The Real Option to Lapse and the Valuation of Death-Protected Investments

Contents
1 Introduction. 3
2 The General Model 8
2.1 Underlying Asset and Dynamics . . . . . . . . . . . . . . . . . 8
2.2 Lapsation and Deferred surrender Charges. . . . . . . . . . . . 9
2.3 TheMaturity and Death Guarantees . . . . . . . . . . . . . . 10
2.4 The Death Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 The Hedge Portfolio . . . . . . . . . . . . . . . . . . . . . . . 11
2.6 Formulation as American Contingent Claim . . . . . . . . . . 12
2.7 Alternative Risk-Neutral Representation. . . . . . . . . . . . . 14
2.8 The Discounted Value of the Continuous Insurance Charges . 14
3 Analytic Solution. 16
3.1 Constant DSC and Hazard Rate with Guarantee at Death only. 16
4 Numerical Examples. 19
5 Conclusion. 23
6 Appendix 30
6.1 Generalizing themodel . . . . . . . . . . . . . . . . . . . . . . 30
6.2 Commentary on section 2.4 of the paper . . . . . . . . . . . . 33
6.3 Commentary on section 2.6 of the paper . . . . . . . . . . . . 34
6.4 Commentary on Section 3: Snell’s envelope . . . . . . . . . . . 36
6.5 Finite Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.6 Commentary on Section 3: the free BVP . . . . . . . . . . . . 40
6.7 CASE 1: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.8 CASE 2: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.9 Commentary on Section 3: αL and αH . . . . . . . . . . . . . 47
6.10 αL, αM, and αH in general . . . . . . . . . . . . . . . . . . . . 49
6.11 Commentary on Section 3: TEDF . . . . . . . . . . . . . . . . 53