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.     

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.     

INTENSITY-BASED VALUATION OF RESIDENTIAL MORTGAGES: AN ANALYTICALLY TRACTABLE MODEL

This paper presents an analytically tractable valuation model for residential mortgages. The random mortgage prepayment time is assumed to have an intensity process of the form h _t = h ₀( t ) +γ ( k − r _t )⁺ , where h ₀( t ) is a deterministic function of time, r _t is the short rate, and γ and k are scalar parameters. The first term models exogenous prepayment independent of interest rates (e.g., a multiple of the PSA prepayment function). The second term models refinancing due to declining interest rates and is proportional to the positive part of the distance between a constant threshold level and the current short rate. When the short rate follows a CIR diffusion, we are able to solve the model analytically and find explicit expressions for the present value of the mortgage contract, its principal-only and interest-only parts, as well as their deltas. Mortgage rates at origination are found by solving a non-linear equation. Our solution method is based on explicitly constructing an eigenfunction expansion of the pricing semigroup, a Feynman-Kac semigroup of the CIR diffusion killed at an additive functional that is a linear combination of the integral of the CIR process and an area below a constant threshold and above the process sample path (the so-called area functional). A sensitivity analysis of the term structure of mortgage rates and calibration of the model to market data are presented.     

RISK-MINIMIZING HEDGING STRATEGIES UNDER RESTRICTED INFORMATION

We construct risk-minimizing hedging strategies in the case where there are restrictions on the available information. the underlying price process is a d -dimensional F-martingale, and strategies φ= (ϑ, η) are constrained to have η G-predictable and η G'-adapted for filtrations η G C G'C F. We show that there exists a unique (ηG, G')-risk-minimizing strategy for every contingent claim H ε E ² (_T, P ) and provide an explicit expression in terms of η G-predictable dual projections. Previous results of Föllmer and Sondermann (1986) and Di Masi, Platen, and Runggaldier (1993) are recovered as special cases. Examples include a Black-Scholes model with delayed information and a jump process model with discrete observations.     

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.     

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.     

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.     

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.     

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.     

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."     

PRICING FOR LARGE POSITIONS IN CONTINGENT CLAIMS

Approximations to utility indifference prices are provided for a contingent claim in the large position size limit. Results are valid for general utility functions on the real line and semi‐martingale models. It is shown that as the position size approaches infinity, the utility function's decay rate for large negative wealths is the primary driver of prices. For utilities with exponential decay, one may price like an exponential investor. For utilities with a power decay, one may price like a power investor after a suitable adjustment to the rate at which the position size becomes large. In a sizable class of diffusion models, limiting indifference prices are explicitly computed for an exponential investor. Furthermore, the large claim limit arises endogenously as the hedging error for the claim vanishes.     

HEDGING WITH ENERGY

In the setting of diffusion models for price evolution, we suggest an easily implementable approximate evaluation formula for measuring the errors in option pricing and hedging due to volatility misspecification. The main tool we use in this paper is a (suitably modified) classical inequality for the L ² norm of the solution, and the derivatives of the solution, of a partial differential equation (the so-called "energy" inequality). This result allows us to give bounds on the errors implied by the use of approximate models for option valuation and hedging and can be used to justify formally some "folk" belief about the robustness of the Black and Scholes model. Surprisingly enough, the result can also be applied to improve pricing and hedging with an approximate model. When statistical or a priori information is available on the "true" volatility, the error measure given by the energy inequality can be minimized w.r.t. the parameters of the approximating model. The method suggested in this paper can help in conjugating statistical estimation of the volatility function derived from flexible but computationally cumbersome statistical models, with the use of analytically tractable approximate models calibrated using error estimates.     

Arbitrage Values Generally Depend On A Parametric Rate of Return

Let X denote a positive Markov stochastic integral, and let S ( t , μ) = exp(μ t ) X ( t ) represent the price of a security at time t with infinitesimal rate of return μ. Contingent claim (option) pricing formulas typically do not depend on μ. We show that if a contingent claim is not equivalent to a call option having exercise price equal to zero, then security prices having this property—option prices do not depend on μ—must satisfy: for some V (0, T ), In( S ( t , μ) X ( V )) is Gaussian on a time interval [ V, T ], and hence S ( t , μ) has independent observed returns. With more assumptions, V = 0, and there exist equivalent martingale measures.     

