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.     

An Empirical Analysis of the Pricing of Bank Issued Options versus Options Exchange Options

Since 1998, large investment banks have become active as issuers of options, generally referred to as call warrants or bank‐issued options. This has led to an interesting situation in the Netherlands, where simultaneously call warrants are traded on the stock exchange, and long‐term call options are traded on the options exchange. Both entitle their holders to buy shares of common stock. We start with a direct comparison between call warrants and call options, written on the same stock and with the same exercise price, but where the call option has a longer time to maturity. In 13 out of 16 cases we find that the call warrants are priced higher, which is a clear violation of basic option pricing rules. In the second part of the analysis we use option pricing models to compare the pricing of call warrants and call options. If implied standard deviations from options are used to price the call warrants, we find that the call warrants are strongly overpriced during the first five trading days. The average overpricing is between 25 and 30%. Only a small part of the overpricing can be explained by rational arguments such as transaction costs. We suggest that the overvaluation can be explained by a combination of an active financial marketing by the banks and the framing effect.     

The risk-shifting effect and the value of a warrant

The exercise of a warrant leads to the well-known dilution phenomenon, the effects of which have been extensively studied over the last four decades. In contrast, the existing literature has paid inadequate attention to the volatility spillover between stockholders and warrant holders. This ‘risk-shifting effect’ has significant implications for warrant pricing, since any formula that assumes a constant volatility of stock returns produces a bias. In this paper we show that a CEV process with a specific elasticity parameter properly models the stochastic volatility of stock returns for a firm with warrants outstanding. In addition, we propose an approximate analytical formula, exclusively based on observable market variables, that is able to absorb the risk-shifting bias.     

A Reduced‐Form Model for Warrant Valuation

This paper studies warrant valuation using a reduced‐form model. Analogous to the credit risk literature, structural models require complete information about the asset value process and the firm’s liabilities. In contrast, reduced‐form models require only information about the firm’s stock price process. We introduce a reduced‐form model where the warrant holder is a price taker, and we relate our model to structural models appearing in the literature.     

IS THE BOOK‐TO‐MARKET RATIO A MEASURE OF RISK?

We develop a leverage‐based alternative to traditional asset pricing models to investigate whether the book‐to‐market ratio acts as a proxy for risk. We argue that the book‐to‐market ratio should act as a proxy because of the expected relations between (1) financial risk and measures of capital structure based on the market value of equity and (2) asset risk and measures of capital structure based on the book value of equity. We find no relation between average stock returns and the book‐to‐market ratio in all‐equity firms after controlling for firm size, and an inverse relation between average stock returns and the book‐to‐market ratio in firms with a negative book value of equity.     

The financing decisions of REITs: The case of warrant extensions

In this paper, we examine the warrant price and stock price reactions to the announcement of warrant life extensions by REITs. As predicted by option pricing theory, warrant prices increase in response to these extensions. The stocks of REITs making the extension announcements experience average abnormal returns that are not significantly different from zero. Thus, total firm value appears to increase, with the gains accruing primarily to the warrantholders.     

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.     

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.     

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。     

Information asymmetry in warrants and their underlying stocks on the stock exchange of Thailand

This paper examines the informational role of warrants based on the unique order data from the Stock Exchange of Thailand, where both warrants and stocks are traded under the same market structure and where warrants are as liquid as stocks. The estimated probability of informed trading (PIN) in warrants is found to be statistically higher than their underlying stocks regardless of order submission type and order size. The PIN explains a substantial portion of the cross-sectional variation in the opening spread beyond trading volume and minimum tick size. We find evidence that a signed warrant trade contains information about the future stock price and that warrants with a higher PIN have greater predictive powers.     

The Risk‐Return Relation in International Stock Markets

We investigate the risk‐return relation in international stock markets using realized variance constructed from MSCI (Morgan Stanley Capital International) daily stock price indices. In contrast with the capital asset pricing model, realized variance by itself provides negligible information about future excess stock market returns; however, we uncover a positive and significant risk‐return tradeoff in many countries after controlling for the (U.S.) consumption‐wealth ratio. U.S. realized variance is also significantly related to future international stock market returns; more importantly, it always subsumes the information content of its local counterparts. Our results indicate that stock market variance is an important determinant of the equity premium.     

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.     

