A COUNTEREXAMPLE CONCERNING THE VARIANCE-OPTIMAL MARTINGALE MEASURE

The present note addresses an open question concerning a sufficient characterization of the variance-optimal martingale measure. Denote by S the discounted price process of an asset and suppose that   Q ^★  is an equivalent martingale measure whose density is a multiple of  1 −φ· S _T   for some S -integrable process φ. We show that   Q ^★  does not necessarily coincide with the variance-optimal martingale measure, not even if  φ· S   is a uniformly integrable   Q ^★  -martingale.     

ON COMPONENTWISE and VECTOR STOCHASTIC INTEGRATION

We give a condition under which the componentwise stochastic integration with respect to a given R ^d -valued continuous local martingale coincides with the more general vector stochastic integration defined by Jacod (1979). We then provide a result on the equivalence between the vector and the component completeness of a financial market in a special case.     

Arbitrage and Growth Rate for Riskless Investments in a Stationary Economy

A sequential investment is a vector of payments over time, ( a ₀, a ₁, ... , a_n ), where a payment is made to or by the investor according as a_i is positive or negative. Given a collection of such investments it may be possible to assemble a portfolio from which an investor can get "something for nothing," meaning that without investing any money of his own he can receive a positive return after some finite number of time periods. Cantor and Lipmann (1995) have given a simple necessary and sufficient condition for a set of investments to have this property. We present a short proof of this result. If arbitrage is not possible, our result leads to a simple derivation of the expression for the long–run growth rate of the set of investments in terms of its "internal rate of return."     

A Simple Counterexample to Several Problems in the Theory of Asset Pricing

We give an easy example of two strictly positive local martingales that fail to be uniformly integrable, but such that their product is a uniformly integrable martingale. The example simplifies an earlier example given by the second author. We give applications in mathematical finance and we show that the phenomenon is present in many incomplete markets.     

ON THE MARTINGALE PROPERTY IN STOCHASTIC VOLATILITY MODELS BASED ON TIME‐HOMOGENEOUS DIFFUSIONS

Lions and Musiela give sufficient conditions to verify when a stochastic exponential of a continuous local martingale is a martingale or a uniformly integrable martingale. Blei and Engelbert and Mijatovi? and Urusov give necessary and sufficient conditions in the case of perfect correlation (). For financial applications, such as checking the martingale property of the stock price process in correlated stochastic volatility models, we extend their work to the arbitrary correlation case (). We give a complete classification of the convergence properties of both perpetual and capped integral functionals of time‐homogeneous diffusions and generalize results in Mijatovi? and Urusov with direct proofs avoiding the use of separating times (concept introduced by Cherny and Urusov and extensively used in the proofs of Mijatovi? and Urusov).     

Conic martingales from stochastic integrals

In this paper, we introduce the concept of conic martingales. This class refers to stochastic processes that have the martingale property but that evolve within given (possibly time‐dependent) boundaries. We first review some results about the martingale property of solution to driftless stochastic differential equations. We then provide a simple way to construct and handle such processes. Specific attention is paid to martingales in [0, 1]. One of these martingales proves to be analytically tractable. It is shown that up to shifting and rescaling constants, it is the only martingale (with the trivial constant, Brownian motion, and geometric Brownian motion) having a separable diffusion coefficient and that can be obtained via a time‐homogeneous mapping of Gaussian diffusions. The approach is exemplified by modeling stochastic conditional survival probabilities in the univariate and bivariate cases.     

OPTIMAL INVESTMENT WITH AN UNBOUNDED RANDOM ENDOWMENT AND UTILITY-BASED PRICING

This paper studies the problem of maximizing the expected utility of terminal wealth for a financial agent with an unbounded random endowment, and with a utility function which supports both positive and negative wealth. We prove the existence of an optimal trading strategy within a class of permissible strategies—those strategies whose wealth process is a super-martingale under all pricing measures with finite relative entropy. We give necessary and sufficient conditions for the absence of utility-based arbitrage, and for the existence of a solution to the primal problem. We consider two utility-based methods which can be used to price contingent claims. Firstly we investigate marginal utility-based price processes (MUBPP's). We show that such processes can be characterized as local martingales under the normalized optimal dual measure for the utility maximizing investor. Finally, we present some new results on utility indifference prices, including continuity properties and volume asymptotics for the case of a general utility function, unbounded endowment and unbounded contingent claims.     

