Fair Terms and Fair Pricing for Multiple Warrant Issues

Abstract

Some firms utilize one or more tranches of warrant issues to supplement their capital base. Unlike exchange-traded options, the exercise of warrants requires the issuance of stock by the company, resulting in a form of dilution. Some previous studies of warrant valuation relied on “the value of the firm,” which is nonobservable, making it difficult to apply the corresponding valuation formula. This paper derives closed-form formulas to value single and multiple tranches of warrants based on the underlying stock price, its volatility, and other known parameter values. The paper first establishes the equivalence of the Black-Scholes formula for both call options and warrants in the case of a single tranche. Thereafter, it considers the impact on the value of previously issued warrants that results when a new tranche of warrants is subsequently issued, showing in each case that fair treatment of the first-issued warrant holders requires an adjustment (due to dilution) in the terms of those warrants and a corresponding modification in the warrants’ value once a second tranche of warrants is issued. To promote such fair treatment, terms of a warrant indenture would specify the nature of the adjustment required when future warrants are issued or exercised, analogous to the antidilution terms related, for example, to stock dividends. Unlike multiple issues of traded options, which are valued independently of one another, multiple warrant issues will be shown to have prices dependent on other warrants outstanding. Also examined is the sensitivity of the fair-value adjustment to changes in the underlying variables, and the theoretical fair-value prices are compared with Black-Scholes prices and with market prices of warrants in the case of two publicly traded companies, each with two warrant issues outstanding. As warrant issues modify the equity structure of a firm, the methodology of valuing warrants presented here will be useful to investment actuaries in situations in which a comprehensive market value for all of a firm’s securities is called for. In addition, risk management practices may sometimes include the use of warrant transactions to hedge stock positions similar to the way that call options are used for that purpose. This may include hedging the risk in equity-linked insurance contracts when the equity position includes stock in companies that have one or more warrant issues that are traded. The methods developed here are also applicable to multiple issues of executive stock options (ESOs) or to combinations of warrant issues and ESOs.     

Pricing Warrants: An Empirical Study of the Black-Scholes Model and Its Alternatives

This paper uses a sample of over 25,000 daily warrant prices to empirically investigate potential problems with the commonly used warrant pricing model proposed by Black and Scholes as an extension of their call option model. One problem seems to be especially important: the constant variance assumption of the dilution adjusted Black-Scholes model appears to cause biases in model prices for almost all warrants and over the entire sample period. We show that more accurate price forecasts are obtained with a specific form of the constant elasticity of variance model.     

Is warrant really a derivative? Evidence from the Chinese warrant market

This paper studies the Chinese warrant market that has been developing since August 2005. Empirical evidence shows that the market prices of warrants are much higher systematically than the Black-Scholes prices with historical volatility. The prices of a warrant and its underlying asset do not support the monotonicity, perfect correlation and option redundancy properties. The cumulated delta-hedged gains for almost all expired warrants are negative. The negative gains are mainly driven by the volatility risk, and the trading values of the warrants for puts and the market risk for calls. The investors are trading some other risks in addition to the underlying risks.     

In-the-Money Warrant Extensions

This paper examines the warrant price and stock price reactions to the extension of the expiration date of in-the-money warrants. The warrant prices increase significantly in response to the announcement, consistent with option pricing theory. Shareholders experience no significant abnormal returns at the announcement, contrary to the conjecture that an extension will transfer wealth from shareholders. There is support for the idea that firms extend warrant life because the existing assets' cash flow obviates the need for additional financing. The data show that both the stocks and the warrants perform poorly in the month following the extension announcement.     

WARRANT PRICING USING OBSERVABLE VARIABLES

The classical warrant pricing formula requires knowledge of the firm value and of the firm‐value process variance. When warrants are outstanding, the firm value itself is a function of the warrant price. Firm value and firm‐value variance are then unobservable variables. I develop an algorithm for pricing warrants using stock prices, an observable variable, and stock return variance. The method also enables estimation of firm‐value variance. A proof of existence of the solution is provided.     

Unit Ipos: The Who,When, And Why Of Warrant Amendment

Although unit initial public offering (IPO) firms reserve the right to amend the original terms of their warrants, only some choose to extend the exercise period, lower the exercise price, or both. We examine the extent of warrant amendment among unit IPOs and find that the decision to amend is related to the need for cash and is generally employed when share prices are closer to the original warrant exercise prices. Furthermore, extension is less likely when the firm is riskier, whereas higher levels of insider ownership significantly reduce the likelihood that a firm will lower the exercise price.     