PSEUDODIFFUSIONS AND QUADRATIC TERM STRUCTURE MODELS

The non-Gaussianity of processes observed in financial markets and the relatively good performance of Gaussian models can be reconciled by replacing the Brownian motion with Lévy processes whose Lévy densities decay as  exp(−λ| x |)  or faster, where  λ > 0  is large. This leads to asymptotic pricing models. The leading term, P ₀, is the price in the Gaussian model with the same instantaneous drift and variance. The first correction term depends on the instantaneous moments of order up to 3, that is, the skewness is taken into account, the next term depends on moments of order 4 (kurtosis) as well, etc. In empirical studies, the asymptotic formula can be applied without explicit specification of the underlying process: it suffices to assume that the instantaneous moments of order greater than 2 are small w.r.t. moments of order 1 and 2, and use empirical data on moments of order up to 3 or 4. As an application, the bond-pricing problem in the non-Gaussian quadratic term structure model is solved. For pricing of options near expiry, a different set of asymptotic formulas is developed; they require more detailed specification of the process, especially of its jump part. The leading terms of these formulas depend on the jump part of the process only, so that they can be used in empirical studies to identify the jump characteristics of the process.     

PRICING AND HEDGING AMERICAN OPTIONS ANALYTICALLY: A PERTURBATION METHOD

This paper studies the critical stock price of American options with continuous dividend yield. We solve the integral equation and derive a new analytical formula in a series form for the critical stock price. American options can be priced and hedged analytically with the help of our critical-stock-price formula. Numerical tests show that our formula gives very accurate prices. With the error well controlled, our formula is now ready for traders to use in pricing and hedging the S&P 100 index options and for the Chicago Board Options Exchange to use in computing the VXO volatility index.     

商品期货套期保值对企业价值的影响

以1999-2010年我国有色金属生产及加工行业的59家上市公司为样本,本文研究了商品期货套期保值对企业价值造成的影响。通过分析套保对稳定企业价值的影响,证明期货套保有助于降低价格风险对企业股价的影响,即有助于稳定企业价值;通过分析套保对提升企业价值的影响,证明国内企业参与期货套保并不能显著提高所有公司的托宾Q值,但是大规模企业由于发挥规模效应,相比较小规模参与者而言较易于提升公司价值;进一步研究表明公司规模和财务状况是影响企业是否参与套保的关键因素。     

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.     

Optimal investment,derivative demand,and arbitrage under price impact

This paper studies the optimal investment problem with random endowment in an inventory‐based price impact model with competitive market makers. Our goal is to analyze how price impact affects optimal policies, as well as both pricing rules and demand schedules for contingent claims. For exponential market makers preferences, we establish two effects due to price impact: constrained trading and nonlinear hedging costs. To the former, wealth processes in the impact model are identified with those in a model without impact, but with constrained trading, where the (random) constraint set is generically neither closed nor convex. Regarding hedging, nonlinear hedging costs motivate the study of arbitrage free prices for the claim. We provide three such notions, which coincide in the frictionless case, but which dramatically differ in the presence of price impact. Additionally, we show arbitrage opportunities, should they arise from claim prices, can be exploited only for limited position sizes, and may be ignored if outweighed by hedging considerations. We also show that arbitrage‐inducing prices may arise endogenously in equilibrium, and that equilibrium positions are inversely proportional to the market makers' representative risk aversion. Therefore, large positions endogenously arise in the limit of either market maker risk neutrality, or a large number of market makers.     

Order imbalance and the dynamics of index and futures prices

This study uses transaction records of index futures and index stocks, with bid/ask price quotes, to examine the impact of stock market order imbalance on the dynamic behavior of index futures and cash index prices. Spurious correlation in the index is purged by using an estimate of the “true” index with highly synchronous and active quotes of individual stocks. A smooth transition autoregressive error correction model is used to describe the nonlinear dynamics of the index and futures prices. Order imbalance in the cash stock market is found to affect significantly the error correction dynamics of index and futures prices. Order imbalance impedes error correction particularly when the market impact of order imbalance works against the error correction force of the cash index, explaining why real potential arbitrage opportunities may persist over time. Incorporating order imbalance in the framework significantly improves its explanatory power. The findings indicate that a stock market microstructure that allows a quick resolution of order imbalance promotes dynamic arbitrage efficiency between futures and underlying stocks. The results also suggest that the unloading of cash stocks by portfolio managers in a falling market situation aggravates the price decline and increases the real cost of hedging with futures. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:1129–1157, 2007     

Quantiles of the Euler Scheme for Diffusion Processes and Financial Applications

In this paper we briefly present the results obtained in our paper ( Talay and Zheng 2002a ) on the convergence rate of the approximation of quantiles of the law of one component of  ( X_t )  , where  ( X_t )  is a diffusion process, when one uses a Monte Carlo method combined with the Euler discretization scheme. We consider the case where  ( X_t )  is uniformly hypoelliptic (in the sense of Condition (UH) below), or the inverse of the Malliavin covariance of the component under consideration satisfies the condition (M) below. We then show that Condition (M) seems widely satisfied in applied contexts. We particularly study financial applications: the computation of quantiles of models with stochastic volatility, the computation of the VaR of a portfolio, and the computation of a model risk measurement for the profit and loss of a misspecified hedging strategy.