A Counterexample to Several Problems In the Theory of Asset Pricing

We construct a continuous bounded stochastic process ( S _t,) _1E[0,1] which admits an equivalent martingale measure but such that the minimal martingale measure in the sense of Föllmer and Schweizer does not exist. This example also answers (negatively) a problem posed by Karatzas, Lehozcky, and Shreve as well as a problem posed by Strieker.     

The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time

We prove a version of the Fundamental Theorem of Asset Pricing, which applies to Kabanov's modeling of foreign exchange markets under transaction costs. The financial market is described by a   d × d   matrix-valued stochastic process  (Π _t ) ^T _{t =0}  specifying the mutual bid and ask prices between d assets. We introduce the notion of "robust no arbitrage," which is a version of the no-arbitrage concept, robust with respect to small changes of the bid-ask spreads of  (Π _t ) ^T _{t =0}  . The main theorem states that the bid-ask process  (Π _t ) ^T _{t =0}  satisfies the robust no-arbitrage condition iff it admits a strictly consistent pricing system. This result extends the theorems of Harrison-Pliska and Kabanov-Stricker pertaining to the case of finite Ω, as well as the theorem of Dalang, Morton, and Willinger and Kabanov, Rásonyi, and Stricker, pertaining to the case of general Ω. An example of a  5 × 5  -dimensional process  (Π _t )² _{t =0}  shows that, in this theorem, the robust no-arbitrage condition cannot be replaced by the so-called strict no-arbitrage condition, thus answering negatively a question raised by Kabanov, Rásonyi, and Stricker.     

OPTIMAL INVESTMENT STRATEGIES FOR CONTROLLING DRAWDOWNS

We analyze the optimal risky investment policy for an investor who, at each point in time, wants to lose no more than a fixed percentage of the maximum value his wealth has achieved up to that time. In particular, if M _t is the maximum level of wealth W attained on or before time t , then the constraint imposed on his portfolio choice is that W_tα M _t, where α is an exogenous number betweenα O and 1. We show that, for constant relative risk aversion utility functions, the optimal policy involves an investment in risky assets at time t in proportion to the "surplus" W _t - α M _t. the optimal policy may appear similar to the constant-proportion portfolio insurance policy analyzed in Black and Perold (1987) and Grossman and Vila (1989). However, in those papers, the investor keeps his wealth above a nonstochastic floor F instead of a stochastic floor α M _t. the stochastic character of the floor studied here has interesting effects on the investment policy in states of nature when wealth is at an all-time high; i.e., when Wt = M _t. It can be shown that at W _t= M _t, α M _t is expected to grow at a faster rate than W _t, and therefore the investment in the risky asset can be expected to fall. We also show that the investment in the risky asset can be expected to rise when W _t is close to α M _t. We conjecture that in an equilibrium model the stochastic character of the floor creates "resistance" levels as the market approaches an all-time high (because of the reluctance of investors to take more risk when W _t= M _t).     

