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

2025-07-11

AI Summary

This paper addresses the pricing and hedging of prepayment options embedded in fixed-rate mortgages, triggered by homeowner relocation—a risk traditionally modeled deterministically. It innovatively treats housing market activity as a stochastic process to capture the uncertainty in relocation timing. Methodologically, the prepayment option is recast as a European receiver swaption with a random maturity; pricing employs analytical swaption formulas under stochastic interest rates, while hedging integrates Delta-Gamma replication within a scenario-based framework and introduces non-traditional actuarial hedging to manage non-tradable risk factors. Results show that housing-market covariance is the primary driver of price divergence between stochastic and deterministic models. The proposed approach enables more accurate valuation, quantifies the impact of housing-market volatility on financial institutions’ risk exposure, and enhances hedging efficacy in complex, multi-source uncertainty environments.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: User modeling for targeted and personalized online advertising

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.

Problem

• Model prepayment option value as European swaption with stochastic maturity
• Analyze stochastic housing market impact on relocation time distribution
• Hedge prepayment risk using Delta-Gamma and actuarial methods

Innovation

• European-type swaption with stochastic maturity
• Stochastic housing market activity modeling
• Delta-Gamma replication for hedging