Stable Laws and the Present Value of Fixed Cash Flows

Abstract

In this paper, the authors consider the present value of a series of fixed cash flows under stochastic interest rates. To model these interest rates, they don’t use the common lognormal model, but stable laws, which better fit in with reality. Their main intention is to derive a result for the distribution function of such a present value. However, due to the dependencies between successive discounted payments, the calculation of an exact analytical distribution is impossible. Therefore, use is made of the methodology of comonotonic random variables and the convex ordering of risks, introduced by the same authors in some previous papers.

The present paper starts with a brief overview of properties and features of stable laws, and of the possible application of the concept of convex ordering to sums of risks, which is also the situation for a present value of future payments. Afterwards, the authors show how, for the present value under investigation, an approximation in the form of a convex upper bound can be derived. This upper bound has an easier structure than the original present value, and they derive elegant calculation formulas for the distribution of this bound. Finally, they provide some numerical examples that illustrate the precision of the approximation. Due to the design of the present value and the construction of the upper bound, these illustrations show great promise concerning the accuracy of the approximation.     

Pricing bounds and approximations for discrete arithmetic Asian options under time-changed Lévy processes

We derive efficient and accurate analytical pricing bounds and approximations for discrete arithmetic Asian options under time-changed Lévy processes. By extending the conditioning variable approach, we derive the lower bound on the Asian option price and construct an upper bound based on the sharp lower bound. We also consider the general partially exact and bounded (PEB) approximations, which include the sharp lower bound and partially conditional moment matching approximation as special cases. The PEB approximations are known to lie between a sharp lower bound and an upper bound. Our numerical tests show that the PEB approximations to discrete arithmetic Asian option prices can produce highly accurate approximations when compared to other approximation methods. Our proposed approximation methods can be readily applied to pricing Asian options under most common types of underlying asset price processes, like the Heston stochastic volatility model nested in the class of time-changed Lévy processes with the leverage effect.     

Pricing Discrete Dynamic Fund Protections

Abstract

The authors investigate the pricing of discretely monitored dynamic fund protections when the fund price follows a lognormal process or a constant elasticity of variance (CEV) process. A backward recursive pricing formula is derived. By employing a numerical technique that combines function approximation and numerical quadrature, the authors demonstrate how to complete each recursion level efficiently. Numerical experiments show that the results compare favorably with those obtained by other pricing methods.     

Interval Estimation of Actuarial Risk Measures

Abstract

This article investigates performance of interval estimators of various actuarial risk measures. We consider the following risk measures: proportional hazards transform (PHT), Wang transform (WT), value-at-risk (VaR), and conditional tail expectation (CTE). Confidence intervals for these measures are constructed by applying nonparametric approaches (empirical and bootstrap), the strict parametric approach (based on the maximum likelihood estimators), and robust parametric procedures (based on trimmed means).

Using Monte Carlo simulations, we compare the average lengths and proportions of coverage (of the true measure) of the intervals under two data-generating scenarios: “clean” data and “contaminated” data. In the “clean” case, data sets are generated by the following (similar shape) parametric families: exponential, Pareto, and lognormal. Parameters of these distributions are selected so that all three families are equally risky with respect to a fixed risk measure. In the “contaminated” case, the “clean” data sets from these distributions are mixed with a small fraction of unusual observations (outliers). It is found that approximate knowledge of the underlying distribution combined with a sufficiently robust estimator (designed for that distribution) yields intervals with satisfactory performance under both scenarios.     

Lognormal Approximation of Complex Path-Dependent Pension Scheme Payoffs

Abstract

This paper analyzes an explicit return smoothing mechanism which has recently been introduced as part of a new type of pension savings contract that has been offered by Danish life insurers. We establish the payoff function implied by the return smoothing mechanism and show that its probabilistic properties are accurately approximated by a suitably adapted lognormal distribution. The quality of the lognormal approximation is explored via a range of simulation-based numerical experiments, and we point to several other potential practical applications of the paper's theoretical results.     

