This paper gives analytical formulas for lookback and barrier options on underlying assets that are exposed to a counterparty risk. The counterparty risk induces a drop in the asset price, but the asset can still be traded after this default time. A novel technique is developed to valuate the lookback and barrier options by first conditioning on the predefault and the postdefault time and then obtain the unconditional analytic formulas for their prices.

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A sub-fractional version of the well-known Merton's model is proposed in this paper. Default probability, values of bonds and equity and credit spreads are derived under some suitable assumptions.

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