Option Pricing in Discrete‐Time Incomplete Market Models

Various aspects of pricing of contingent claims in discrete time for incomplete market models are studied. Formulas for prices with proportional transaction costs are obtained. Some results concerning pricing with concave transaction costs are shown. Pricing by the expected utility of terminal wealth is also considered.     

Optimal No‐Arbitrage Bounds on S&P 500 Index Options and the Volatility Smile

This article shows that the volatility smile is not necessarily inconsistent with the Black–Scholes analysis. Specifically, when transaction costs are present, the absence of arbitrage opportunities does not dictate that there exists a unique price for an option. Rather, there exists a range of prices within which the option's price may fall and still be consistent with the Black–Scholes arbitrage pricing argument. This article uses a linear program (LP) cast in a binomial framework to determine the smallest possible range of prices for Standard & Poor's 500 Index options that are consistent with no arbitrage in the presence of transaction costs. The LP method employs dynamic trading in the underlying and risk‐free assets as well as fixed positions in other options that trade on the same underlying security. One‐way transaction‐cost levels on the index, inclusive of the bid–ask spread, would have to be below six basis points for deviations from Black–Scholes pricing to present an arbitrage opportunity. Monte Carlo simulations are employed to assess the hedging error induced with a 12‐period binomial model to approximate a continuous‐time geometric Brownian motion. Once the risk caused by the hedging error is accounted for, transaction costs have to be well below three basis points for the arbitrage opportunity to be profitable two times out of five. This analysis indicates that market prices that deviate from those given by a constant‐volatility option model, such as the Black–Scholes model, can be consistent with the absence of arbitrage in the presence of transaction costs. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:1151–1179, 2001     

Optimal arbitrage strategies on stock index futures under position limits

Assuming the absence of market frictions, deterministic interest rates, and certainty in dividend payouts from the stocks in the index basket, an arbitrageur can lock in the profit of a positive (negative) arbitrage basis in a stock index futures by adopting a short (long) futures strategy. In addition, the arbitrageur may improve the arbitrage profit by adopting the so‐called early unwinding strategy of liquidating the position before maturity, or more aggressively from the long position directly to the short position or vice versa. In this study, we examine the optimal arbitrage strategies in stock index futures with position limits and transaction costs. In our analysis, the index arbitrage basis is assumed to follow the Brownian Bridge process. The model formulation of the option value functions leads to a coupled system of variational inequalities. We determine the values of the arbitrage opportunities and the optimal threshold values of the arbitrage basis at which the arbitrageur should optimally close an existing position or open a new index arbitrage position. In particular, we examine the impact of transaction costs on the index arbitrage strategies. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:394–406, 2011     

A Study of Arbitrage Efficiency Between the FTSE‐100 Index Futures and Options Contracts

Despite the importance of the London markets and the significance of the relationship for market makers, little published research is available on arbitrage between the FTSE‐100 Index futures and the FTSE‐100 European index options contracts. This study uses the put–call–futures parity condition to throw light on the relationship between options and futures written against the FTSE Index. The arbitrage methodology adopted in this study avoids many of the problems that have affected prior research on the relationship between options or futures prices and the underlying index. The problems that arise from nonsynchroneity between options and futures prices are reduced by the matching of options and futures prices within narrow time intervals with time‐stamped transaction data. This study allows for realistic trading and market‐impact costs. The feasibility of strategies such as execute‐and‐hold and early unwinding is examined with both ex‐post and ex‐ante simulation tests that take into consideration possible execution time lags for the arbitrage trade. This study reveals that the occurrence of matched put–call–futures trios exhibits a U‐shaped intraday pattern with a concentration at both open and close, although the magnitude of observed mispricings has no discernible intraday pattern. Ex‐post arbitrage profits for traders facing transaction costs are concentrated in at‐the‐money options. As in other major markets, despite important microstructure differences, opportunities are generally rapidly extinguished in less than 3 min. The results suggest that arbitrage opportunities for traders facing transaction costs are small in number and confirm the efficiency of trading on the London International Financial Futures and Options Exchange. © 2002 John Wiley & Sons, Inc. Jrl Fut Mark 22:31–58, 2002     

Bounds on Derivative Prices in an Intertemporal Setting with Proportional Transaction Costs and Multiple Securities

The observed discrepancies of derivative prices from their theoretical, arbitrage-free values are examined in the presence of transaction costs. Analytic upper and lower bounds on the reservation write and purchase prices, respectively, are obtained when an investor's preferences exhibit constant relative risk aversion between zero and one. The economy consists of multiple primary securities with stationary returns, a constant rate of interest, and any number of American or European derivatives with, possibly, path-dependent arbitrary payoffs.     

