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.     

A NOTE ON ARBITRAGE AND CLOSED CONVEX CONES

We give an example of a subspace K of     such that     , where     denotes the closure with respect to convergence in probablity. On the other hand, the cone   C ≔ K − L ^∞₊  is dense in   L ^∞  with respect to the weak-star topology  σ( L ^∞, L ¹)  . This example answers a question raised by I. Evstigneev. The topic is motivated by the relation of the notion of no arbitrage and the existence of martingale measures in Mathematical Finance.     

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.     

STOCHASTIC VOLATILITY MODELS, CORRELATION, AND THE q-OPTIMAL MEASURE

The aim of this paper is to study the minimal entropy and variance-optimal martingale measures for stochastic volatility models. In particular, for a diffusion model where the asset price and volatility are correlated, we show that the problem of determining the q -optimal measure can be reduced to finding a solution to a representation equation. The minimal entropy measure and variance-optimal measure are seen as the special cases   q = 1  and   q = 2  respectively. In the case where the volatility is an autonomous diffusion we give a stochastic representation for the solution of this equation. If the correlation ρ between the traded asset and the autonomous volatility satisfies  ρ² < 1/ q   , and if certain smoothness and boundedness conditions on the parameters are satisfied, then the q -optimal measure exists. If  ρ²≥ 1/ q   , then the q -optimal measure may cease to exist beyond a certain time horizon. As an example we calculate the q -optimal measure explicitly for the Heston model.     

RISK INDIFFERENCE PRICING IN JUMP DIFFUSION MARKETS

We study the risk indifference pricing principle in incomplete markets: The (seller's)  risk indifference price        is the initial payment that makes the  risk  involved for the seller of a contract equal to the risk involved if the contract is not sold, with no initial payment. We use stochastic control theory and PDE methods to find a formula for       and similarly for      . In particular, we prove that  where    p _low   and    p _up   are the lower and upper hedging prices, respectively.     

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.     

ESTIMATION OF VALUE AT RISK AND RUIN PROBABILITY FOR DIFFUSION PROCESSES WITH JUMPS

In this paper we give upper bounds for both the Value at Risk   VaR _α,  0 < α < 1  , and for ruin probabilities associated with the supremum of a process driven by a Brownian motion and a compound Poisson process. We obtain lower bounds for the same Value at Risk, and for different cases we discuss the behavior of the bounds for small α. We prove our bounds are "asymptotically" optimal, as α tends to zero. The ruin probabilities obtained are related to other bounds found in recent literature.     

Step Options

Motivated by risk management problems with barrier options, we propose a flexible modification of the standard knock‐out and knock‐in provisions and introduce a family of path‐dependent options: step options . They are parametrized by a finite knock‐out (knock‐in) rate , ρ. For a down‐and‐out step option, its payoff at expiration is defined as the payoff of an otherwise identical vanilla option discounted by the knock‐out factor exp(-ρτ_B^‐) or max(1‐ρτ^-B,0), where &\tau;_B^‐ is the total time during the contract life that the underlying price was lower than a prespecified barrier level ( occupation time ). We derive closed‐form pricing formulas for step options with any knock‐out rate in the range $[0,∞). For any finite knock‐out rate both the step option's value and delta are continuous functions of the underlying price at the barrier. As a result, they can be continuously hedged by trading the underlying asset and borrowing. Their risk management properties make step options attractive "no‐regrets" alternatives to standard barrier options. As a by‐product, we derive a dynamic almost‐replicating trading strategy for standard barrier options by considering a replicating strategy for a step option with high but finite knock‐out rate. Finally, a general class of derivatives contingent on occupation times is considered and closed‐form pricing formulas are derived.     

