Pricing and hedging the prepayment option of mortgages under stochastic housing market activity

Leonardo Perotti, Lech A. Grzelak and Cornelis W. Oosterlee

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Abstract: Prepayment risk embedded in fixed-rate mortgages forms a significant fraction of a financial institution's exposure. The embedded prepayment option bears the same interest rate risk as an exotic interest rate swap with a suitable stochastic notional. Focusing on penalty-free prepayment because of the contract owner's relocation to a new house, we model the prepayment option value as an European-type interest rate receiver swaption with stochastic maturity matching the stochastic time of relocation. This is a convenient representation since it allows us to compute the prepayment option value in terms of well-known pricing formulas for European-type swaptions. We investigate the effect of a stochastic housing market activity as the explanatory variable for the distribution of the relocation time, as opposed to the conventional assumption of a deterministic housing market activity. We prove that the housing market covariance drives the prepayment option price difference between the stochastic setting and its deterministic counterpart. The prepayment option exposure is hedged using market instruments based on Delta-Gamma replication. Furthermore, since the housing market activity is a non-tradable risk factor, we perform non-standard actuarial hedging focusing on controlling the prepayment option exposure yield by risky housing market scenarios.

Date: 2025-07
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