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     

MODIFIED LELAND’S STRATEGY FOR A CONSTANT TRANSACTION COSTS RATE

In 1985 Leland suggested an approach to price contingent claims under proportional transaction costs. Its main idea is to use the classical Black–Scholes formula with a suitably adjusted volatility for a periodical revision of the portfolio whose terminal value approximates the pay‐off. Unfortunately, if the transaction costs rate does not depend on the number of revisions, the approximation error does not converge to zero as the frequency of revisions tends to infinity. In the present paper, we suggest a modification of Leland’s strategy ensuring that the approximation error vanishes in the limit.     

Leland's Approach to Option Pricing: The Evolution of a Discontinuity

A claim of Leland (1985) states that in the presence of transaction costs a call option on a stock S , described by geometric Brownian motion, can be perfectly hedged using Black–Scholes delta hedging with a modified volatility. Recently Kabanov and Safarian (1997) disproved this claim, giving an explicit (up to an integral) expression of the limiting hedging error, which appears to be strictly negative and depends on the path of the stock price only via the stock price at expiry S _T . We prove in this paper that the limiting hedging error, considered as a function of S _T , exhibits a removable discontinuity at the exercise price. Furthermore, we provide a quantitative result describing the evolution of the discontinuity: Hedging errors, plotted over the price at expiry, show a peak near the exercise price. We determine the rate at which that peak becomes narrower (producing the discontinuity in the limit) as the lengths of the revision intervals shrink.     

Robustness of the Black and Scholes Formula

Consider an option on a stock whose volatility is unknown and stochastic. An agent assumes this volatility to be a specific function of time and the stock price, knowing that this assumption may result in a misspecification of the volatility. However, if the misspecified volatility dominates the true volatility, then the misspecified price of the option dominates its true price. Moreover, the option hedging strategy computed under the assumption of the misspecified volatility provides an almost sure one-sided hedge for the option under the true volatility. Analogous results hold if the true volatility dominates the misspecified volatility. These comparisons can fail, however, if the misspecified volatility is not assumed to be a function of time and the stock price. The positive results, which apply to both European and American options, are used to obtain a bound and hedge for Asian options.     

Flight-to-liquidity: Evidence from China's stock market

In an order-driven and strictly regulated stock market, illiquidity risks' effects on asset pricing should be highlighted, particularly in such extreme market conditions as those in China. This paper utilizes panel data from China's stock market in an attempt to answer whether the illiquidity risk in various dimensions—including price impacts, the transaction speed, trading volume, transaction costs, and asymmetric information—can explain stock returns. We find that almost all dimensions of stock illiquidity are positively associated with excess stock returns. More importantly, smaller, less-liquid stocks suffer more liquidity costs, providing a strong evidence for “flight-to-liquidity.” Additionally, the transaction costs and asymmetric information, denoted by bid-ask spreads, robustly account for these illiquidity effects on stock pricing and differ from the findings in the U.S. market. We also find that the “flight-to-liquidity” can partially explain the idiosyncratic volatility puzzle, investors' gambling, and herding psychologies. This study provides substantial policy implications in regulation and portfolio management for emerging markets.     

Should Stochastic Volatility Matter to the Cost-Constrained Investor?

Significant strides have been made in the development of continuous-time portfolio optimization models since Merton (1969) . Two independent advances have been the incorporation of transaction costs and time-varying volatility into the investor's optimization problem. Transaction costs generally inhibit investors from trading too often. Time-varying volatility, on the other hand, encourages trading activity, as it can result in an evolving optimal allocation of resources. We examine the two-asset portfolio optimization problem when both elements are present. We show that a transaction cost framework can be extended to include a stochastic volatility process. We then specify a transaction cost model with stochastic volatility and show that when the risk premium is linear in variance, the optimal strategy for the investor is independent of the level of volatility in the risky asset. We call this the Variance Invariance Principle.     

A brighter future with lower transactions costs?