Equity with Warrants in Private Placements

We examine private equity with warrant (unit) placements and compare them with private equity placements. Firms making unit placements are smaller, younger, riskier, and characterized by higher information asymmetry than equity‐placing firms. Furthermore, unit‐placing firms experience good pre‐placement stock performance; however, their post‐placement performance is poor and worse than that of equity‐placing firms. We also find that very few of the placed warrants are in the money at expiration. Our results are consistent with the window of opportunity hypothesis and the theory that warrants are especially desirable to a clientele of overoptimistic investors.     

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.     

The valuation of compound options

This paper presents a theory for pricing options on options, or compound options. The method can be generalized to value many corporate liabilities. The compound call option formula derived herein considers a call option on stock which is itself an option on the assets of the firm. This perspective incorporates leverage effects into option pricing and consequently the variance of the rate of return on the stock is not constant as Black-Scholes assumed, but is instead a function of the level of the stock price. The Black-Scholes formula is shown to be a special case of the compound option formula. This new model for puts and calls corrects some important biases of the Black-Scholes model.     

Exchange Rate Exposure at the Firm and Industry Level

Previous work on the exposure of firms to exchange rate risk has primarily focused on U.S. firms and, surprisingly, found stock returns were not significantly affected by exchange‐rate fluctuations. The equity market premium for exposure to currency risk was also found to be insignificant. In this paper we examine the relation between Japanese stock returns and unanticipated exchange‐rate changes for 1,079 firms traded on the Tokyo stock exchange over the 1975–1995 period. Second, we investigate whether exchange‐rate risk is priced in the Japanese equity market using both unconditional and conditional multifactor asset pricing testing procedures. We find a significant relation between contemporaneous stock returns and unanticipated yen fluctuations. The exposure effect on multinationals and high‐exporting firms, however, is found to be greater in comparison to low‐exporting and domestic firms. Lagged‐exchange rate changes on firm value are found to be statistically insignificant implying that investors are able to assess the impact of exchange‐rate changes on firm value with no significant delay. The industry level analysis corroborates the cross‐sectional findings for Japanese firms in that they are sensitive to contemporaneous unexpected exchange‐rate fluctuations. The co‐movement between stock returns and changes in the foreign value of the yen is found to be positively associated with the degree of the firm's foreign economic involvement and inversely related to its size and debt to asset ratio. Asset pricing tests show that currency risk is priced. We find corroborating evidence in support of the view that currency exposure is time varying. Our results indicate that the foreign exchange‐rate risk premium is a significant component of Japanese stock returns. The combined evidence from the currency exposure and asset pricing analyses, suggests that currency risk is priced and, therefoe, has implications for corporate and portfolio managers.     

Are liquidity and corporate control priced by shareholders? Empirical evidence from Swiss dual class shares

Stadard asset pricing models generally exclude corporate control and liquidity considerations as joint explanatory factors of the stock price formation process. This empirical study investigates their influence on Swiss Bearer and Registered share prices issued by the same firm. It is shown that the statistical properties of both shares' returns differ without implying profitable arbitrage opportunities. A multifactor model of the ‘premium’ between Bearer and Registered stock prices is then proposed and tested. The results show that the freely negotiable equity book value, the existence of dominant shareholder positions and ownership transfer regime changes are significant variables in explaining the dual class share price differential.     

Common risk factors in bank stocks

This paper provides evidence on the risk factors that are priced in bank equities. Alternative empirical models with precedent in the nonfinancial asset pricing literature are tested, including the single-factor CAPM, three-factor Fama–French model, and ICAPM. Our empirical results indicate that an unconditional two-factor ICAPM model that includes the stock market excess return and shocks to the slope of the yield curve is useful in explaining the cross-section of bank stock returns. However, we find no evidence that firm specific factors such as size and book-to-market ratios are priced in bank stock returns. These results have a number of important implications for the estimation of the banks’ cost of capital as well as regulatory initiatives to utilize market discipline to evaluate bank risk under Basel II.     

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.     

Bought deals: The value of underwriter certification in seasoned equity offerings

The bought deal is the predominant method of underwriting SEOs in Canada. Offer prices are set and underwriters commit to purchase offerings several days earlier for bought deals than for firm commitment issues, implying stronger underwriter certification for bought deal issues. Consistent with the certification hypothesis, this study finds a significantly smaller negative stock price reaction around the announcement of bought deals compared to firm commitment issues. Bought deals are further shown to have smaller offer price discounts and smaller underwriting fees, implying superior pricing and thus, higher quality offerings. These findings suggest that investment banks’ underwriting method of choice is informative of issue quality.