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Callable convertible bonds under liquidity constraints and hybrid priorities

David Hobson, Gechun Liang, Edward Wang

Abstract

This paper investigates the callable convertible bond problem in the presence of a liquidity constraint modelled by Poisson signals. We assume that neither the bondholder nor the firm has absolute priority when they stop the game simultaneously, but instead, a proportion $m\in[0,1]$ of the bond is converted to the firm's stock and the rest is called by the firm. The paper thus generalizes the special case studied in [Liang and Sun, Dynkin games with Poisson random intervention times, SIAM Journal on Control and Optimization, 57 (2019), 2962-2991] where the bondholder has priority ($m=1$), and presents a complete solution to the callable convertible bond problem with liquidity constraint. The callable convertible bond is an example of a Dynkin game, but falls outside the standard paradigm since the payoffs do not depend in an ordered way upon which agent stops the game. We show how to deal with this non-ordered situation by introducing a new technique which may be of interest in its own right, and then apply it to the bond problem.

Callable convertible bonds under liquidity constraints and hybrid priorities

Abstract

This paper investigates the callable convertible bond problem in the presence of a liquidity constraint modelled by Poisson signals. We assume that neither the bondholder nor the firm has absolute priority when they stop the game simultaneously, but instead, a proportion of the bond is converted to the firm's stock and the rest is called by the firm. The paper thus generalizes the special case studied in [Liang and Sun, Dynkin games with Poisson random intervention times, SIAM Journal on Control and Optimization, 57 (2019), 2962-2991] where the bondholder has priority (), and presents a complete solution to the callable convertible bond problem with liquidity constraint. The callable convertible bond is an example of a Dynkin game, but falls outside the standard paradigm since the payoffs do not depend in an ordered way upon which agent stops the game. We show how to deal with this non-ordered situation by introducing a new technique which may be of interest in its own right, and then apply it to the bond problem.

Paper Structure

This paper contains 21 sections, 38 theorems, 101 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dynkin games without the order condition
  3. The callable convertible bond: problem specification and auxiliary results
  4. Specification of the convertible bond problem
  5. Some auxiliary results
  6. Auxiliary optimal stopping problem for the bondholder
  7. The auxiliary optimal stopping problem
  8. Case $m=1$, $\mu<r$
  9. Case $m\in[0, 1)$, $\mu< \lambda+r$ and case $m=0$, $\mu>\lambda+r$
  10. The optimal stopping boundary via a fixed point
  11. Solving the optimal stopping problem
  12. Case $m=0, \mu=\lambda+r$, $K \geq P$
  13. Auxiliary optimal stopping problems for the firm
  14. The auxiliary optimal stopping problem
  15. Case $m\in(0,1]$, $\mu<\lambda+r$ and $K\in(P,\hat{K}(m))$
  16. ...and 6 more sections

Key Result

Lemma 2.5

\newlabellem:gamma Suppose that the payoff processes $U,M,L$ satisfy the generalised order condition. Then, the random variable $\gamma$ is in $\mathcal{T}$ and satisfies the following properties: (1) $L_{\gamma}\geq M_\gamma\geq U_{\gamma}$ on the set $\{\gamma<\eta\}$. (2) For any $\tau \in \mat

Figures (10)

  • Figure 4.1: The function $\hat{K}(m)$ for $\mu<r$, whence $\hat{m}>1$.
  • Figure 4.2: The function $\hat{K}(m)$ for $\mu\in[r,\lambda+r)$, whence $\hat{m} \in (0,1]$.
  • Figure 4.3: Function $f^m(x)$ for $\mu<r$
  • Figure 4.4: Function $f^m(x)$ for $\mu=r$
  • Figure 4.5: Function $f^m(x)$ for $\mu\in(r,\lambda+r)$
  • ...and 5 more figures

Theorems & Definitions (80)

  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Corollary 2.8
  • proof
  • ...and 70 more