The profitability of index futures arbitrage: Evidence from bid-ask quotes

Previous studies investigated the profitability of stock index futures based on transaction price data, and could overstate the frequency of arbitrage opportunities and size of arbitrage profits. This article obtains a data base for the Hong Kong index futures and index options market that contains both real-time transaction prices and bid-ask quotes; the article further examines the bias of identifying arbitrage opportunities based on transaction prices. The article finds the percentage of observations violating no-arbitrage bounds is significantly reduced when bid-ask quotes are employed instead of transaction prices. This suggests studies that implement arbitrage strategies based on transaction prices employ prices from the wrong side of the spread. This article finds a relationship between the frequency of violations (evaluated from transaction prices) and the size of bid-ask spreads in the futures and options markets. This phenomenon indicates that a larger mispricing, which may arise when the bid-ask spread is wider, does not necessarily imply profitable arbitrage opportunity. © 1998 John Wiley & Sons, Inc. Jrl Fut Mark 18:743–763, 1998     

Randomized Stopping Times and American Option Pricing with Transaction Costs

In a general discrete-time market model with proportional transaction costs, we derive new expectation representations of the range of arbitrage-free prices of an arbitrary American option. The upper bound of this range is called the upper hedging price, and is the smallest initial wealth needed to construct a self-financing portfolio whose value dominates the option payoff at all times. A surprising feature of our upper hedging price representation is that it requires the use of randomized stopping times (Baxter and Chacon 1977), just as ordinary stopping times are needed in the absence of transaction costs. We also represent the upper hedging price as the optimum value of a variety of optimization problems. Additionally, we show a two-player game where at Nash equilibrium the value to both players is the upper hedging price, and one of the players must in general choose a mixture of stopping times. We derive similar representations for the lower hedging price as well. Our results make use of strong duality in linear programming.     

A Stochastic Control Approach to Risk Management Under Restricted Information

We use techniques from discrete-time stochastic control under partial state information to determine a shortfall-risk minimizing investment strategy in the case when there is only restricted information on the underlying market model and transaction costs as well as shortselling constraints are present. The approach is adaptive in the sense that it takes into account all the information on the underlying model that becomes successively available to an economic agent by observing the prices in the market. As an immediate byproduct of the approach it is possible to determine the entire shortfall distribution corresponding to the optimal strategy and to various values of the initial capital.     

Customs-related Transaction Costs,Firm Size and International Trade Intensity

customs are generally perceived as a time-consuming impediment to international trade. However, few studies have empirically examined the determinants and the impact of this type of government-imposed transaction costs. This paper analyses the role of firm size as a determinant of customs-related transaction costs, as well as the effect of firm size on the relationship between these costs and the international trade intensity of firms. The results of this study indicate that customs-related transaction costs repress international trade activities of firms, even at low levels of these costs. The paper identifies transaction-related economies of scale, simplified customs procedures and advanced information and communication technology as main determinants of customs-related transaction costs. It is shown that when these factors are taken into account, firm size has no effect on customs-related transaction costs. Policy implications are considered for firm strategy and public policy.     

Box‐spread arbitrage efficiency of Nifty index options: The Indian evidence

This study examines the market efficiency for the European style Nifty index options using the box‐spread strategy. Time‐stamped transactions data are used to identify the mispricing and arbitrage opportunities for options with this modelfree approach. Profit opportunities, after accounting for the transaction costs, are quite frequent, but do not persist even for two minutes. The mispricing is higher for the contracts with higher liquidity (immediacy) risk captured by the moneyness (the difference between the strike prices and the spot price) and the volatility of the underlying. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:544–562, 2009     

Do retail gasoline prices rise more readily than they fall?: A threshold cointegration approach

This paper revisits the controversy over whether retail gasoline prices respond to increases in upstream prices more rapidly than decreases. Using threshold and momentum models of cointegration and daily data at different stages in the distribution chain, we find that transmission between upstream and downstream prices is mostly asymmetric in the momentum model: increases in upstream prices are passed on to downstream prices more quickly than decreases. We distinguish between small and large shocks and show that the asymmetry is more pronounced for small shocks, which may be due to consumer search costs.     

On Level Curves of Value Functions in Optimization Models of Expected Utility

We study the level sets of value functions in expected utility stochastic optimization models. We consider optimal portfolio management models in complete markets with lognormally distributed prices as well as asset prices modeled as diffusion processes with nonlinear dynamics. Besides the complete market cases, we analyze models in markets with frictions like correlated nontraded assets and diffusion stochastic volatilities. We derive, for all the above models, equations that their level curves solve and we relate their evolution to power transformations of derivative prices. We also study models with proportional transaction costs in a finite horizon setting and we derive their level curve equation; the latter turns out to be a Variational Inequality with mixed gradient and obstacle constraints.     

Why Doesn't the Black‐‐Scholes Model Fit Japanese Warrants and Convertible Bonds?

In this paper, we investigate the systematic departures of traded prices of Japanese equity warrants and convertible bonds from their theoretical Black–Scholes values. We briefly consider transactions costs and the dilution adjustment as potential explanations of the discrepancy. However, our major focus is on shifts in volatility of the prices of the underlying stocks as a function of the stock price changes; such shifts are not taken into account in the Black–Scholes values. We assume that the pseudo‐probability distributions of prices of stocks of cross‐sections of companies which are roughly similar in size are identical. This simple assumption, which can be generalized, enables us to infer the implied probability distribution and binomial tree for stock price changes using the Derman and Kani (1994), Rubinstein (1994) and Shimko (1993) approach. The cross‐section of warrant prices implies an inverse volatility smile and a positively skewed probability density for stock prices. Rubinstein's identifying assumptions generate an implied binomial tree in which the relative size of up‐steps and down‐steps, and thus volatility, changes systematically as stock prices change. We briefly consider potential explanations for the implied behaviour, and for the difference in the smile pattern between index options and the warrants and convertibles.     

