Optimal exercise strategies for corporate warrants

Abstract

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.

Keywords:

• Warrants
• Exercise strategy
• Option pricing
• Financial structure

Notes

†Note that is continuous in time t because the derivative is continuous. Thus, the magnitude of immediately before maturity provides us with a reasonable impression about the shape of the volatility at a certain point of time t slightly before maturity T.

‡Note that in figure 3 has a different shape than the volatility in figure 4 from Crouhy and Galai (1994). The reason for this difference is that we focus on the marginal point of time before exercise, but Crouhy and Galai regard the volatility for warrants with a finite time to maturity. Thus, in their diagram the volatility is continuous and can increase or decrease with the firm value.

†For the derivation of optimal strategies see also Schmeidler (1973) and Blonski (1999).

†According to Bühler and Koziol (2002), the debt value always increases with the number of prematurely converted convertible bonds. Thus, this property of debt of a firm with warrants is fundamentally different from the behaviour of debt of a firm with convertible bonds. For this reason, the analysis of the optimal exercise volume and the asset values, which will be conducted in this section, will show some major differences compared to the corresponding analysis of convertible bonds.

†Since the stock value can decrease with the firm value V ₀ in special cases, we must regard the absolute value of the derivative. As a consequence, the volatility can be zero when is zero.

‡The corresponding volatility of the ex dividend stock return immediately before exercise is . Thus, for both volatilities coincide. Otherwise, the relative difference is m E/V ₀ which diminishes if V ₀ is large.

†Strictly speaking, the size operator around the derivative is superfluous in the unrestricted case as the stock value cannot decrease in the firm value V ₀.

†These findings are consistent with the numerical results of Schulz and Trautmann (1994) obtained for the application of the ‘option-like’ warrant valuation.

‡This ‘option-like’ approach is suggested by Schulz and Trautmann (1994). The simplification of the ‘option-like’ approach is to use the simple Black--Scholes formula for call option values with a constant stock return volatility. Of course, the ‘option-like’ warrant value computed in this way can deviate from the warrant value of a firm value model, because of the assumed stock value process in the ‘option-like’ approach which is not consistent with that in the firm value approach. Note that whenever consistent processes for the dynamics of the underlying are assumed, the warrant value coincides with the ‘option-like’ call value. This effect is also analysed by Hanke and Pötzelberger (2000, 2002).

†This effect is because one individual warrant holder has the same effect as a call holder. If she or he exercises, the exercise value is obtained per warrant due to the missing price impact of one individual warrant holder.