\mbox{NA}_{1}\), a supermartingale numéraire under the original probability \(P\) becomes a local martingale numéraire for equivalent probabilities arbitrarily close to \(P\) in the total variation distance.     

Filtration shrinkage,the structure of deflators,and failure of market completeness

Finance and Stochastics - We analyse the structure of local martingale deflators projected on smaller filtrations. In a general continuous-path setting, we show that the local martingale parts in...     

No-arbitrage up to random horizon for quasi-left-continuous models

This paper studies the impact, on no-arbitrage conditions, of stopping the price process at an arbitrary random time. As price processes, we consider the class of quasi-left-continuous semimartingales, i.e., semimartingales that do not jump at predictable stopping times. We focus on the condition of no unbounded profit with bounded risk (called NUPBR), also known in the literature as no arbitrage of the first kind. The first principal result describes all the pairs of quasi-left-continuous market models and random times for which the resulting stopped model fulfils NUPBR. Furthermore, for a subclass of quasi-left-continuous local martingales, we construct explicitly martingale deflators, i.e., strictly positive local martingales whose product with the price process stopped at a random time is a local martingale. The second principal result characterises the random times that preserve NUPBR under stopping for any quasi-left-continuous model. The analysis carried out in the paper is based on new stochastic developments in the theory of progressive enlargements of filtrations.     

Option pricing in the presence of natural boundaries and a quadratic diffusion term

This paper uses a probabilistic change-of-numeraire technique to compute closed-form prices of European options to exchange one asset against another when the relative price of the underlying assets follows a diffusion process with natural boundaries and a quadratic diffusion coefficient. The paper shows in particular how to interpret the option price formula in terms of exercise probabilities which are calculated under the martingale measures associated with two specific numeraire portfolios. An application to the pricing of bond options and certain interest rate derivatives illustrates the main results.     

A NO-ARBITRAGE MARTINGALE ANALYSIS FOR JUMP-DIFFUSION VALUATION

This study presents a jump-diffusion valuation framework using the no-arbitrage martingale approach. Equilibrium conditions needed to support a jump-diffusion pricing standard process are derived. The results are a generalized jump-diffusion security market line and its corresponding equilibrium valuation relation that prices both jump and diffusion risk. To value options, a fundamental formula is derived that includes existing jump-diffusion option valuation formulas as special cases. 1 find Merton's (1976a) assumption of diversifiable jump risk to be consistent with no-arbitrage only when the aggregate consumption flow does not jump. Simulation shows that Merton's formula undervalues/overvalues options on hedging/cyclical assets. When the jump arrival frequency is larger, the mispricing is larger/smaller for in-the-money/out-of-the-money options.     

Hedging American Options in Merton's Model: A Locally Risk Minimizing Approach

An interesting problem, related to American options in incomplete markets, is the possibility to select a preferable equivalent martingale measure in order to compute the prices. With this in mind, we consider a particular option that may be viewed as a finite collection of suitable European options and for which the minimal martingale measure permits the minimization of the local risk. Since this option is an approximation of the American put, the stability result presented, concerning the portfolio decomposition, also suggests an argument in favor of the minimal martingale measure in the American case.     

On <Emphasis Type="Italic">q</Emphasis>-optimal martingale measures in exponential Lévy models

We give a sufficient condition to identify the q-optimal signed and the q-optimal absolutely continuous martingale measures in exponential Lévy 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. C.N. gratefully acknowledges financial support by UniCredit, Markets and Investment Banking. However, this paper does not reflect the opinion of UniCredit, Markets and Investment Banking, it is the personal view of the authors.     

Joint analysis and estimation of stock prices and trading volume in Barndorff-Nielsen and Shephard stochastic volatility models

We introduce a variant of the Barndorff-Nielsen and Shephard stochastic volatility model where the non-Gaussian Ornstein–Uhlenbeck process describes some measure of trading intensity like trading volume or number of trades instead of unobservable instantaneous variance. We develop an explicit estimator based on martingale estimating functions in a bivariate model that is not a diffusion, but admits jumps. It is assumed that both the quantities are observed on a discrete grid of fixed width, and the observation horizon tends to infinity. We show that the estimator is consistent and asymptotically normal and give explicit expressions of the asymptotic covariance matrix. Our method is illustrated by a finite sample experiment and a statistical analysis of IBM? stock from the New York Stock Exchange and Microsoft Corporation? stock from Nasdaq during a history of five years.     

Stochastic orders in dynamic reinsurance markets

We consider a dynamic reinsurance market, where the traded risk process is driven by a compound Poisson process and where claim amounts are unbounded. These markets are known to be incomplete, and there are typically infinitely many martingale measures. In this case, no-arbitrage pricing theory can typically only provide wide bounds on prices of reinsurance claims. Optimal martingale measures such as the minimal martingale measure and the minimal entropy martingale measure are determined, and some comparison results for prices under different martingale measures are provided. This leads to a simple stochastic ordering result for the optimal martingale measures. Moreover, these optimal martingale measures are compared with other martingale measures that have been suggested in the literature on dynamic reinsurance markets.Received: March 2004, Mathematics Subject Classification (2000): 62P05, 60J75, 60G44JEL Classification: G10     

Diversified Portfolios in Continuous Time

We study a financial market containing an infinite number ofassets, where each asset price is driven by an idiosyncraticrandom source as well as by a systematic noise term. Introducing"asymptotic assets" which correspond to certain infinitely welldiversified portfolios we study absence of (asymptotic) arbitrage,and in this context we obtain continuous time extensions ofatemporal APT results. We also study completeness and derivativepricing, showing that the possibility of forming infinitelywell diversified portfolios has the property of completing themarket. It also turns out that models where the all risk isof diffusion type are qualitatively quite different from modelswhere one risk is of diffusion type and the other is of Poissontype. We also present a simple martingale based theory for absenceof asymptotic arbitrage. JEL Classification: G12, G13,     

Lévy term structure models: No-arbitrage and completeness

The Lévy 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.Received: May 2004, Mathematics Subject Classification (2000): 60H30, 91B28, 60G51JEL Classification: E43, G13Work supported in part by the European Communitys Human Potential Programme under contract HPRN-CT-2000-00100, DYNSTOCH.