Analysis of Error with Malliavin Calculus: Application to Hedging

The aim of this paper is to compute the quadratic error of a discrete time-hedging strategy in a complete multidimensional model. This result extends that of Gobet and Temam (2001) and Zhang (1999) . More precisely, our basic assumption is that the asset prices satisfy the d -dimensional stochastic differential equation   dXⁱ_t = Xⁱ_t ( bⁱ ( X_t ) dt +σ _{i , j} ( X_t ) dW^j_t )  . We precisely describe the risk of this strategy with respect to n , the number of rebalancing times. The rates of convergence obtained are     for any options with Lipschitz payoff and  1/ n ^1/4  for options with irregular payoff.     

MARKOWITZ'S PORTFOLIO OPTIMIZATION IN AN INCOMPLETE MARKET

In this paper, for a process S , we establish a duality relation between K_p , the     - closure of the space of claims in     , which are attainable by "simple" strategies, and     , all signed martingale measures     with     , where   p ≥ 1, q ≥ 1  and     . If there exists a     with     a.s., then K_p consists precisely of the random variables     such that ϑ is predictable S -integrable and     for all     . The duality relation corresponding to the case   p = q = 2  is used to investigate the Markowitz's problem of mean–variance portfolio optimization in an incomplete market of semimartingale model via martingale/convex duality method. The duality relationship between the mean–variance efficient portfolios and the variance-optimal signed martingale measure (VSMM) is established. It turns out that the so-called market price of risk is just the standard deviation of the VSMM. An illustrative example of application to a geometric Lévy processes model is also given.     

ON THE ERROR IN THE MONTE CARLO PRICING OF SOME FAMILIAR EUROPEAN PATH-DEPENDENT OPTIONS

This paper studies the relative error in the crude Monte Carlo pricing of some familiar European path-dependent multiasset options. For the crude Monte Carlo method it is well known that the convergence rate   O ( n ^−1/2)  , where n is the number of simulations, is independent of the dimension of the integral. This paper also shows that for a large class of pricing problems in the multiasset Black-Scholes market the constant in   O ( n ^−1/2)  is independent of the dimension. To be more specific, the constant is only dependent on the highest volatility among the underlying assets, time to maturity, and degree of confidence interval.     

RISK MEASURES ON ORLICZ HEARTS

Coherent, convex, and monetary risk measures were introduced in a setup where uncertain outcomes are modeled by bounded random variables. In this paper, we study such risk measures on Orlicz hearts. This includes coherent, convex, and monetary risk measures on L^p -spaces for  1 ≤ p < ∞  and covers a wide range of interesting examples. Moreover, it allows for an elegant duality theory. We prove that every coherent or convex monetary risk measure on an Orlicz heart which is real-valued on a set with non-empty algebraic interior is real-valued on the whole space and admits a robust representation as maximal penalized expectation with respect to different probability measures. We also show that penalty functions of such risk measures have to satisfy a certain growth condition and that our risk measures are Luxemburg-norm Lipschitz-continuous in the coherent case and locally Luxemburg-norm Lipschitz-continuous in the convex monetary case. In the second part of the paper we investigate cash-additive hulls of transformed Luxemburg-norms and expected transformed losses. They provide two general classes of coherent and convex monetary risk measures that include many of the currently known examples as special cases. Explicit formulas for their robust representations and the maximizing probability measures are given.     

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.     

Asymptotics of the price oscillations of a European call option in a tree model

It is well known that the price of a European vanilla option computed in a binomial tree model converges toward the Black-Scholes price when the time step tends to zero. Moreover, it has been observed that this convergence is of order 1/ n in usual models and that it is oscillatory. In this paper, we compute this oscillatory behavior using asymptotics of Laplace integrals, giving explicitly the first terms of the asymptotics. This allows us to show that there is no asymptotic expansion in the usual sense, but that the rate of convergence is indeed of order 1/ n in the case of usual binomial models since the second term (in     ) vanishes. The next term is of type   C ₂( n )/ n   , with   C ₂( n )  some explicit bounded function of n that has no limit when n tends to infinity.     