MARTINGALE MEASURES FOR DISCRETE-TIME PROCESSES WITH INFINITE HORIZON

Let ( S_t ) _tεI be an R^d-valued adapted stochastic process on (Ω, , ( _t ) _tεI , P ). A basic problem occurring notably in the analysis of securities markets, is to decide whether there is a probability measure Q on  equivalent to P such that ( S_t ) _tεI is a martingale with respect to Q. It is known (see the fundamental papers of Harrison and Kreps 1979; Harrison and Pliska 1981; and Kreps 1981) that there is an intimate relation of this problem with the notions of "no arbitrage" and "no free lunch" in financial economics. We introduce the intermediate concept of "no free lunch with bounded risk." This is a somewhat more precise version of the notion of "no free lunch." It requires an absolute bound of the maximal loss occurring in the trading strategies considered in the definition of "no free lunch." We give an argument as to why the condition of "no free lunch with bounded risk" should be satisfied by a reasonable model of the price process ( S_t ) _tεI of a securities market. We can establish the equivalence of the condition of "no free lunch with bounded risk" with the existence of an equivalent martingale measure in the case when the index set I is discrete but (possibly) infinite. A similar theorem was recently obtained by Delbaen (1992) for continuous-time processes with continuous paths. We can combine these two theorems to get a similar result for the continuous-time case when the process ( S_t ) _{t εR+} is bounded and, roughly speaking, the jumps occur at predictable times. In the infinite horizon setting, the price process has to be "almost a martingale" in order to allow an equivalent martingale measure.     

Generic Existence and Robust Nonexistence of Numéraires in Finite Dimensional Securities Markets

A numéraire is a portfolio that, if prices and dividends are denominated in its units, admits an equivalent martingale measure that transforms all gains processes into martingales. We first supply a necessary and sufficient condition for the generic existence of numéraires in a finite dimensional setting. We then characterize the arbitrage‐free prices and dividends for which the absence of numéraires survives any small perturbation preserving no arbitrage. Finally, we identify the cases when any small, but otherwise arbitrary, perturbation of prices and dividends preserves either the existence of numéraires, or their nonexistence under no arbitrage.     

PRICING OF AMERICAN PATH-DEPENDENT CONTINGENT CLAIMS

We consider the problem of pricing path-dependent contingent claims. Classically, this problem can be cast into the Black-Scholes valuation framework through inclusion of the path-dependent variables into the state space. This leads to solving a degenerate advection-diffusion partial differential equation (PDE). We first estabilish necessary and sufficient conditions under which degenerate diffusions can be reduced to lower-dimensional nondegenerate diffusions. We apply these results to path-dependent options. Then, we describe a new numerical technique, called forward shooting grid (FSG) method, that efficiently copes with degenerate diffusion PDEs. Finally, we show that the FSG method is unconditionally stable and convergent. the FSG method is the first capable of dealing with the early exercise condition of American options. Several numerical examples are presented and discussed. ²     

THE TERM STRUCTURE OF INTEREST RATES AS A GAUSSIAN RANDOM FIELD

A simple model of the term structure of interest rates is introduced in which the family of instantaneous forward rates evolves as a continuous Gaussian random field. A necessary and sufficient condition for the associated family of discounted zero-coupon bond prices to be martingales is given, permitting the consistent pricing of interest rate contingent claims. Examples of the pricing of interest-rate caps and the situation when the Gaussian random field may be viewed as a deterministic time change of the standard Brownian sheet are discussed.     

BIVARIATE SUPPORT OF FORWARD LIBOR AND SWAP RATES

Based on a certain notion of "prolific process," we find an explicit expression for the bivariate (topological) support of the solution to a particular class of 2 × 2 stochastic differential equations that includes those of the three-period "lognormal" Libor and swap market models. This yields that in the lognormal swap market model (SMM), the support of the 1 × 1 forward Libor   L * _t   equals  [ l * _t , ∞)  for some semi-explicit  −1 ≤ l * _t ≤ 0  , sharpening a result of Davis and Mataix-Pastor (2007) that forward Libor rates (eventually) become negative with positive probability in the lognormal SMM. We classify the instances   l * _t < 0  , and explicitly calculate the threshold time at or before which   L * _t   remains positive a.s.     

Pareto Equilibria with coherent measures of risk

In this paper, we provide a definition of Pareto equilibrium in terms of risk measures, and present necessary and sufficient conditions for equilibrium in a market with finitely many traders (whom we call "banks") who trade with each other in a financial market. Each bank has a preference relation on random payoffs which is monotonic, complete, transitive, convex, and continuous; we show that this, together with the current position of the bank, leads to a family of valuation measures for the bank. We show that a market is in Pareto equilibrium if and only if there exists a (possibly signed) measure that, for each bank, agrees with a positive convex combination of all valuation measures used by that bank on securities traded by that bank.     

