A pricing formula for delayed claims: Appreciating the past to value the future

We consider the valuation of contingent claims with delayed dynamics in a Black \& Scholes complete market model. We find a pricing formula that can be decomposed into terms reflecting the market values of the past and the present, showing how the valuation of future cashflows cannot abstract away from the contribution of the past. As a practical application, we provide an explicit expression for the market value of human capital in a setting with wage rigidity

Managing Capital Market Risk for Retirement

We offer an overview of solutions available to pension plans to manage capital market risk in order to meet their obligations. We outline the main drivers behind the evolution of asset-liability management (ALM) for pension plans and the emergence of liability-driven investment (LDI) in the last decade. We look at some of the most popular pension de-risking tools and at recent innovations prompted by the Global Financial Crisis. We offer examples based on the rise of cross-asset correlation, the use of hybrid products to mitigate tail risk, and the increasing relevance of counterparty risk mitigation tools such as collateralization. We conclude by outlining some of the main challenges ahead, including developments in pension regulation, centralized clearing of over-the-counter (OTC) instruments, and risk taking incentives in delegated asset management for long term retirement obligations

The cost of counterparty risk and collateralization in longevity swaps

Derivative longevity risk solutions, such as bespoke and indexed longevity swaps, allow pension schemes and annuity providers to swap out longevity risk, but introduce counterparty credit risk, which can be mitigated if not fully eliminated by collateralization. We examine the impact of bilateral default risk and collateral rules on the marking to market of longevity swaps, and show how longevity swap rates must be determined endogenously from the collateral flows associated with the marking-to-market procedure. For typical interest rate and mortality parameters, we find that the impact of collateralization is modest in the presence of symmetric default risk, but more pronounced when default risk and/or collateral rules are asymmetric. Our results suggest that the overall cost of collateralization is comparable with, and often much smaller than, that found in the interest-rate swaps market (as a result of the offsetting effects of interest rate and longevity risks), which may then provide the appropriate reference framework for the credit enhancement of both indemnity-based and indexed longevity risk solutions.longevity swap; counterparty risk; default risk; collateral; marking-to-market

Wage Rigidity and Retirement in Optimal Portfolio Choice

We study an agent's lifecycle portfolio choice problem with stochastic labor income, borrowing constraints and a finite retirement date. Similarly to arXiv:2002.00201, wages evolve in a path-dependent way, but the presence of a finite retirement time leads to a novel, two-stage infinite dimensional stochastic optimal control problem, which we fully solve obtaining explicitly the optimal controls in feedback form. This is possible as we find an explicit solution to the associated Hamilton-Jacobi-Bellman (HJB) equation, which is an infinite dimensional PDE of parabolic type. The identification of the optimal feedbacks is more delicate than in arXiv:2002.00201 to the two-stage structure and to the presence of time-dependent state constraints, which appear to be new in the infinite dimensional stochastic control literature. The explicit solution allows us to study the properties of optimal strategies and discuss their implications for portfolio choice. Importantly, we discuss not only the optimal allocations for the case of labor income spanned by the traded assets, but also provide novel insights into the case in which wages are also driven by idiosyncratic shocks.Comment: 30 pages, 1 figur

The role of counterparty risk and collateral in longevity swaps *

Abstract Longevity swaps allow pension schemes and annuity providers to swap out longevity risk, but introduce counterparty credit risk, which can be mitigated or eliminated by collateralization. In this study, we examine the impact of bilateral default risk and collateralization rules on the marking to market of longevity swaps. In particular, we show how different rules for posting collateral during the life of the swap may affect longevity swap rates

The cost of counterparty risk and collateralization in longevity swaps

Derivative longevity risk solutions, such as bespoke and indexed longevity swaps, allow pension schemes and annuity providers to swap out longevity risk, but introduce counterparty credit risk, which can be mitigated if not fully eliminated by collateralization. We examine the impact of bilateral default risk and collateral rules on the marking to market of longevity swaps, and show how longevity swap rates must be determined endogenously from the collateral flows associated with the marking-to-market procedure. For typical interest rate and mortality parameters, we find that the impact of collateralization is modest in the presence of symmetric default risk, but more pronounced when default risk and/or collateral rules are asymmetric. Our results suggest that the overall cost of collateralization is comparable with, and often much smaller than, that found in the interest-rate swaps market (as a result of the offsetting effects of interest rate and longevity risks), which may then provide the appropriate reference framework for the credit enhancement of both indemnity-based and indexed longevity risk solutions

Optimal portfolio choice with path dependent labor income: the infinite horizon case

We consider an infinite horizon portfolio problem with borrowing constraints, in which an agentreceives labor income which adjusts to financial market shocks in a path dependent way. Thispath-dependency is the novelty of the model, and leads to an infinite dimensional stochasticoptimal control problem. We solve the problem completely, and find explicitly the optimalcontrols in feedback form. This is possible because we are able to find an explicit solutionto the associated infinite dimensional Hamilton-Jacobi-Bellman (HJB) equation, even if stateconstraints are present. To the best of our knowledge, this is the first infinite dimensionalgeneralization of Merton’s optimal portfolio problem for which explicit solutions can be found.The explicit solution allows us to study the properties of optimal strategies and discuss theirfinancial implications