Optimal policies of call with notice period requirement

Abstract

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.

Appendices

Appendix A – Proof of Proposition 1

• 1. When q = 0, the callable American warrant is never exercised prematurely so that ${W(S, \tau)\leq c(S, \tau)}$, where c(S, τ) is the price function of the European call counterpart. For ${\tau \leq \tau_n}$, we have ${c_n(S, \tau_n) > c(S, \tau)\geq W(S, \tau)}$ so that the issuer never calls the warrant prematurely.

• 2. To show that ${S^{\ast}_{\rm call}(\tau)}$ always exists for τ > τ _n , we prove by contradiction. Assume that there exists some τ₀ > τ _n such that ${W(S, \tau) < c_n(S, \tau_n)}$ for all S and ${\tau_n < \tau \leq \tau_0}$, that is, the warrant is never called. Combining with the result in part (1), the callable American warrant then becomes a European call option for ${\tau \leq \tau_0}$ so that

  $$ W(S, \tau) = c(S, \tau; X)\quad \hbox{for}\quad \tau \leq \tau_0. $$

  It suffices to show that W(S, τ) >  c _n (S, τ _n ) for some S and ${\tau_n < \tau\leq \tau_0}$. For ${\tau_n < \tau\leq \tau_0}$, the put-call parity relation gives

  $$ \eqalign{ c_n(S, \tau_n)-W(S, \tau)\cr=\hskip.3pc c_n(S, \tau_n)-c(S, \tau)\cr=\hskip.3pc Ke^{-r\tau_n}+c(S, \tau_n; K+X)-c(S, \tau; X)\cr=\hskip.3pc X(e^{-r\tau}-e^{-r\tau_n})+p(S, \tau_n; K+X)-p(S, \tau; X), }<!endaligned> $$

  where ${p(S, \tau; X)}$ denotes the price function of a European put with strike␣price X. Since the put price function tends to zero as S tends to infinity, p(S, τ; X) and p(S, τ _n ; K + X) become sufficiently small when S is sufficiently large. Since we have

  $$ \lim_{S\to \infty}[c_n(S, \tau_n)-W(S, \tau)]=X(e^{-r\tau}-e^{-r\tau_n}) < 0\quad \hbox{for}\quad \tau > \tau_n, $$

  we then deduce that there exist some sufficiently large value of S such that

  $$ W(S, \tau) > c_n(S, \tau_n)\quad \hbox{for}\quad \tau_n < \tau\leq \tau_0. $$

  A contradiction is encountered since we have assumed ${W(S, \tau) < c_n(S, \tau_n)}$ for all S and ${\tau_n < \tau\leq \tau_0}$. This would imply that there exists critical stock price ${S^{\ast}_{\rm call}}$ such that when S reaches the level ${S^{\ast}_{\rm call}, W(S, \tau)}$ becomes equal to ${c_n(S,\tau_n)}$.

• 3. The monotonically decreasing property of ${S^{\ast}_{\rm call}(\tau)}$ is derived from the monotonically increasing property of the price function W(S, τ) with respect to τ. To show the unboundedness of ${S^{\ast}_{\rm call}(\tau^+_n)}$, we prove by contradiction. Suppose ${S^{\ast}_{\rm call}(\tau^+_n)}$ is finite, by continuity of the price function, we have

  $$ W(S,\tau_n)=c_n(S, \tau_n)\quad \hbox{for}\quad S > S^{\ast}_{\rm call}(\tau^+_n). $$

  This leads to a contradiction since the issuer should not call the warrant at τ = τ _n . Hence, ${S^{\ast}_{\rm call}\to \infty}$ as ${\tau \to \tau^+_n}$. By setting q = 0 in the asymptotic formula for ${S^{\ast}_{\rm call}(\infty)}$ in Proposition 2 [see Eq. (16)] and observing that μ₊ = 1 when q = 0, we obtain Eq. (9).

Appendix B – Proof of Proposition 2

The proofs of the existence of ${S^{\ast}_{\rm call}(\tau)}$ for all τ and the decreasing property of ${S^{\ast}_{\rm call}(\tau)}$ can be established using similar arguments as presented in Appendix A.