The Transaction Cost Economics of Market‐based Exchange: The Impact of Reputation and External Verification Agencies

In this analysis, we empirically investigate market‐based transaction costs. We measure market‐based transaction costs indirectly by examining variations in market prices when selling Morgan Silver Dollars on eBay. We find that the reputation of both the seller and the coin‐rating agency employed significantly influences the price premium obtained. Moreover, we find that the use of a coin‐rating agency with a poor reputation proved more damaging than the use of no coin‐rating agency. Thus, we find support for the suggestion that the reputation of sellers and third‐party verification agencies have a significant influence on perceived market‐based transaction costs.     

浙商对推进市场与产业互动的作用

本文通过建立“市场交换演进模型”，及其对不同市场均衡状态的演进比较，从理论上回答了：（1）在具有众多浙商与消费者分散交易，以及具有垄断势力大商企购销经营共存的市场中，浙商推动了市场供给曲线右移，放大了市场销售量，导致了市场交换模式的演进；（2）众多浙商的存在，构造了可降低市场交易成本的机制，增进了社会福利；（3）浙商担当起了发现市场、拓展市场、推动市场与产业互动、促进浙江产业集群成长的作用。     

Too many options? Theory and evidence on option exchange design

A model of option exchange design is proposed and tested. The model allows investors to choose among several exchange‐traded options based on a trade‐off between standardization costs and liquidity/transaction costs. It employs a spatial economics approach to provide results for the existence of markets for particular option contracts on the exchange, a comparison of exchange design by a social planner and a profit‐maximizing monopolist (corresponding to the idea that most derivatives exchanges centralize the design and creation of option contracts), and comparative statics that can potentially aid decision makers in the design of option exchanges. In the empirical work, open interest is analyzed for Chicago Board Options Exchange (CBOE) options on the stocks in the S&P 100 index. In accordance with the model's predictions, open interest forms a previously undocumented seesaw pattern across strike prices, clustering around certain strike prices, and dropping off for the adjacent strike prices. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:533–570, 2006     

ARBITRAGE IN SECURITIES MARKETS WITH SHORT-SALES CONSTRAINTS

In this paper we derive the implications of the absence of arbitrage in securities markets models where traded securities are subject to short-sales constraints and where the borrowing and lending rates differ. We show that a securities price system is arbitrage free if and only if there exists a numeraire and an equivalent probability measure for which the normalized (by the numeraire) price processes of traded securities are supermartingales. Also, the tightest arbitrage bounds that can be inferred on the price of a contingent claim without knowing agents'preferences are equal to its largest and smallest expected normalized payoff with respect to the supermartingale measures. In the case where the underlying security price follows a diffusion process and where short selling is possible but costly, we derive partial differential equations that must be satisfied by the arbitrage bounds on derivative securities prices, and we determine optimal hedging strategies. We compute the arbitrage bounds on common securities numerically for several values of the borrowing and short-selling costs and show that they can be quite sharp.     

沪深300股指期货定价实证研究

股指期货是一种发展迅速的金融衍生产品,而合约定价问题是其重要研究方向之一。股指期货定价的基本方法是利用无套利定价原理得出的持有成本模型;而如果综合交易费用、融资成本、存贷款利差、保证金等市场因素,则可以得到股指期货的无套利定价区间。使用这两种模型对中国金融期货交易所的沪深300股指期货仿真交易合约进行实证分析,结果发现,实际交易价格和理论价格有较大偏差,市场中存在大量套利机会,定价效率有待提高。为此可以考虑的建议包括允许融资融券交易、推出沪深300指数ETF等。     

Digitalisierung: neue Aufgreifkriterien der Fusionskontrolle in der 9. GWB-Novelle

The Ninth amendment of the German Competition Act introduces new merger control thresholds. These thresholds are designed to close an enforcement gap that is especially found in parts of the digital economy. As the takeover of Facebook and WhatsApp revealed, traditional turnover-based thresholds are inapplicable to business models prevalent in some parts of the digital economy. The amendment addresses these limitations and specifically introduces a transaction value-based threshold that will enable the Federal Competition Authority to carry out merger control in certain mergers with high transaction values and low turnovers.     

LIMIT THEOREMS FOR PARTIAL HEDGING UNDER TRANSACTION COSTS

We study shortfall risk minimization for American options with path‐dependent payoffs under proportional transaction costs in the Black–Scholes (BS) model. We show that for this case the shortfall risk is a limit of similar terms in an appropriate sequence of binomial models. We also prove that in the continuous time BS model, for a given initial capital, there exists a portfolio strategy which minimizes the shortfall risk. In the absence of transactions costs (complete markets) similar limit theorems were obtained by Dolinsky and Kifer for game options. In the presence of transaction costs the markets are no longer complete and additional machinery is required. Shortfall risk minimization for American options under transaction costs was not studied before.