No-Arbitrage Valuation of Contingent Claims Depending on an Untradeable Asset

We consider an incomplete market situation with the presence of an untradeable asset and several tradeable assets. By an untradeable asset we mean an asset that cannot be traded on a public market. Typical examples of untradeable assets include real options and private credit/debt investments. We then exploit the relationship between the untradeable asset and tradeable assets to evaluate contingent claims depending on the untradeable asset. Under a multidimensional generalized Black-Scholes (GBS) framework, we study two different methods for pricing these kinds of contingent claims. The first is mean-variance hedging (MVH). The second is the method proposed in Jarrow (2023). We illustrate the two methods by applying them to two particular contingent claims: The option to defer and the spread option. No-arbitrage prices and admissible replicating trading strategies are derived. Lastly, we run simulations to test the performance of these replicating trading strategies.