Modeling and replication of the prepayment option of mortgages including behavioral uncertainty

Prepayment risk embedded in fixed-rate mortgages forms a significant fraction of a financial institution's exposure, and it receives particular attention because of the magnitude of the underlying market. The embedded prepayment option bears the same interest rate risk as an exotic interest rate swap with a suitable stochastic notional. We investigate the effect of relaxing the assumption of a deterministic relationship between the market interest rate incentive and the prepayment rate. A non-hedgeable risk factor is modeled to capture the uncertainty in mortgage owners’ behavior, leading to an incomplete market. We prove under natural assumptions that including behavioral uncertainty reduces the exposure's value. We statically replicate the embedded prepayment option exposure with swaps and swaptions. We show that a replication based solely on swaps cannot easily control the right tail of the exposure distribution, while including swaptions enables that. The replication framework is flexible and focuses on different regions in the exposure distribution. Since a non-hedgeable risk factor entails the existence of multiple equivalent martingale measures, pricing and optimal replication are not unique. We investigate the effect of a market price of risk misspecification, and we provide a methodology to generate robust hedging strategies. Such strategies, obtained as solutions to a saddle-point problem, allow us to bound the exposure against a misspecification of the pricing measure.