Assessing the quality of a security market: a new approach to transaction-cost measurement

I discuss a new method for measuring the deviations betweenactual transaction prices and implicit efficient prices. Theapproach decomposes security transaction prices into random-walkand stationary components. The random-walk component may beidentified with the efficient price. The stationary component,the difference between the efficient price and the actual transactionprice, is termed the pricing error. Its dispersion is a naturalmeasure of market quality. I describe practical strategies forestimating these quantities. For a sample of NYSE stocks, theaverage pricing error standard deviation estimate is roughly0.33 percent of the stock price. If the pricing error is normallydistributed and if it is always a positive cost incurred bythe transaction initiators, the corresponding average transactioncost for these traders is 0.26 percent of the stock price. Thedispersion of the pricing error is also found to be elevatedat the beginning and end of the trading session.     

认购权证与标的股票间的价格相关关系实证研究

为系统分析我国认购权证市场与其标的股票市场间的价格相关关系，发现权证市场与股票市场间的联系。本文以12组样本认购权证目收盘价和标的股票的目收盘价为观测对象，运用ADF单位根检验、Johansen协整检验、向量误差修正模型（VECM）、Granger因果检验和Hasbrouck方差分解方法，考察了两个时序数列间的长期均衡关系、短期动态关系、Granger因果关系和两市在价格发现功能中作用的大小、反映信息的效率，得出了研究结论。     

Price and volume effects associated with derivative warrant issuance on the Stock Exchange of Hong Kong

This paper examines the price and volume effects of underlying stocks around the announcement date of derivative warrants issued in Hong Kong. In general, the results indicate that underlying stocks are subject to extra buying pressure a few days before the new derivative warrant issuance, which is consistent with our hypothesis about the hedging effect created by the merchant banks that initiate the warrant issuance. Since the prices of underlying stocks peak on the first day after the warrant announcement and are stable thereafter, the information effect associated with the warrant issuance appears to be weak and does not last long. In addition, we find that underlying stocks have abnormal increases in price and volume during the last 5 minutes of trading on the warrant issuance day. This might be due to investors’ buying behavior precipitated by information leakage about the successful warrant issuance and/or to the price manipulation by merchant banks in order to attain a better payoff from the warrant issuing business.     

沪深权证价格偏离分析

沪深证券市场权证理论价格与权证市场价格的偏离现象主要不是理论定价公式的输入变量误差导致的,而是因为股票收益率与权证收益率之间的相关系数与理论预期不一致,乃至相冲突所导致的。权证理论价格与市场价格偏离的主要原因是在当前的交易制度安排下,由于市场限制(摩擦)抑制了套利交易,导致权证市场投机交易占主导,以致权证市场价格偏离长期均衡水平。     

Some Hypotheses on the Valuation of Stock Warrants

Stock warrants are often portrayed as securities that yield supernormal returns while their counterpart stocks only earn moderate yields. Those returns seem inconsistent with efficient capital markets. A present-value warrant model is developed in this paper. This model assumes that warrants grow at a rate appropriate to their risk class. The model suggests that warrant values vary directly with growth of the stock and the time remaining to exercise, and vary inversely with the warrant holder's cost of capital.     

股权分置改革中的公司认股证定价探讨——以长电权证定价为例

由于公司认股证期末需要通过发行新股以应对权证持有人行权要求,故不能直接套用Black-Scholes公式.长电权证作为首只公司认股证,目前部分研究机构对其定价过程中的问题存在一定认识误区.本文将理清认股证定价模型原理,并针对长电权证的实际情况修正公司认股证模型,运用MC模拟的方法加以验证并提出考虑红利时的处理方法.     

Warrant valuation and exercise strategy

This paper demonstrates that warrant valuation and exercise strategy differ fundamentally from call option valuation. Simultaneous exercise of warrants is shown to be suboptimal and a monopolist owning all warrants can achieve a higher value. Unless warrants are perfectly divisible, no satisfactory equilibrium exists for the valuation and exercise of widely held warrants. The problems encountered appear to be quite general and stem from necessary assumptions about future corporate dividend policy and capital structure. Such assumptions are necessary for any model of corporate security valuation.     