Dynamic Minimization of Worst Conditional Expectation of Shortfall

In a complete financial market model, the shortfall-risk minimization problem at the terminal date is treated for the seller of a derivative security F . The worst conditional expectation of the shortfall is adopted as the measure of this risk, ensuring that the minimized risk satisfies certain desirable properties as the dynamic measure of risk, as proposed by Cvitanić and Karatzas (1999) . The terminal value of the optimized portfolio is a binary functional dependent on F and the Radon-Nikodym density of the equivalent local martingale measure. In particular, it is observed that there exists a positive number x * that is less than the replicating cost x_F of F , and that the strategy minimizing the expectation of the shortfall is optimal if the hedger's capital is in the range [ x *, x_F ].     

A unified approach to systemic risk measures via acceptance sets

We specify a general methodological framework for systemic risk measures via multidimensional acceptance sets and aggregation functions. Existing systemic risk measures can usually be interpreted as the minimal amount of cash needed to secure the system after aggregating individual risks. In contrast, our approach also includes systemic risk measures that can be interpreted as the minimal amount of cash that secures the aggregated system by allocating capital to the single institutions before aggregating the individual risks. An important feature of our approach is the possibility of allocating cash according to the future state of the system (scenario‐dependent allocation). We also provide conditions that ensure monotonicity, convexity, or quasi‐convexity of our systemic risk measures.     

RISK MEASURES AND CAPITAL REQUIREMENTS FOR PROCESSES

In this paper we propose a generalization of the concepts of convex and coherent risk measures to a multiperiod setting, in which payoffs are spread over different dates. To this end, a careful examination of the axiom of translation invariance and the related concept of capital requirement in the one-period model is performed. These two issues are then suitably extended to the multiperiod case, in a way that makes their operative financial meaning clear. A characterization in terms of expected values is derived for this class of risk measures and some examples are presented.     

Put Option Premiums and Coherent Risk Measures

This note defines the premium of a put option on the firm as a measure of insolvency risk. The put premium is not a coherent risk measure as defined by Artzner et al. (1999). It satisfies all the axioms for a coherent risk measure except one, the translation invariance axiom. However, it satisfies a weakened version of the translation invariance axiom that we label translation monotonicity. The put premium risk measure generates an acceptance set that satisfies the regularity Axioms 2.1–2.4 of Artzner et al. (1999). In fact, this is a general result for any risk measure satisfying the same risk measure axioms as the put premium. Finally, the coherent risk measure generated by the put premium's acceptance set is the minimal capital required to protect the firm against insolvency uniformly across all states of nature.     

A REPRESENTATION RESULT FOR CONCAVE SCHUR CONCAVE FUNCTIONS

A representation result is provided for concave Schur concave functions on   L ^∞(Ω)  . In particular, it is proven that any monotone concave Schur concave weakly upper semicontinuous function is the infinimum of a family of nonnegative affine combinations of Choquet integrals with respect to a convex continuous distortion of the underlying probability. The method of proof is based on the concave Fenchel transform and on Hardy and Littlewood's inequality. Under the assumption that the probability space is nonatomic, concave, weakly upper semicontinuous, law-invariant functions are shown to coincide with weakly upper semicontinuous concave Schur concave functions. A representation result is, thus, obtained for weakly upper semicontinuous concave law-invariant functions.     

SPARSE CALIBRATIONS OF CONTINGENT CLAIMS

The two problems of determining the existence of arbitrage among a finite set of options and of calculating the supremum price of an option consistent with other options prices have been reduced to finding an appropriate model of bounded size in many special cases. We generalize this result to a class of arbitrage-free  m -period markets with    d  + 1   basic securities and with no prior measure. We show there are no dominating trading strategies for a given set of  l  contingent claims if and only if their bid-ask prices are asymptotically consistent with models supported by at most   ( l  +  d  + 1)( d  + 1) ^{m −1}   points, if    m  ≥ 1  . An example showing the tightness of our bound is given.