Explaining the volatility smile: non-parametric versus parametric option models

We employ a “non-parametric” pricing approach of European options to explain the volatility smile. In contrast to “parametric” models that assume that the underlying state variable(s) follows a stochastic process that adheres to a strict functional form, “non-parametric” models directly fit the end distribution of the underlying state variable(s) with statistical distributions that are not represented by parametric functions. We derive an approximation formula which prices S&P 500 index options in closed form which corresponds to the lower bound recently proposed by Lin et al. (Rev Quant Financ Account 38(1):109–129, 2012). Our model yields option prices that are more consistent with the data than the option prices that are generated by several widely used models. Although a quantitative comparison with other non-parametric models is more difficult, there are indications that our model is also more consistent with the data than these models.     

ON THE RATIONALITY OF COMMON STOCK RETURN VOLATILITY

This paper derives the relationship between the population unconditional variance of common stock returns and the variance of expected returns conditional on a well-specified information set. As a consequence, a lower bound is obtained for the variance of common stock returns. The sample counterpart of this bound is then empirically tested against the sample variance of returns. The paper's main conclusion can be stated as follows: the observed volatility of real (inflation-adjusted) common stock returns is not “irrationally” large. The paper admits of this conclusion because the point estimate of the lower-bound variance derived in this model is actually larger than the point estimate of common stock return volatility. However, since these point estimates are found to have a statistically insignificant difference, equality of the two variances cannot be ruled out. Hence, “rationality” of common stock returns—as implied by a utility-based valuation conditional on a specified information set—cannot be rejected.     

Self-Annuitization and Ruin in Retirement

Abstract

At retirement, most individuals face a choice between voluntary annuitization and discretionary management of assets with systematic withdrawals for consumption purposes. Annuitization–buying a life annuity from an insurance company–assures a lifelong consumption stream that cannot be outlived, but it is at the expense of a complete loss of liquidity. On the other hand, discretionary management and consumption from assets–self-annuitization–preserves flexibility but with the distinct risk that a constant standard of living will not be maintainable.

In this paper we compute the lifetime and eventual probability of ruin (PoR) for an individual who wishes to consume a fixed periodic amount–a self-constructed annuity–from an initial endowment invested in a portfolio earning a stochastic (lognormal) rate of return. The lifetime PoR is the probability that net wealth will hit zero prior to a stochastic date of death. The eventual PoR is the probability that net wealth will ever hit zero for an infinitely lived individual.

We demonstrate that the probability of ruin can be represented as the probability that the stochastic present value (SPV) of consumption is greater than the initial investable wealth. The lifetime and eventual probabilities of ruin are then obtained by evaluating one minus the cumulative density function of the SPV at the initial wealth level. In that eventual case, we offer a precise analytical solution because the SPV is known to be a reciprocal gamma distribution. For the lifetime case, using the Gompertz law of mortality, we provide two approximations. Both involve “moment matching” techniques that are motivated by results in Arithmetic Asian option pricing theory. We verify the accuracy of these approximations using Monte Carlo simulations. Finally, a numerical case study is provided using Canadian mortality and capital market parameters. It appears that the lifetime probability of ruin–for a consumption rate that is equal to the life annuity payout–is at its lowest with a well-diversified portfolio.     

Stochastic Life Annuities

Abstract

This paper gives analytic approximations for the distribution of a stochastic life annuity. It is assumed that returns follow a geometric Brownian motion. The distribution of the stochastic annuity may be used to answer questions such as “What is the probability that an amount F is sufficient to fund a pension with annual amount y to a pensioner aged x?” The main idea is to approximate the future lifetime distribution with a combination of exponentials, and then apply a known formula (due to Marc Yor) related to the integral of geometric Brownian motion. The approximations are very accurate in the cases studied.     

