Discriminatory Prices,Predation and Signal-Jamming in a Horizontal Differentiation Model

We study the effects of predation in an asymmetric duopoly model à la Hotelling in which the incumbent is able to price discriminate while the entrant sets a uniform price. We show that when the discount factor is high (low) enough and the incumbent accommodates entrance the incumbent initially sets a uniform price (discriminatory prices) and then engages in price discrimination. Under certain conditions, the entrant prices aggressively in order to discourage predation from the incumbent: predation actually does not occur and all equilibrium prices are lower with respect to the case in which the threat of predation is absent. In a T-period model, we derive conditions under which the equilibrium prices increase over time until they stabilize at the level that would result in the absence of the threat of predation.     

INFLATION AND THE TIMING OF PRICE CHANGES (*)

Consider an oligopolistic industry in which the firms produce perfect substitutes and incur transaction costs of changing price. Under constant inflation, there is an equilibrium in which the firms change their nominal prices at equally spaced time-points. Increasing the inflation rate increases the size of price changes, increases the frequency of price changes, and increases the average real price paid by buyers. The analysis derives its significance from its implications for the welfare costs of inflation: the inflation-induced distortions in relative prices are larger the more sticky nominal prices are; and are larger the greater the tendency for firms to stagger (rather than synchronize) their price changes.     

Mean‐field games with differing beliefs for algorithmic trading

Even when confronted with the same data, agents often disagree on a model of the real world. Here, we address the question of how interacting heterogeneous agents, who disagree on what model the real world follows, optimize their trading actions. The market has latent factors that drive prices, and agents account for the permanent impact they have on prices. This leads to a large stochastic game, where each agents performance criteria are computed under a different probability measure. We analyze the mean‐field game (MFG) limit of the stochastic game and show that the Nash equilibrium is given by the solution to a nonstandard vector‐valued forward–backward stochastic differential equation. Under some mild assumptions, we construct the solution in terms of expectations of the filtered states. Furthermore, we prove that the MFG strategy forms an ε‐Nash equilibrium for the finite player game. Finally, we present a least square Monte Carlo based algorithm for computing the equilibria and show through simulations that increasing disagreement may increase price volatility and trading activity.     

EQUILIBRIUM STATE PRICES IN A STOCHASTIC VOLATILITY MODEL1

In a stochastic volatility model, the no-free-lunch assumption does not induce a unique arbitrage price because of market incompleteness. In this paper, we consider a contingent claim on the primitive asset, traded in zero net supply. Given a system of Arrow-Debreu state prices, we provide necessary and sufficient conditions for consistency with an intertemporal additive equilibrium model that we fully characterize. We show that the risk premia corresponding to the minimal martingale of Föllmer and Schweizer (1991) are consistent with logarithmic preferences, while the Hull and White model (1987) (volatility risk premium independent of the asset price) is consistent with a class of utility functions including constant relative risk aversion (CRRA) ones.     

A Stochastic Extension of the Miller-Modigliani Framework1

This paper deals with the problem of the financial valuation of a firm and its shares of stock with general financing policies in a partial equilibrium framework. the model assumes a time-dependent discount rate and a general stochastic environment in a discrete-time setting. the fundamental valuation approach under the assumption of risk neutrality is used to obtain the time path of share price, the number of outstanding shares, and the value of the firm. These are shown to be the unique conditional expectations of certain stochastic processes. A broad class of firms for which the solution formula yields finite-valued solutions is characterized. the results are extended to the non-risk-neutral case. A regularity condition, which is both necessary and sufficient for the share price to equal the capitalization of future dividends accruing to the share, is obtained. As a mathematical aside, it is shown in the appendix that in the absence of this condition, the so-called stream of dividends approach is meaningless in the sense that it does not yield any financial valuation.     

Cost share‐induced technological change and Kaldor’s stylized facts

This paper presents a theory of induced technological change in which firms pursue a random, local, and bounded search for productivity‐enhancing innovations. Firms implement profitable innovations at fixed prices, which then spread through the economy. After diffusion, all firms adjust prices and wages. The model is consistent with a variety of price‐setting behaviors, which determine equilibrium positions characterized by constant cost shares and productivity growth rates. A fixed mark‐up can yield Marx‐biased technological change. Target‐return pricing yields Harrod‐neutral technological change with a fixed wage share as a stable equilibrium, consistent with Kaldor's stylized facts, while allowing for deviations from equilibrium, as observed in the longer historical record.     