Recently, the OMX Nordic Exchange reduced the exchange fee for trading the OMXS 30 index futures with more than 22%. The reduction in exchange fees provides this study with a unique opportunity to investigate the effects of a change in fixed transaction costs on futures market liquidity, trading activity, volatility, futures pricing efficiency, and the futures exchange's revenues. The results show a ceteris paribus increase in futures trading volume with 19%, a 27% decrease in futures bid–ask spread, and a 27% increase in volatility, as a result of the futures exchange fee reduction, whereas the pricing efficiency of the futures contract and the exchange's revenues are unaffected by the change in transaction costs. The exchange fee reduction has improved futures market liquidity at the cost of higher volatility. Moreover, the attractiveness and competitiveness of the futures exchange has increased relative alternative trading venues, without a loss of revenues in the process. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:775–796, 2009     

Multidimensional Variance‐Optimal Hedging in Discrete‐Time Model—A General Approach

One of the well‐known approaches to the problem of option pricing is a minimization of the global risk, considered as the expected quadratic net loss. In the paper, a multidimensional variant of the problem is studied. To obtain the existence of the variance‐optimal hedging strategy in a model without transaction costs, we can apply the result of Monat and Stricker. Another possibility is a generalization of the nondegeneracy condition that appeared in a paper of Schweizer, in which a one‐dimensional problem is solved. The relationship between the two approaches is shown. A more difficult problem is the existence of an optimal solution in the model with transaction costs. A sufficient condition in a multidimensional case is formulated.     

INVESTING WITH LIQUID AND ILLIQUID ASSETS

We find optimal trading policies for long‐term investors with constant relative risk aversion and constant investment opportunities, which include one safe asset, liquid risky assets, and an illiquid risky asset trading with proportional costs. Access to liquid assets creates a diversification motive, which reduces illiquid trading, and a hedging motive, which both reduces illiquid trading and increases liquid trading. A further tempering effect depresses the liquid asset's weight when the illiquid asset's weight is close to ideal, to keep it near that level by reducing its volatility. Multiple liquid assets lead to portfolio separation in four funds: the safe asset, the myopic portfolio, the illiquid asset, and its hedging portfolio.     

Volatility trade design

Despite the fact that they are heavily traded, discussed in every derivatives text, and necessary to aligning implied volatilities with volatility expectations, volatility trades such as straddles, strangles, and option/asset combinations have received scant attention in the finance research literature. Using a unique data set for the Eurodollar options market, the trading and structure of seven volatility trades—straddles, strangles, option/asset combinations, guts, butterflies, iron butterflies, and condors—are examined. We find that both traders' choices among the seven strategies and the designs they choose for the individual strategies indicate that volatility traders seek designs with (1) low deltas, (2) low transaction costs, and (3) high gammas and vegas. Among other things, these three presumed objectives explain why butterflies, guts, and condors are rarely traded; covered call and put writing is rare; and straddles are the most popular volatility trade. These objectives also explain the usual design of straddles, strangles, and asset/option combinations and the straddle–strangle choice. Our data also indicate that, in constructing their spreads, traders rely on heuristics that lead to relatively low deltas and high gammas and vegas, but not always the lowest delta and highest gamma/vega constructions implied by more sophisticated models. We find little evidence of trading based on the shape of the smile, that is, little evidence that trades are designed to long (short) strikes with low (high) implied volatilities. We find that some volatility trade structures—those that (1) receive considerable attention in finance textbooks, (2) have been posited by finance researchers, or (3) are recognized by the exchanges—are rarely employed by traders, whereas others are quite common. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:243–279, 2005     

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.     

An Empirical Investigation of Trade Credit Demand

This paper investigates the motivations for a firm's demand for trade credit. Demand for credit is modelled as a function of transaction costs motivations, financing motivations, operational considerations, seller compliance issues and supplier marketing, whilst controlling for the firm's business environment and for firm characteristics such as size and industry. This paper builds on previous studies by considering a wider range of factors that can affect trade credit demand holistically on a single sample. It appears that the use of trade credit is widespread, and that it is generally perceived as an important short term financing option, although the availability of trade credit is not a major influence on supplier choice. The level of a firm's trade credit demand is found to be significantly influenced by transaction costs, financing, operational issues, marketing activities by suppliers, the firm's investment in trade debtors and firm size.     

