We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Utility Analysis and the Consumption Function: An Interpretation of Cross-Section Data

This study examines the link between corporate social responsibility (CSR) and bank debt. Our focus on banks exploits their specialized role as delegated monitors of the firm. Using a sample of 3996 loans to US firms, we find that firms with social responsibility concerns pay between 7 and 18 basis points more than firms that are more responsible. Lenders are more sensitive to CSR concerns in the absence of security. We document a mixed reaction to discretionary CSR investments. Low-quality borrowers that engage in discretionary CSR spending face higher loan spreads and shorter maturities, but lenders are indifferent to CSR investments by high-quality borrowers.

The impact of corporate social responsibility on the cost of bank loans

In this paper we consider the evolution of the post-60 mortality curve in the UK and its impact on the pricing of the risk associated with mortality improvements over time: so-called longevity risk. We introduce a two-factor stochastic model for the development of this curve through time. The flrst factor afiects mortality-rate dynamics at all ages in the same way, whereas the second factor afiects mortality-rate dynamics at higher ages much more than at lower ages. The paper then examines the pricing of longevity bonds with difierent terms to maturity referenced to difierent cohorts. We flnd that longevity risk over relativelyshort time horizons is very low, but at horizons in excess of 10 years it begins to pick up very rapidly. We propose a method for risk-adjusting the market price of a longevity bond which includes an allowance for parameter risk in the stochastic mortality model. We utilise the pricing information contained in the November 2004 European Investment Bank longevity bond to make inferences about the likely market prices of the risks in the model. Based on these, we investigate how future issues might be priced to ensure an absence of arbitrage between bonds with difierent characteristics.

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A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty

This paper presents new information on the expected present discounted value of payouts on individual life annuities. The annuity we examine is the single premium immediate life annuity, an insurance product that pays out a nominal level sum as long as the covered person lives, in exchange for an initial lump-sum premium. This annuity offers protection against the risk of someone outliving his saving, given uncertainty about longevity. For reasonable estimates of behavioral parameters, we calculate that individual annuities are currently priced so that retirees without bequest motives should find these policies of substantial value in configuring their portfolios to smooth retirement consumption. We also find that the expected present discounted value of payouts, relative to the initial cost of the annuity, has increased over the last decade. These findings bear on the policy debate regarding the role of individual choice and self-reliance in retirement planning.

New Evidence on the Money's Worth of Individual Annuities

Arithmetic Asian options are difficult to price and hedge as they do not have closed-form analytic solutions. The main theoretical reason for this difficulty is that the payoff de? pends on the finite sum of correlated lognormal variables, which is not lognormal and for which there is no recognizable probability density function. We use elementary techniques to derive the probability density function of the infinite sum of correlated lognormal ran? dom variables and show that it is reciprocal gamma distributed, under suitable parameter restrictions. A random variable is reciprocal gamma distributed if its inverse is gamma dis? tributed. We use this result to approximate the finite sum of correlated lognormal variables and then value arithmetic Asian options using the reciprocal gamma distribution as the state-price density function. We thus obtain a closed-form analytic expression for the value of an arithmetic Asian option, where the cumulative density function of the gamma distri? bution, G(d) in our formula, plays the exact same role as N(d) in the Black-Scholes/Merton formula. In addition to being theoretically justified and exact in the limit, we compare our method against other algorithms in the literature and show our method is quicker, at least as accurate, and, in our opinion, more intuitive and pedagogically appealing than any pre? viously published result, especially when applied to high yielding currency options.

Asian options, the sum of lognormals, and the reciprocal gamma distribution

Annuitization and asset allocation

Most US-based insurance companies offer holders of their tax-sheltered savings plans (VAs), the long-term option to annuitise their policy at a pre-determined rate over a pre-specified period of time. Currently, there is approximately one trillion dollars invested in such policies, with guaranteed annuitisation rates, in addition to any guaranteed minimum death benefit. The insurance company has essentially granted the policyholder an option on two underlying stochastic variables; future interest rates and future mortality rates. Although the (put) option on interest rates is obvious, the (put) option on mortality rates is not. Motivated by this product, this paper attempts to value (options on) mortality-contingent claims, by stochastically modelling the future hazard-plus-interest rate. Heuristically, we treat the underlying life annuity as a defaultable coupon-bearing bond, where the default occurs at the exogenous time of death. From an actuarial perspective, rather than considering the force of mortality (hazard rate) at time t for a person now age x , as a number μ x ( t ), we view it as a random variable forward rate
				 μ ˜ x ( t ) , whose expectation is the force of mortality in the classical sense (
				 μ x ( t ) = E [ μ ˜ x ( t ) ] ). Our main qualitative observation is that both mortality and interest rate risk can be hedged, and the option to annuitise can be priced by locating a replicating portfolio involving insurance, annuities and default-free bonds. We provide both a discrete and continuous-time pricing framework.