Risk premiums on inventory assets: the case of crude oil and natural gas

This study tested for the presence of risk premiums on crude oil and natural gas. The econometric analysis followed from a stochastic model in which the equilibrium value of inventories depends on a convenience yield and an option value related to price uncertainty. The empirical findings provide rather strong support for the presence of risk premiums and also evidence for the existence of convenience yields. The risk premiums rose sharply with greater price volatility and help to explain why prices for immediate sales often exceed prices for future delivery. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:109–126, 2001     

THE STOCHASTIC VOLATILITY MODEL OF BARNDORFF‐NIELSEN AND SHEPHARD IN COMMODITY MARKETS

We consider the non‐Gaussian stochastic volatility model of Barndorff‐Nielsen and Shephard for the exponential mean‐reversion model of Schwartz proposed for commodity spot prices. We analyze the properties of the stochastic dynamics, and show in particular that the log‐spot prices possess a stationary distribution defined as a normal variance‐mixture model. Furthermore, the stochastic volatility model allows for explicit forward prices, which may produce a hump structure inherited from the mean‐reversion of the stochastic volatility. Although the spot price dynamics has continuous paths, the forward prices will have a jump dynamics, where jumps occur according to changes in the volatility process. We compare with the popular Heston stochastic volatility dynamics, and show that the Barndorff‐Nielsen and Shephard model provides a more flexible framework in describing commodity spot prices. An empirical example on UK spot data is included.     

On Feedback Effects from Hedging Derivatives

This paper proposes a new explanation for the smile and skewness effects in implied volatilities. Starting from a microeconomic equilibrium approach, we develop a diffusion model for stock prices explicitly incorporating the technical demand induced by hedging strategies. This leads to a stochastic volatility endogenously determined by agents' trading behavior. By using numerical methods for stochastic differential equations, we quantitatively substantiate the idea that option price distortions can be induced by feedback effects from hedging strategies.     

Resource-based FDI and expropriation in developing economies

Globally, foreign direct investment (FDI) assets are expropriated more in resource extraction industries compared to other sectors. Despite the higher apparent risk of expropriation in resources, countries more likely to expropriate also have a larger share of FDI in the resource sector. An incomplete markets model of FDI is developed to account for this puzzle. The type of government regime is stochastic, with low penalty regimes facing a relatively low, exogenous cost of expropriating FDI, and country risk is measured by the variation in these costs across different regimes. The key innovation of the model is that the government, before the regime type is known, is able to charge different prices to domestic and foreign investors for mineral rights. Granting cheap access increases FDI and reduces the country's share of resource rents, increasing the temptation to expropriate in a relatively low penalty regime. In very high-risk countries, subsidizing resource FDI increases the total value of output by raising investment, and the net gains from expropriating in a low penalty regime outweigh the rents foregone under a high penalty one. However, a stochastic resource output price results in relatively low-risk countries restricting FDI inflows to the resource sector instead — “windfall profits” in this sector raise incentives to expropriate when prices are high, yet minimization of the ex ante risk of expropriation is preferred owing to the relatively high penalty for expropriating. These results imply a higher average share of resource-based FDI in countries most likely to expropriate, while resources account for a high share of expropriated assets compared to the sector's global share of FDI. We show that the model is able to reconcile observed patterns of foreign investment and expropriation for a sample of 38 developing and emerging economies.     

Bounds on European Option Prices under Stochastic Volatility

In this paper we consider the range of prices consistent with no arbitrage for European options in a general stochastic volatility model. We give conditions under which the infimum and the supremum of the possible option prices are equal to the intrinsic value of the option and to the current price of the stock, respectively, and show that these conditions are satisfied in most of the stochastic volatility models from the financial literature. We also discuss properties of Black–Scholes hedging strategies in stochastic volatility models where the volatility is bounded.     

A maximal affine stochastic volatility model of oil prices

This study develops and estimates a stochastic volatility model of commodity prices that nests many of the previous models in the literature. The model is an affine three‐factor model with one state variable driving the volatility and is maximal among all such models that are also identifiable. The model leads to quasi‐analytical formulas for futures and options prices. It allows for time‐varying correlation structures between the spot price and convenience yield, the spot price and its volatility, and the volatility and convenience yield. It allows for expected mean‐reversion in the short term and for an increasing expected long‐term price, and for time‐varying risk premia. Furthermore, the model allows for the situation in which options' prices depend on risk not fully spanned by futures prices. These properties are desirable and empirically important for modeling many commodities, especially crude oil. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:101–133, 2010     

TAKEOVERS OF DIFFUSELY HELD FIRMS: A NONSTANDARD APPROACH

A common theme in the literature on corporate control is that, when share ownership is diffuse, the free-rider problem prevents raiders from making acquisitions at tender prices below the postacquisition share price. In this paper, we address this question by formulating a nonstandard model of takeovers of diffusely held firms. It is demonstrated that, even when individual shareholdings are infinitesimal relative to firm size, takeovers succeed with positive probability and equilibria exist in which the raider earns substantial per share profits. Further, the Nash equilibria of the game are characterized with regard to raider profit, the aggregate fraction of shares tendered, and the relation between raider profit and the degree of randomization exhibited by shareholder tendering strategies.     