Non-monotonic pricing kernel and an extended class of mixture of distributions for option pricing

We derive closed form European option pricing formulae under the general equilibrium framework for underlying assets that have an \(N\) -mixture of transformed normal distributions. The component distributions need not belong to the same class but must all be transformed normal. An important implication of our results is that the mixture of distributions is consistent with a “what appears to be abnormal” non-monotonic (asset specific) pricing kernel for the S&P 500 and that the representative agent has a “logical” monotonic decreasing marginal utility. We show that a mixture of two lognormal distributions is sufficient to produce this result and also implied volatility smiles of a wide variety of shapes.     

Studies in risk theory with numerical illustrations concerning distribution functions and stop loss premiums. Part I

Abstract

The aim of this paper is to analyse two functions that are of general interest in the collective risk theory, namely F, the distribution function of the total amount of claims, and II, the Stop Loss premium. Section 2 presents certain basic formulae. Sections 17-18 present five claim distributions. Corresponding to these claim distributions, the functions F and II were calculated under various assumptions as to the distribution of the number of claims. These calculations were performed on an electronic computer and the numerical method used for this purpose is presented in sections 9, 19 and 20 under the name of the C-method which method has the advantage of furnishing upper and lower limits of the quantities under estimation. The means of these limits, in the following regarded as the “exact” results, are given in Tables 4-20. Sections 11-16 present certain approximation methods. The N-method of section 11 is an Edgeworth expansion, while the G-method given in section 12 is an approximation by a Pearson type III curve. The methods presented in sections 13-16, and denoted AI-A4, are all applications and modifications of the Esscher method. These approximation methods have been applied for the calculation of F and II in the cases mentioned above in which “exact” results were obtained. The results are given in Tables 4-20. The object of this investigation was to obtain information as to the precision of the approximation methods in question, and to compare their relative merits. These results arc discussed in sections 21-24.     

A Welfare-Based Analysis of International Monetary Policy Spillovers at the Zero Lower Bound

We study the welfare effects of international monetary policy spillovers at the zero lower bound, focusing on the effects of forward guidance in a large economy (“foreign”) on its smaller trading partner (“home”) in scenarios motivated by the global financial crisis. We find that foreign forward guidance has an overall prosper-thy-neighbor effect on the home economy as long as it increases foreign welfare. This finding holds under alternative assumptions about exchange rate pass-through or the degree of international financial integration. However, foreign forward guidance may worsen the trade-off between the stabilization objectives of home monetary policy.     

Target volatility option pricing in the lognormal fractional SABR model

We examine in this article the pricing of target volatility options in the lognormal fractional SABR model. A decomposition formula of Itô's calculus yields an approximation formula for the price of a target volatility option in small time by the technique of freezing the coefficient. A decomposition formula in terms of Malliavin derivatives is also provided. Alternatively, we also derive closed form expressions for a small volatility of volatility expansion of the price of a target volatility option. Numerical experiments show the accuracy of the approximations over a reasonably wide range of parameters.     

An Elementary Upper Bound for the Loading of a Stoploss Cover

Abstract

The famous reinsurance treaty, called stoploss cover, is reconsidered. By applying the well-known Cauchy-Schwarz inequality an elementary bound is derived for the variance of the claims amount taken by the reinsurer. This gives an upper bound for the loading of the treaty in case of using the standard deviation or variance principle.     

Optimal Reinsurance and Investment for a Jump Diffusion Risk Process under the CEV Model

Abstract

We consider an optimal reinsurance-investment problem of an insurer whose surplus process follows a jump-diffusion model. In our model the insurer transfers part of the risk due to insurance claims via a proportional reinsurance and invests the surplus in a “simplified” financial market consisting of a risk-free asset and a risky asset. The dynamics of the risky asset are governed by a constant elasticity of variance model to incorporate conditional heteroscedasticity. The objective of the insurer is to choose an optimal reinsurance-investment strategy so as to maximize the expected exponential utility of terminal wealth. We investigate the problem using the Hamilton-Jacobi-Bellman dynamic programming approach. Explicit forms for the optimal reinsuranceinvestment strategy and the corresponding value function are obtained. Numerical examples are provided to illustrate how the optimal investment-reinsurance policy changes when the model parameters vary.     