Mortality derivatives and the option to annuitise

Financial valuation OF GMWBs: We develop a variety of methods for assessing the cost and value of a very popular ‘rider’ available to North American investors on variable annuity (VA) policies called a Guaranteed Minimum Withdrawal Benefit (GMWB). The GMWB promises to return the entire initial investment, albeit spread over an extended period of time, regardless of subsequent market performance. First, we take a static approach that assumes individuals behave passively and holds the product to maturity. We show how the product can be decomposed into a Quanto Asian Put plus a generic term-certain annuity. At the other extreme of consumer behavior, the dynamic approach leads to an optimal stopping problem akin to pricing an American put option, albeit complicated by the non-traditional payment structure. Our main result is that the No Arbitrage hedging cost of a GMWB ranges from 73 to 160 basis points of assets. In contrast, most products in the market only charge 30–45 basis points. Although we suggest a number of behavioral reasons for the apparent under-pricing of this feature in a typically overpriced VA market, we conclude by arguing that current pricing is not sustainable and that GMWB fees will eventually have to increase or product design will have to change in order to avoid blatant arbitrage opportunities.

Financial valuation of guaranteed minimum withdrawal benefits

The authors use risk-neutral option pricing theory to value the guaranteed minimum death benefit (GMDB) in variable annuities (VAs) and some recently introduced mutual funds. A variety of death benefits, such as return-of-premium, rising floors, and "ratches," are analyzed. Specifically, the authors compute the fair insurance risk fee, charged to assets, that funds the embedded option. The authors derive analytic option prices for a simplified exponential mortality model and robust numerical estimates in the case of a properly calibrated Gompertz model. The authors label this contingent claim a Titanic option because its payoff structure is in between European and American style but is triggered by death. The authors' main objective is to compare theoretical estimates against a cross-section of insurance risk charges, as reported by Morningstar, Inc. The authors' main conclusion is that a simple return-of-premium death benefit is worth between one and ten basis points, depending on gender, purchase age, and asset volatility. In contrast, the median Mortality and Expense risk charge for return-of-premium variable annuities is 115 basis points. Presumably, the remaining markup can be attributed to profits, model imperfections, or, more cynically, to an implicit payment for the tax-deferral privilege. INTRODUCTION In the United States, a variable annuity (VA) policy is a collection of investment subaccounts that are wrapped with a life insurance contract. The raison d'etre of the VA is the convenient deferral of all income and capital gains taxation until the funds are withdrawn or annuitized at retirement. The tax deferral on investment gains--of increasing long-term benefit--make VAs a popular alternative to fully taxable mutual funds, even though they involve higher management fees. In fact, despite the recent reduction in the long-term capital gains tax rate, sales of variable annuities amounted to $121 billion during 1999 and $100 billion during 1998. Interestingly, an estimated 55 percent of these variable annuity purchases were conducted within qualified, i.e., tax-sheltered, IRAs, 401(k)s, 403(b)s, and 457 retirement savings plans for which the tax deferral is redundant. [1] However, in addition to the tax deferral--and in contrast to most mutual funds--the variable annuity provides a unique death benefit, possibly justifying its sale within tax-sheltered plans. The variable annuity death benefit stipulates that at least the original investment will be returned to the estate or beneficiary of the policy, regardless of the performance of the underlying assets in the account. This guaranty will be in-the-money if the VA policyholder (annuitant) dies during a "bear" market and at the same time the investment is showing a paper loss. Recently, some innovative mutual funds have joined with insurance companies to offer a similar death benefit on their investments. Presumably, this innovation has been partially driven by an attempt to compete with the VA market and its implicit investment protection. Of course, the tax deferral can only be provided by VAs. The coincidence of death and market downturn, as unlikely as it may seem, is the impetus for this study. Specifically, the authors are interested in the economic value and fair cost of this death benefit. Before embarking on this endeavor, it is important to mention that not all VA contracts and death-protected mutual funds (DPMF) are created equal. Many innovative insurance companies sell VA products with death benefits that are more "generous" than others. By generous the authors mean that the guaranteed minimum death benefit is higher than a simple return of premium plus minimal interest. For example, some products guarantee a more lucrative death benefit that locks in accrued investment gains during the life of the contract. As such, the VA and death-protected mutual fund market is not homogenous, and one must be careful with any generalizations about these products. …

Moshe A. Milevsky

Papers

Asian options, the sum of lognormals, and the reciprocal gamma distribution

Annuitization and asset allocation

Mortality derivatives and the option to annuitise

Financial valuation of guaranteed minimum withdrawal benefits

The Titanic Option: Valuation of the Guaranteed Minimum Death Benefit in Variable Annuities and Mutual Funds