关于权证定价修正模型的比较研究——基于交易成本和股息分红因素的分析框架

在我国股权分置改革中，权证推动了证券市场的金融创新。鉴于权证定价可以借鉴期权理论，国外B—S模型对我国权证市场的创新和风险管理具有一定的参考意义。本文采用B—S定价模型定价宝钢认购权证和长电认购权证，分别从交易成本和股息分红的角度进行了相应的模型调整，以改进、完善适应我国权证市场的定价方法。     

Warrant pricing: a review of empirical research

Recently, several warrant pricing studies have become available for different models as well as for different countries. The most important conclusions that can be drawn from reviewing these studies are: (1) it is not necessary to make a correction on option valuation models for the dilution effect; (2) the only model that systematically outperforms the Black-Scholes (1973) type models is the Square Root model; (3) US and German warrants seem to be priced correctly, while deviations are found for English and Japanese warrants (underpriced by the market) and Swiss and Dutch warrants (overpriced by the market).     

Impact of Imperfect Information on the Optimal Exercise Strategy for Warrants

In this paper, we determine the optimal exercise strategy for corporate warrants if investors suffer from imperfect information and we evaluate the impact of this friction on the value of a warrant. For this purpose, we address both exercises at maturity, where imperfect information about the firm value is present, and exercises before maturity which are impacted by imperfect information about the size of the dividend. We model imperfect information so that all warrant holders know that they obtain biased signals of the true state without observing the signals of other warrant holders. The optimal exercise strategy follows from a complex game among warrant holders in which every individual warrant holder must account for the potential signals of the other warrant holders and their resulting exercise decisions. The main findings are that due to imperfect information warrant holders optimally start to exercise their warrants later than without imperfect information. Moreover, a simple block exercise strategy is always an equilibrium strategy for a high degree of imperfect information before maturity, even though a partial exercise can be the unique strategy without imperfect information. Remarkably, imperfect information does not necessarily result in a lower warrant value. As long as a warrant holder has a signal that allows for correct exercise decisions, then imperfect information enhances the warrant value due to suboptimal exercises by other investors.     

Warrant exercise and bond conversion in competitive markets

We develop a theory of warrants held by competitive warrantholders not constrained to exercise their warrants as one block; the theory also applies to convertible bonds held by competitive bondholders not constrained to convert their bonds as one block. We prove that the warrant (bond) price in each of the competitive equilibria is less than or equal to the price in an economy with the block constraint; and for at least one competitive equilibrium the warrant (bond) price equals the warrant (bond) price in the block-constrained economy. We illustrate the paths of competitive warrant exercise and bond conversion and conclude that under realistic assumptions they can be long.     

Determinants of Contract Choice: The Use of Warrants to Compensate Underwriters of Seasoned Equity Issues

The issuer's decision to include warrants as compensation to underwriters is studied for a sample of 1,991 negotiated firm commitment issues of seasoned equity. Using a two-stage logit model to correct for self-selection bias, we find direct evidence that warrant compensation functions as a bond, substituting for reputational capital and enabling the underwriter to certify the issue price. To a lesser degree, the decision also is affected by regulations on underwriter compensation and on the use of underwriter warrants. Issuers' decisions are consistent with an objective of minimizing total underwriting cost, including cash compensation, warrants, and underpricing.     

The Valuation Effects of Warrant Extensions

In this paper, we examine the warrant price and stock price reactions to the announcement of warrant life extensions. As predicted by option-pricing theory, warrant prices increase in response to an extension. Our principal finding is that the stocks of firms making the extension announcements experience positive abnormal returns on average. We interpret the evidence as supportive of an anticipation hypothesis in which the market perceives the decision to extend the warrants' expiration date as a favorable indication for the stock price before the subsequent expiration.     

投资者是非理性的吗——卖空限制下我国权证价格偏离探析

本文研究了影响我国权证价格偏离的主要因素。以往文献的重售动机理论和便利收益理论不能诠释权证定价偏差的不对称性。我们的研究提出理性避险动机是权证价格偏离的主导因素,即我国投资者并不是非理性的进行重售投机,也并没有忽视权证的避险性能。本文避免了以往文献的Black-Scholes模型依赖,在平价准则的基础上构造了非模型的误差度量方法,从而剔除了模型假设导致的价格偏差,保证了研究结果的有效性和准确性。