Optimal exercise strategies for corporate warrants

In this paper, we analyse the optimal exercise strategies for corporate warrants issued by levered firms. For the analysis, we distinguish between two exercise variants, namely the traditional block exercise and competitive exercise in equilibrium. We find that the optimal exercise date under the block condition can be before or after an optimal exercise in equilibrium. Surprisingly, optimal block exercise can occur even without any dividend payments in contrast to the competitive exercise. As a consequence, the asset values and the stock volatility under block exercise fundamentally deviate from those under the competitive exercise variant. Moreover, the value of a warrant in the block case and its exercise strategy do not coincide with those of a corresponding call option which contrasts with the assumption of ‘option-like’ warrant valuation.     

Optimal policies of call with notice period requirement

When an American warrant or a convertible bond is called by its issuer, the holder is usually given a notice period to decide whether to sell the derivative back to the issuer at the call price or to exercise the conversion right. Several earlier papers have shown that such notice period requirement may substantially affect the optimal call policy adopted by the issuer. In this paper, we perform theoretical studies on the impact of the notice period requirement on issuer’s optimal call policy for American warrants and convertible bonds. We also examine how the optimal call policy of the issuer interacts with holder’s optimal conversion policy.     

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.     

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.     

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。     

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.     

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.     

Real and nominal interest rates: a discrete-time moel and its continuous-time limit

I provide a general equilibrium theory of the term structureof real interest rates in a discrete-time economy. I derivethe prices for one-perid and two-period real bonds and a simplerecursive formula for general k-period bonds, and prove thatthe price formula with appropriately specified parameters conversesto that of the Cox, Ingersoll, and Ross model (1985). In addition,I consider the behavior of nominal bond pries in a partial equilibriumsetting in which an exogenous price level process is correlatedwith the real economy. Finally, I provide an illustrative empiricalinvestigation of the model. The results indicate a significantcorrelation between the price level and the growth rate of consumption,which does not support the 'money neutrality' assumption underlyingCox, Ingersoll and Ross's nominal bond prices and related empiricalstudies, such as Gibbons and Ramaswamy (1992), Heston (1991),and Pearson and Sun (1991).     

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 Exercise,Dividends, and Reinvestment Policy

In this paper, we examine sequential exercise strategies by warrantholders and the gain from hoarding warrants. We analyze several obstacles to acquiring large blocks in order to exploit sequential strategies. First, we identify several reinvestment policies for which sequential exercise is not advantageous, thereby eliminating the gain from hoarding. However, sequential exercise strategies may be advantageous for monopoly or oligopoly warrantholders, even absent dividends, because using exercise proceeds to repurchase stock or to expand the firm's scale increases the riskiness of an equity share. Second, oligopoly warrantholders can receive a smaller warrant value than perfectly competitive warrantholders, suggesting a potential cost to unsuccessful hoarding.     

The nominal duration of TIPS bonds

This paper studies the price responsiveness (effective duration) of U.S. government issued inflation-indexed bonds, known by the acronym TIPS (Treasury Inflation-Protected Securities), to changes in nominal interest rates, real interest rates, and expected inflation. Using the TIPS pricing formula derived by Laatsch and Klein [Q. Rev. Econ. Finance 43 (2002) 405], we first confirm that TIPS bonds have zero sensitivity to changes solely in expected inflation. By changes solely in expected inflation, we mean that the real rate remains unchanged and the nominal rate changes in accordance with the established Fisher [Publ. Am. Econ. Assoc. 11 (1896)] effect. We show that the first derivative of the TIPS price is zero whenever the real rate is held constant. Thus, the first partial derivative of the TIPS bond pricing formula with respect to expected inflation is zero and the first partial derivative of the TIPS bond price with respect to nominal rates is also zero, given, in each case, that we hold the real rate constant. We then temporarily shift the analysis to zero-coupon TIPS bonds and zero-coupon ordinary Treasury bonds. We prove that the nominal duration of zero-coupon TIPS bonds equals that of zero-coupon ordinary Treasury bonds when the real rate changes but expected inflation is held constant.However, if expected inflation changes and the change in the nominal rate does not yield a constant real rate, zero-coupon TIPS prices will change and they will change by a smaller percentage than will zero-coupon ordinary Treasury bonds. We analyze TIPS responsiveness to changes in nominal rates under such conditions. We derive an approximation to effective duration that demonstrates that the effective durations of various maturity zero-coupon TIPS bonds are approximately linear functions in time to maturity of the effective duration of the one-year zero-coupon TIPS bond, ceteris paribus.Nominal effective duration of TIPS bonds is certainly of interest to fixed income portfolio managers that might have a desire to include such bonds in their portfolio. After all, the greater portion of a typical fixed income portfolio is in traditional, noninflation protected bonds whose major risk exposure is to changes in nominal rates. To properly assess the role of TIPS bonds in the portfolio, portfolio managers need information as to how TIPS bonds respond to the changes in nominal rates that are driving the price behavior of the bulk of the portfolio's assets. Prior to concluding the paper, we demonstrate how portfolio managers can calculate the nominal durations of coupon TIPS bonds using the zero-coupon duration formula we derive.     