Volatility forecasts embedded in the prices of crude-oil options

This paper evaluates the ability of alternative option-implied volatility measures to forecast crude-oil return volatility. We find that a corridor implied volatility measure that aggregates information from a narrow range of option contracts consistently outperforms forecasts obtained by the popular Black–Scholes and model-free volatility expectations, as well as those generated by a realized volatility model. This measure ranks favorably in regression-based tests, delivers the lowest forecast errors under different loss functions, and generates economically significant gains in volatility timing exercises. Our results also show that the Chicago Board Options Exchange's “oil-VIX” index performs poorly, as it routinely produces the least accurate forecasts.     

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.     

Vehicle currency use in international trade

We explore the major driving forces for currency invoicing in international trade with a simple model and a novel dataset covering 24 countries. We contrasts a “coalescing” effect, where exporters minimize the movements of their prices relative to their competitors', with incentives to hedge macroeconomic volatility and transaction costs. The key determinants of invoice currency choice are industry features and country size, with some role for foreign-exchange bid–ask spreads. The coalescing effect also goes a long way to explaining the well-known dominance of the dollar. Trade flows to the United States are predominantly invoiced in dollar, as foreign exporters face competition with U.S. firms. The use of the dollar in trade flows that do not involve the United States reflects trade in homogeneous products where firms need to keep their price in line with their competitors'.     

DERIVATIVE ASSET PRICING WITH TRANSACTION COSTS1

In the modern theory of finance, the valuation of derivative assets is commonly based on a replication argument. When there are transaction costs, this argument is no longer valid. In this paper, we try to address the general problem of finding the optimal portfolio among those which dominate a given derivative asset at maturity. We derive an interval for its price. the upper bound is the minimum amount one has to invest initially in order to obtain proceeds at least as valuable as the derivative asset. the lower bound is the maximum amount one can borrow initially against the proceeds of the derivative asset. We show that, in some instances, this interval may be strictly bounded above by the price of the replicating strategy. Prima facie, the cost of a dominating strategy should appear to be higher than that of the replicating one. But because trading is costly, it may pay to weigh the benefits of replication against those of potential savings on transaction costs.     

Locational Competition: A Neglected Paradigm in the International Division of Labour

Krugman's verdict that competitiveness of countries is a largely meaningless concept is a serious misjudgement of the economics profession. Countries compete for the mobile factors of production, most importantly for capital and technology. The exit‐option of these factors and of firms changes the calculus of national governments. This paper sets out the main elements of the concept of competition between locations – locational competition – and analyses its impact on welfare and employment of the capital‐exporting country. It also looks at whether competition between countries necessarily results in a race to the bottom or whether it can function as a controlling mechanism for governments and as a discovery device. The paper discusses under which conditions common rules are needed to reduce transaction costs and to prevent strategic, opportunistic behaviour of countries and which common rules thus reduce transaction costs. Finally, it deals with the question whether one institutional equilibrium in the world economy can be expected or whether many national equilibriums can coexist.     

Semistatic hedging and pricing American floating strike lookback options

We price an American floating strike lookback option under the Black–Scholes model with a hypothetic static hedging portfolio (HSHP) composed of nontradable European options. Our approach is more efficient than the tree methods because recalculating the option prices is much quicker. Applying put–call duality to an HSHP yields a tradable semistatic hedging portfolio (SSHP). Numerical results indicate that an SSHP has better hedging performance than a delta-hedged portfolio. Finally, we investigate the model risk for SSHP under a stochastic volatility assumption and find that the model risk is related to the correlation between asset price and volatility.     

Optimal partial hedging of options with small transaction costs

This study uses asymptotic analysis to derive optimal hedging strategies for option portfolios hedged using an imperfectly correlated hedging asset with small fixed and/or proportional transaction costs, obtaining explicit formulae in special cases. This is of use when it is impractical to hedge using the underlying asset itself. The hedging strategy holds a position in the hedging asset whose value lies between two bounds, which are independent of the hedging asset's current value. For low absolute correlation between hedging and hedged assets, highly risk‐averse investors and large portfolios, hedging strategies and option values differ significantly from their perfect market equivalents. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 31:855–897, 2011