We compute the asymptotic limit ${S^{\ast}_{\rm call}(\infty)}$ by solving the price function W _∞(S) of the perpetual callable American warrant. The governing equation of W _∞(S) is given by

$$ \frac{\sigma^2}{2}S^2\frac{\hbox{d}^2W_\infty}{\hbox{d}S^2}+(r-q)S\frac{\hbox{d}W_\infty} {\hbox{d}S}-rW_\infty=0,\quad 0 < S < S^{\ast}_{\rm call}(\infty), $$

with boundary conditions

$$ W_\infty(0) =0\quad \hbox{and}\quad W_\infty(S^{\ast}_{\rm call}(\infty))=c_n(S^{\ast}_{\rm call}(\infty),\tau_n) $$

and smooth pasting condition

$$ \frac{\hbox{d}W_\infty}{\hbox{d}S}(S^{\ast}_{\rm call}(\infty))=e^{-q\tau_n}N(d_1(S^{\ast}_{\rm call}(\infty))), $$

where

$$ d_1(S)=\frac{\ln\frac{S}{K+X}+\left(r-q+\frac{\sigma^2}{2}\right)\tau_n}{\sigma\sqrt{\tau_n}}. $$

The general solution to the price function W _∞(S) takes the form (Kwok, 2007)

$$ W_\infty(S)=\alpha S^{\mu_+}, $$

where α is an arbitrary constant and μ₊ is the positive root of the auxiliary equation:

$$ \frac{\sigma^2}{2}\mu^2+\left(r-q-\frac{\sigma^2}{2}\right)\mu-r=0. $$

The arbitrary constant α and ${S^{\ast}_{\rm call}(\infty)}$ are determined by solving simultaneously

$$ \eqalign{ \alpha [S^{\ast}_{\rm call}(\infty)]^{\mu_+}=c_n(S^{\ast}_{\rm call}(\infty), \tau_n)\cr \alpha{\mu_+}[S^{\ast}_{\rm call}(\infty)]^{\mu_+-1}=e^{-q\tau_n}N(d_1(S^{\ast}_{\rm call}(\infty))). }<!endaligned> $$

By eliminating α and using the price formula of ${c_n(S, \tau_n)}$ in Eq. (8), the asymptotic lower bound ${S^{\ast}_{\rm call}(\infty)}$ is determined by solving

$$ \eqalign{ Ke^{-r\tau_n} +\left(1-\frac{1}{\mu_+}\right)S^{\ast}_{\rm call}(\infty)e^{-q\tau_n}N(d_1(S^{\ast}_{\rm call}(\infty))\cr -(K+X)e^{-r\tau_n}N(d_2(S^{\ast}_{\rm call}(\infty)))=0, }<!endaligned> $$

where

$$ d_2(S)=d_1(S)-\sigma\sqrt{\tau_n}. $$

By setting q = 0, the above equation for ${S^{\ast}_{\rm call}(\infty)}$ reduces to Eq. (9).

Appendix C – Proof of Proposition 3

• 1. Suppose we let ${\widetilde{W}(S, \tau)=B(S, \tau)-X}$, where B(S, τ) is the price of a non-callable convertible bond. Without the embedded callable right in the bond, we always have ${{\cal L}B\geq 0}$. It then follows that ${\widetilde{W}(S, \tau)}$ satisfies the following linear complementarity formulation

  $$ \eqalign{ {\cal L}\widetilde{W}\geq -rX\quad \hbox{and}\quad \widetilde{W}\geq S-X\cr ({\cal L}\widetilde{W}+rX)[\widetilde{W}-(S-X)]=0 }<!endaligned> $$

  and

  $$ \widetilde{W}(S, 0) =(S-X)^+. $$

  The above formulation differs from that of an American call only by the source term −rX. By using the comparison principle, we deduce that ${\widetilde{W}(S, \tau)\leq C(S, \tau)}$, where C(S,τ) is the price function of the American call. The price curve of ${\widetilde{W}(S, \tau)}$ always stays below that of C(S, τ) so that it intersects the intrinsic value line S − X at a lower critical stock price, hence ${S^{\ast}_b(\tau)\leq S^{\ast}(\tau)}$ for all ${\tau \geq 0}$.

• 2. It is obvious that ${S^{\ast}_b(\tau)\geq Xe^{-r\tau}}$ since the lower bound of the bond value is given by max(Xe ^−rτ,  S). As τ → 0⁺, we have

  $$ S^{\ast}_b(0^+)\geq X. $$

  (i)

  On the other hand, suppose ${S^{\ast}_b(0^+) > X}$, then there exists some stock price level S satisfying ${X < S < S^{\ast}_b(0^+)}$ such that the bond remains alive at τ→ 0⁺. By continuity, the bond value is equal to S as τ→ 0⁺. Substituting B(S,␣0⁺) = S into the Black-Scholes equation, we have ${\frac{\partial B}{\partial \tau}(S, 0^+)= -qS < 0}$. A contradiction is encountered since this implies that the bond value falls below the intrinsic value. Hence, we deduce that

  $$ S^{\ast}_b(0^+)\leq X. $$

  (ii)

  Combining the above two inequalities, we obtain ${S^{\ast}_b(0^+)=X}$. When τ → ∞, the convertible bond becomes essentially equivalent to the American warrant with zero strike price. Recall that ${S^{\ast}(\infty)=\frac{\mu_+}{\mu_+-1}X}$, so we obtain ${S^{\ast}_b(\infty)=0}$ since X is taken to be 0.