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15
Numerical methods for controlled HamiltonJacobiBellman PDEs in finance
 Journal of Computational Finance
"... Many nonlinear option pricing problems can be formulated as optimal control problems, leading to HamiltonJacobiBellman (HJB) or HamiltonJacobiBellmanIsaacs (HJBI) equations. We show that such formulations are very convenient for developing monotone discretization methods which ensure convergenc ..."
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Cited by 31 (13 self)
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Many nonlinear option pricing problems can be formulated as optimal control problems, leading to HamiltonJacobiBellman (HJB) or HamiltonJacobiBellmanIsaacs (HJBI) equations. We show that such formulations are very convenient for developing monotone discretization methods which ensure convergence to the financially relevant solution, which in this case is the viscosity solution. In addition, for the HJB type equations, we can guarantee convergence of a Newtontype (Policy) iteration scheme for the nonlinear discretized algebraic equations. However, in some cases, the Newtontype iteration cannot be guaranteed to converge (for example, the HJBI case), or can be very costly (for example for jump processes). In this case, we can use a piecewise constant control approximation. While we use a very general approach, we also include numerical examples for the specific interesting case of option pricing with unequal borrowing/lending costs and stock borrowing fees.
Pricing Longevity Bonds using Implied Survival Probabilities. Working paper
, 2006
"... For annuity providers, longevity risk, i.e. the risk that future mortality trends differ from those anticipated, constitutes an important risk factor. In order to manage this risk, new financial products will be needed. One of the basic building blocks for such mortality backed securities is the so ..."
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Cited by 13 (3 self)
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For annuity providers, longevity risk, i.e. the risk that future mortality trends differ from those anticipated, constitutes an important risk factor. In order to manage this risk, new financial products will be needed. One of the basic building blocks for such mortality backed securities is the socalled survivor or longevity bond, the future payments of which depend on the survival rates of a certain population. We propose a methodology for the modeling and pricing of longevity bonds. We generalize the ideas of Lin and Cox (2005) and show how to derive implied survival probabilities from annuity market quotes. Taking those implied survival probabilities as a starting point, we derive the price and the dynamics of longevity bonds by applying the HeathJarrowMorton framework for mortality modeling building on an idea proposed by Miltersen and Persson (2005). We show how the models within our framework can be calibrated and applied for pricing mortality derivatives.
On the pricing of longevitylinked securities
 Insurance: Mathematics and Economics
, 2010
"... For annuity providers, longevity risk, i.e. the risk that future mortality trends differ from those anticipated, constitutes an important risk factor. In order to manage this risk, new financial products, socalled longevity derivatives, may be needed, even though a first attempt to issue a longevi ..."
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Cited by 11 (0 self)
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For annuity providers, longevity risk, i.e. the risk that future mortality trends differ from those anticipated, constitutes an important risk factor. In order to manage this risk, new financial products, socalled longevity derivatives, may be needed, even though a first attempt to issue a longevity bond in 2004 was not successful. While different methods of how to price such securities have been proposed in recent literature, no consensus has been reached. This paper reviews, compares and comments on these different approaches. In particular, we use data from the United Kingdom to derive prices for the proposed first longevity bond and an alternative security design based on the different methods. ∗An earlier version of this paper entitled “Pricing Longevity Bonds Using Implied Survival Probabilities ” was presented at the
Maximal use of central differencing for HamiltonJacobiBellman PDEs in finance
 SIAM JOURNAL ON NUMERICAL ANALYSIS
, 2008
"... In order to ensure convergence to the viscosity solution, the standard method for discretizing HJB PDEs uses forward/backward differencing for the drift term. In this paper, we devise a monotone method which uses central weighting as much as possible. In order to solve the discretized algebraic eq ..."
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Cited by 9 (5 self)
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In order to ensure convergence to the viscosity solution, the standard method for discretizing HJB PDEs uses forward/backward differencing for the drift term. In this paper, we devise a monotone method which uses central weighting as much as possible. In order to solve the discretized algebraic equations, we have to maximize a possibly discontinuous objective function at each node. Nevertheless, convergence of the overall iteration can be guaranteed. Numerical experiments on two examples from the finance literature show higher rates of convergence for this approach compared to the use of forward/backward differencing only.
Canonical Valuation of MortalityLinked Securities
 Journal of Risk and Insurance
, 2010
"... A fundamental question in the study of mortalitylinked securities is how to place a value on them. This is still an open question, partly because there is a lack of liquidly traded longevity indexes or securities from which we can infer the market price of risk. This paper develops a framework for ..."
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Cited by 2 (0 self)
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A fundamental question in the study of mortalitylinked securities is how to place a value on them. This is still an open question, partly because there is a lack of liquidly traded longevity indexes or securities from which we can infer the market price of risk. This paper develops a framework for pricing mortalitylinked securities, on the basis of the theory of canonical valuation. This framework is largely nonparametric, helping us avoid parameter and model risk, which may be significant in other pricing methods. The framework is then applied to a mortalitylinked security, and the results are compared against those derived from the Wang transform and some modelbased methods.