我国认购权证负溢价问题研究

从理论上讲,认购权证的价格应高于其内在价值,即认购权证溢价应为正。然而从实际情况来看,我国的认购权证在2007年却大量出现负溢价的情况。本文将从套利者、投资者、投机者的角度以及权证市场参与者共同存在的一些限制等方面对我国认购权证出现负溢价的问题进行解释,并给出诸如以发行美式认购权证等方法解决这一问题的政策建议。     

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 Pricing of French Unit Seasoned Equity Offerings

Units are bundles of common stock and warrants. By issuing units, firms precommit to a future and uncertain seasoned offering at the exercise price of the warrants. This study shows that the issuance of units seasoned offerings in France is accompanied by significant abnormal returns of on average 9–12%, depending on the computing methods. Underpricing increases with the risk of the issuer and the relative size of the future seasoned equity issue linked to warrant exercises. Our results are consistent with our signaling hypothesis.     

The Pricing Formula for Commodity-Linked Bonds with Stochastic Convenience Yields and Default Risk

At the maturity, the owner of a commodity-linked bond has the right to receive the face value of the bond and the excess amount of spot market value of the reference commodity bundle over the prespecified exercise price. This payoff structure is an important characteristic of the commodity-linked bonds. In this paper, we derive closed pricing formulae for the commodity-linked bonds. We assume that the reference commodity price and the value of the firm (bonds' issuer) follow geometric Brownian motions and that the net marginal convenience yield and interest rate follow Ornstein–Uhlenbech processes. In the appendix, we derive pricing formulae for bonds which are the same as the above commodity-linked bonds, except that the reference commodity price in the definition of the payoff at the maturity is replaced by the value of a special asset which depends on the convenience yield. This revised version was published online in August 2006 with corrections to the Cover Date.     

Impact of warrant introductions on the behaviour of underlying stocks: Australian evidence

The present study examines the impact of first‐time introduction of warrants by third party issuers on the trading behaviour of a sample of underlying stocks listed on the Australian Stock Exchange. We investigate the price, liquidity and volatility impact of underlying stocks after warrant issuance and find considerable differences to those found for option listings. Significant negative abnormal returns on both the announcement and listing date of derivative warrants are reported, followed by a negative price drift. Relative trading volume and price volatility of underlying stocks are found to be significantly higher post‐warrant listing. Interestingly, we find that warrant holders are unable to realize gains for the majority of trading days when they are alive, consistent with the view that banks trade profitability from their issue.     

Does gold offer a better protection against losses in sovereign debt bonds than other metals?

It is a commonly held view that gold protects investors’ wealth in the event of negative economic conditions. In this study, we test whether other metals offer similar or better investment opportunities in periods of market turmoil. Using a sample of 13 sovereign bonds, we show that other precious metals, palladium in particular, offer investors greater compensation for their bond market losses than gold. We also find that industrial metals, especially copper, tend to outperform gold and other precious metals as hedging vehicles and safe haven assets against losses in sovereign bonds. However, the outcome of the hedge and safe haven properties is not always consistent across the different bonds. Finally, our analysis suggests that copper is the best performing metal in the period immediately after negative bond price shocks.     

Equilibrium exercise of European warrants

We review recent European warrant pricing theory that accounts for other securities in the capital structure of the firm, besides the stock and a warrant. An individual who owns a European warrant determines his exercise policy in equilibrium taking into account any transfer of wealth between stockholders and holders of other non-expiring securities. As a result, the equilibrium exercise policy differs from the textbook call-like policy. The equilibrium exercise policy depends on the direction of wealth transfer. We derive the competitive equilibrium policy and demonstrate how it can be computed.     

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.     

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.