Equilibrium model with default and dynamic insider information

Abstract

We consider an equilibrium model à la Kyle–Back for a defaultable claim issued by a given firm. In such a market the insider observes continuously in time the value of the firm, which is unobservable by the market makers. Using the construction in Campi et al. (http://hal.archives-ouvertes.fr/hal-00534273/en/, 2011) of a dynamic three-dimensional Bessel bridge, we provide the equilibrium price and the insider’s optimal strategy. As in Campi and Çetin (Finance Stoch. 11:591–602, 2007), the information released by the insider while trading optimally makes the default time predictable in the market’s view at the equilibrium. We conclude the paper by comparing the insider’s expected profits in the static and dynamic private information case. We also compute explicitly the value of the insider’s information in the special cases of a defaultable stock and a bond.

Appendix

Lemma 5.1

Suppose that h is a nondecreasing right-continuous function with at most exponential growth. Let

$$H(t,x):=\int_{-\infty}^{\infty}h(x+y)\frac{1}{\sqrt{2 \pi (1-t)}}\exp \biggl(-\frac{y^2}{2 (1-t)}\biggr)\,dy, $$

and let (ξ _n ) _n≥1 be a convergent sequence such that lim _n→∞ H(t _n ,ξ _n )=a for some a in the range of h or in the interval (inf _x h(x),sup _x h(x)) and for some sequence (t _n ) _n≥1⊆[0,1) converging to 1. Then

$$\lim_{n \rightarrow \infty}\xi_n \in\big[X^a_{{\min}}, X^a_{{\max}}\big], $$

where

$$X^a_{{\min}}:=\inf\bigl\{ x: h(x)\geq a\bigr\} \quad\mbox{\textit{and}} \quad X^a_{{\max}}:=\sup\bigl\{ x: h(x)\leq a\bigr\}. $$

Proof

Suppose that $\lim_{n \rightarrow \infty}\xi_{n}<X^{a}_{{\min}}$. Then there exists some ξ such that we have $\lim_{n \rightarrow \infty}\xi_{n}<\xi<X^{a}_{{\min}}$. Since H is nondecreasing in x, one has

$$\lim_{n \rightarrow \infty}H(t_n, \xi_n)\leq\lim_{n \rightarrow \infty}H(t_n, \xi )=h(\xi)<a, $$

which is a contradiction. Similarly, we obtain that $\lim_{n \rightarrow \infty }\xi_{n}\leq X^{a}_{{\max}}$. □