Annuity Decisions with Systematic Longevity Risk,” Working paper
, 2009
"... In this paper we investigate the effect of systematic longevity risk, i.e., the risk arising from uncertain future survival probabilities, on the attractiveness of different types of annuities. We consider a lifecycle framework with expected utility where an individual faces both investment and lon ..."
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In this paper we investigate the effect of systematic longevity risk, i.e., the risk arising from uncertain future survival probabilities, on the attractiveness of different types of annuities. We consider a lifecycle framework with expected utility where an individual faces both investment and longevity risk. In contrast to existing literature we allow not only for idiosyncratic, but also for systematic longevity risk. When comparing the expected lifetime utility, conditional on the type of annuity which is purchased, we find for a 65year old male that (i) systematic longevity risk reduces the attractiveness of annuities, (ii) when an immediate annuity is purchased, the expected lifetime utility is decreasing in the postponement period, (iii) when in the future purchasing an immediate annuities, the effect of the evolution of the survival probabilities on the optimal fraction of annuitized wealth is large, and (iv) the optimal annuity to purchase at retirement is a deferred annuity which starts to pay after only a short deferral period. However, when the purchase of an annuity with the optimal deferral period is compared to the purchase of an immediate annuity at retirement date, the utility gain is negligibly small.
Stochastic Analysis of Insurance Products
, 2011
"... Dedicated to my family. ii ACKNOWLEDGEMENTS This thesis would not have been possible without the support and encouragement of my Ph.D. advisors Professor Virginia Young and Professor Haitao Li. They have given me enormous freedom to pursue my own interests while providing me the right amount of guid ..."
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Dedicated to my family. ii ACKNOWLEDGEMENTS This thesis would not have been possible without the support and encouragement of my Ph.D. advisors Professor Virginia Young and Professor Haitao Li. They have given me enormous freedom to pursue my own interests while providing me the right amount of guidance to ensure that my work contributes to the mainstream research in actuarial science and mathematical finance. I would like to express my gratitude to Professor Erhan Bayraktar, who taught me everything I know about probability and stochastic analysis. I wish to thank Professor Virginia Young and Professor Erhan Bayraktar for being my dissertation readers and Professor Haitao Li and Professor Kristen S. Moore for serving on my thesis committee. I owe many thanks to the Department of Mathematics, University of Michigan, for providing me the financial support during the last five years.
Longevity Risk in Fair Valuing Level 3 Assets in Securitised Portfolios*
"... Fair value accounting aims to establish a threelevel hierarchy that distinguishes (1) readily observable measurement inputs from (2) less readily observable measurement inputs and (3) unobservable measurement inputs. Level 3 longevity valued assets will pose unique valuation risks once securitised ..."
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Fair value accounting aims to establish a threelevel hierarchy that distinguishes (1) readily observable measurement inputs from (2) less readily observable measurement inputs and (3) unobservable measurement inputs. Level 3 longevity valued assets will pose unique valuation risks once securitised pools of these alternative asset classes come to market as investment vehicles for pension plans and individual retirement accounts. No uniform framework is available to assure consistent fair market valuation and transparency for investor decisionmaking. Applying existing international auditing standards and analytical procedures (IFRS 13) will offer a platform upon which fund managers, their auditors and actuaries can agree upon uniform valuation and presentation guidelines. Application of these quasigovernmental standards will bring future liquidity to otherwise illiquid capital market instruments. This paper presents a valuation methodology consistent with fair value accounting and auditing standards. The methodology incorporates the longevity predictive modelling of Stallard in a form that is compatible with Bayes Factor weighted average valuation techniques based on the study by Kass and Raftery. The methodology is applicable to fair valuation of life settlement portfolios where the combination of too few large death benefit policies and large variances in individual life expectancy estimates currently challenge accurate valuation and periodic revaluation.
Hedging Pure Endowments with Mortality Derivatives
, 2010
"... In recent years, a market for mortality derivatives began developing as a way to handle systematic mortality risk, which is inherent in life insurance and annuity contracts. Systematic mortality risk is due to the uncertain development of future mortality intensities, or hazard rates. In this paper ..."
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In recent years, a market for mortality derivatives began developing as a way to handle systematic mortality risk, which is inherent in life insurance and annuity contracts. Systematic mortality risk is due to the uncertain development of future mortality intensities, or hazard rates. In this paper, we develop a theory for pricing pure endowments when hedging with a mortality forward is allowed. The hazard rate associated with the pure endowment and the reference hazard rate for the mortality forward are correlated and are modeled by diffusion processes. We price the pure endowment by assuming that the issuing company hedges its contract with the mortality forward and requires compensation for the unhedgeable part of the mortality risk in the form of a prespecified instantaneous Sharpe ratio. The major result of this paper is that the value per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation under an equivalent martingale measure. Another important result is that hedging with the mortality forward may raise or lower the price of this pure endowment comparing to its price without hedging, as determined in Bayraktar et al. [2009]. The market price of the reference mortality risk and the correlation between the two portfolios jointly determine the cost of hedging. We demonstrate our results using numerical examples.