A variation of Merton's corporate bond valuation model for firms with illiquid but observable assets

We introduce a new model for pricing corporate bonds, which is a modification of the classical model of Merton. In this new model, we drop the liquidity assumption of the firm's asset value process, and assume that there is a liquidly tradeable asset in the market whose value is correlated with the firm's asset value, and all portfolios can be constructed using solely this asset and the money market account. We formulate the market price of the corporate bond as the product of the price of an optimal replicating portfolio and $\exp (- \kappa \times \mbox {replication error}) $exp⁡(−κ×replication error), where κ is a non-negative constant. The interpretation is that the representative investor uses the price of the optimal replicating portfolio as a benchmark and requests compensation for the non-hedgeable risk. We show that if the replication error is measured relative to the firm's value, the resulting formula is arbitrage free with mild restrictions on the parameters.