The Dynamic Extreme Co-Movement between Chinese Stock Market and Global Stock Markets

ABSTRACT

We use time-varying Symmetrized Joe-Clayton Copula model to study the extreme co-movement (boom or crash together) between the Chinese stock market and major stock markets in the world from 2007 to 2017, including developed markets and stock markets on “Belt and Road Initiative” (hereafter B.R.I.). We find that the extreme co-movement probability between Chinese market and “Belt and Road Initiative” markets is higher than developed markets at both tails. Then we study important “real” and “non-fundamental” factors affecting the excess co-movement probability, including bilateral trade openness, financial integration, and economic policy uncertainty. The results of panel regression analysis show that: the bilateral financial integration has significant effects over the lower tail dependence between Chinese and developed markets, but does not affect the extreme co-movement between Chinese and B.R.I. markets. And the bilateral trade openness is an important factor for the extreme co-movement at both tail between Chinese and global markets. The economic policy uncertainty index, especially China’s economic policy uncertainty, plays a key role in the extreme co-movement between Chinese and developed markets at both tails. However, it has sizable effects only at the upper tail co-movement between Chinese and B.R.I. markets.     

Comment on “A note on the returns from minimum variance investing”

In Scherer (2011) the author, analyzing minimum variance investing, shows that “the minimum variance tends to hold low beta and low residual risk stocks.”The conclusion that low residual risk assets and assets with a low β (below one) create a positive portfolio weight in the minimum variance portfolio does not prove the paper's “conjecture that the minimum variance portfolio is likely to pick up low beta and low residual risk stocks.”     

Analytical approximations for the critical stock prices of American options: a performance comparison

Many efficient and accurate analytical methods for pricing American options now exist. However, while they can produce accurate option prices, they often do not give accurate critical stock prices. In this paper, we propose two new analytical approximations for American options based on the quadratic approximation. We compare our methods with existing analytical methods including the quadratic approximations in Barone-Adesi and Whaley (J Finance 42:301–320, 1987) and Barone-Adesi and Elliott (Stoch Anal Appl 9(2):115–131, 1991), the lower bound approximation in Broadie and Detemple (Rev Financial Stud 9:1211–1250, 1996), the tangent approximation in Bunch and Johnson (J Finance 55(5):2333–2356, 2000), the Laplace inversion method in Zhu (Int J Theor Appl Finance 9(7):1141–1177, 2006b), and the interpolation method in Li (Working paper, 2008). Both of our methods give much more accurate critical stock prices than all the existing methods above.     

Stein’s method and zero bias transformation for CDO tranche pricing

We propose an original approximation method, which is based on Stein’s method and the zero bias transformation, to calculate CDO tranches in a general factor framework. We establish first-order correction terms for the Gaussian and the Poisson approximations respectively and we estimate the approximation errors. The application to the CDO pricing consists of combining the two approximations. This work is partially supported by Fondation de risque.     

A comparison principle between rough and non-rough Heston models—with applications to the volatility surface

We present a number of related comparison results, which allow one to compare moment explosion times, moment generating functions and critical moments between rough and non-rough Heston models of stochastic volatility. All results are based on a comparison principle for certain non-linear Volterra integral equations. Our upper bound for the moment explosion time is different from the bound introduced by Gerhold, Gerstenecker and Pinter [Moment explosions in the rough Heston model. Decisions in Economics and Finance, 2019, 42, 575–608] and tighter for typical parameter values. The results can be directly transferred to a comparison principle for the asymptotic slope of implied variance between rough and non-rough Heston models. This principle shows that the ratio of implied variance slopes in the rough versus non-rough Heston model increases at least with power-law behavior for small maturities.