EXACT SOLUTION OF A MARTINGALE STOCHASTIC VOLATILITY OPTION PROBLEM AND ITS EMPIRICAL EVALUATION

Exact explicit solution of the log-normal stochastic volatility (SV) option model has remained an open problem for two decades. In this paper, I consider the case where the risk-neutral measure induces a martingale volatility process, and derive an exact explicit solution to this unsolved problem which is also free from any inverse transforms. A representation of the asset price shows that its distribution depends on that of two random variables, the terminal SV as well as the time average of future stochastic variances. Probabilistic methods, using the author's previous results on stochastic time changes, and a Laplace–Girsanov Transform technique are applied to produce exact explicit probability distributions and option price formula. The formulae reveal interesting interplay of forces between the two random variables through the correlation coefficient. When the correlation is set to zero, the first random variable is eliminated and the option formula gives the exact formula for the limit of the Taylor series in Hull and White's (1987) approximation. The SV futures option model, comparative statics, price comparisons, the Greeks and practical and empirical implementation and evaluation results are also presented. A PC application was developed to fit the SV models to current market prices, and calculate other option prices, and their Greeks and implied volatilities (IVs) based on the results of this paper. This paper also provides a solution to the option implied volatility problem, as the empirical studies show that, the SV model can reproduce market prices, better than Black–Scholes and Black-76 by up to 2918%, and its IV curve can reproduce that of market prices very closely, by up to within its 0.37%.     

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.     

ANALYTICAL COMPARISONS OF OPTION PRICES IN STOCHASTIC VOLATILITY MODELS

This paper gives an ordering on option prices under various well-known martingale measures in an incomplete stochastic volatility model. Our central result is a comparison theorem that proves convex option prices are decreasing in the market price of volatility risk, the parameter governing the choice of pricing measure. The theorem is applied to order option prices under q -optimal pricing measures. In doing so, we correct orderings demonstrated numerically in Heath, Platen, and Schweizer ( Mathematical Finance , 11(4), 2001) in the special case of the Heston model.     

国内外棉花市场价格的动态关系分析——基于VECM模型

本文运用协整检验、误差修正模型及脉冲响应函数等方法,分析了我国加入世贸组织后国内棉花价格与国际棉花价格之间的动态关系。结果表明,国内棉价与国际棉价具有长期均衡关系,其中国际棉价波动对国内棉价有较强的冲击,对国内市场起引导作用;而国内棉价波动对国际市场影响较小,并在此基础上提出了政策建议。     

An explicitly solvable multi‐scale stochastic volatility model: Option pricing and calibration problems

We introduce an explicitly solvable multiscale stochastic volatility model that generalizes the Heston model. The model describes the dynamics of an asset price and of its two stochastic variances using a system of three Ito stochastic differential equations. The two stochastic variances vary on two distinct time scales and can be regarded as auxiliary variables introduced to model the dynamics of the asset price. Under some assumptions, the transition probability density function of the stochastic process solution of the model is represented as a one‐dimensional integral of an explicitly known integrand. In this sense the model is explicitly solvable. We consider the risk‐neutral measure associated with the proposed multiscale stochastic volatility model and derive formulae to price European vanilla options (call and put) in the multiscale stochastic volatility model considered. We use the thus‐obtained option price formulae to study the calibration problem, that is to study the values of the model parameters, the correlation coefficients of the Wiener processes defining the model, and the initial stochastic variances implied by the “observed” option prices using both synthetic and real data. In the analysis of real data, we use the S&P 500 index and to the prices of the corresponding options in the year 2005. The web site http://www.econ.univpm.it/recchioni/finance/w7 contains some auxiliary material including some animations that helps the understanding of this article. A more general reference to the work of the authors and their coauthors in mathematical finance is the web site http://www.econ.univpm.it/recchioni/finance . © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:862–893, 2009     

Black‐Scholes‐Merton revisited under stochastic dividend yields

European options are priced in a framework à la Black‐Scholes‐Merton, which is extended to incorporate stochastic dividend yield under a stochastic mean–reverting market price of risk. Explicit formulas are obtained for call and put prices and their Greek parameters. Some well‐known properties of the Black‐Scholes‐Merton formula fail to hold in this setting. For example, the delta of the call can be negative and even greater than one in absolute terms. Moreover, call prices can be a decreasing function of the underlying volatility although the latter is constant. Finally, and most importantly, option prices highly depend on the features of the market price of risk, which does not need to be specified at all in the standard Black‐Scholes‐Merton setting. The results are simulated in order to assess the economic impact of assuming that the dividend yield is deterministic when it is actually stochastic, as well as to assess the economic importance of the features of the market price of risk. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:703–732, 2006