Portfolio optimization with insider’s initial information and counterparty risk

Abstract

We study the gain of an insider having private information which concerns the default risk of a counterparty. More precisely, the default time τ is modelled as the first time a stochastic process hits a random threshold L. The insider knows this threshold (as it can be the case for the manager of the counterparty) and this information is modelled by using an initial enlargement of filtration. The standard investors only observe the value of the threshold at the default time and estimate the default event by its conditional density process. The financial market consists of a risk-free asset and a risky asset whose price is exposed to a sudden jump at the default time of the counterparty. All investors aim to maximize the expected utility from terminal wealth given their own information at the initial date. We solve the optimization problem under short-selling and buying constraints and we compare through numerical illustrations the optimal processes for the insider and the standard investors.

Appendix

We recall the canonical decomposition of $\mathbb {G}^{M}$-adapted (resp., $\mathbb {G}^{M}$-predictable) processes (see Jeulin [15], Lemmas 3.13 and 4.4).

Lemma A.1

1. For t≥0, any $\mathcal {G}_{t}^{M}$-measurable random variable can be written in the form

$$Y_t=1_{\tau>t}Y_t^0(\ell)+1_{\tau\leq t}Y_t^1(\tau), $$

where $Y_{t}^{0}(\cdot)$ and $Y_{t}^{1}(\cdot)$ are $\mathcal {F}_{t}\otimes\mathcal{B}(\mathbb {R}_{+})$-measurable.

2. Any $\mathbb {G}^{M}$-adapted process Y admits a decomposition of the form

$$Y_t=1_{\tau> t}Y_t^0(\ell)+1_{\tau\leq t}Y_t^1(\tau), $$

where Y ⁰(⋅) and Y ¹(⋅) are $\mathbb {F}\otimes\mathcal{B}(\mathbb {R}_{+})$-adapted.³

3. Any $\mathbb {G}^{M}$-predictable process Y admits a decomposition of the form

$$Y_t=1_{\tau\geq t}Y_t^0(\ell)+1_{\tau< t}Y_t^1(\tau), $$

where Y ⁰(⋅) and Y ¹(⋅) are $\mathcal {P}(\mathbb {F})\otimes\mathcal{B}(\mathbb {R}_{+})$-measurable, $\mathcal {P}(\mathbb {F})$ being the predictable σ-algebra associated with the filtration $\mathbb {F}$.

Remark A.2

To compare with the case of a standard investor, we recall that any $\mathcal {G}_{t}$-measurable random variable Z _t can be written as $Z_{t}=1_{\tau>t} Z_{t}^{0}+1_{\tau\leq t}Z_{t}^{1}(\tau)$, where $Z_{t}^{0}$ and $Z_{t}^{1}(\cdot)$ are respectively $\mathcal {F}_{t}$-measurable and $\mathcal {F}_{t}\otimes\mathcal{B}(\mathbb {R}_{+})$-measurable.