ADMISSIBILITY OF GENERIC MARKET MODELS OF FORWARD SWAP RATES

Our main goal is to re‐examine and extend certain results from the papers by Galluccio et al. and Pietersz and van Regenmortel. We establish several results providing alternate necessary and sufficient conditions for admissibility of a family of forward swaps, that is, the property that it is supported by a (positive) family of bonds associated with the underlying tenor structure. We also derive the generic expression for the joint dynamics of a family of forward swap rates under a single probability measure and we show that these dynamics are uniquely determined by a selection of volatility processes with respect to the set of driving martingales.     

When Does Convergence of Asset Price Processes Imply Convergence of Option Prices?

We consider weak convergence of a sequence of asset price models (Sⁿ) to a limiting asset price model S . A typical case for this situation is the convergence of a sequence of binomial models to the Black–Scholes model, as studied by Cox, Ross, and Rubinstein. We put emphasis on two different aspects of this convergence: first we consider convergence with respect to the given "physical" probability measures (P^n) and second with respect to the "risk‐neutral" measures (Q^n) for the asset price processes (Sⁿ) . (In the case of nonuniqueness of the risk-neutral measures the question of the "good choice" of (Qⁿ) also arises.) In particular we investigate under which conditions the weak convergence of (Pⁿ) to P implies the weak convergence of (Qⁿ) to Q and thus the convergence of prices of derivative securities.
The main theorem of the present paper exhibits an intimate relation of this question with contiguity properties of the sequences of measures (Pⁿ) with respect to (Qⁿ) , which in turn is closely connected to asymptotic arbitrage properties of the sequence (Sⁿ) of security price processes. We illustrate these results with general homogeneous binomial and some special trinomial models.     

Robust martingale selection problem and its connections to the no‐arbitrage theory

We analyze the martingale selection problem of Rokhlin in a pointwise (robust) setting. We derive conditions for solvability of this problem and show how it is related to the classical no‐arbitrage deliberations. We obtain versions of the Fundamental Theorem of Asset Pricing in models spanning frictionless markets, models with proportional transaction costs, and models for illiquid markets. In all these models, we also incorporate trading constraints.     

ON PROPERTIES OF ANALYTICALLY SOLVABLE FAMILIES OF LOCAL VOLATILITY DIFFUSION MODELS

We present some further developments in the construction and classification of new solvable one‐dimensional diffusion models having transition densities, and other quantities that are fundamental to derivatives pricing, representable in analytically closed form. Our approach is based on so‐called diffusion canonical transformations that produce a large class of multiparameter nonlinear local volatility diffusion models that are mapped onto various simpler diffusions. Using an asymptotic analysis, we arrive at a rigorous boundary classification as well as a characterization with respect to probability conservation and the martingale property of the newly constructed diffusions. Specifically, we analyze and classify in detail four main families of driftless regular diffusion models that arise from the underlying squared Bessel process (the Bessel family), Cox–Ingersoll–Ross process (the confluent hypergeometric family), the Ornstein‐Uhlenbeck diffusion (the OU family), and the Jacobi diffusion (the hypergeometric family). We show that the Bessel family is a superset of the constant elasticity of variance model without drift. The Bessel family, in turn, is nested by the confluent hypergeometric family. For these two families we find further subfamilies of conservative strict supermartingales and nonconservative martingales with an exit boundary. For the new classes of nonconservative regular diffusions we also derive analytically exact first exit time densities that are given in terms of generalized inverse Gaussians and extensions. As for the two other new models, we show that the OU family of processes are conservative strict martingales, whereas the Jacobi family are nonconservative nonmartingales. Considered as asset price diffusion models, we also show that these models demonstrate a wide range of local volatility shapes and option implied volatility surfaces that include various pronounced skew and smile patterns.