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Energy quanto options are risk management tools that have a payoff similar to the product of the payoffs of two options, each written on an energy-related underlying. These options, as opposed to standardized contracts that only account for price risk, are designed to manage both volumetric and price risk in energy markets. Since the use of such options enables actors in the energy market also to hedge against production volume risk, they are becoming very popular. This paper considers the valuation of such an option on futures when the underlying futures prices are governed by Markov-modulated additive processes, which have independent but non-stationary increments within each regime. We derive a valuation formula by using the Fast Fourier Transform (FFT) technique under the assumption that the joint characteristic function of the log-futures prices is known analytically. We study this approximation under different regime-switching models. Several numerical case studies illustrate that our FFT-based valuation has a high precision and is much faster than Monte Carlo estimates.

The global reforms to public pension schemes over the last thirty years have progressively reduced individuals’ post-retirement social security income. In order to compensate for this, individuals join pension funds and individual plans to increase their wealth at retirement. These types of fully funded plans generally give individuals the opportunity to withdraw the capital accumulated into their scheme or to convert it into an annuity. In this paper, we analyse individuals’ post-retirement choices to allocate the wealth at retirement between consumption, risk-free investments and a life annuity. We develop a discrete time optimisation model, in a deterministic framework, with a constant relative risk aversion (CRRA) utility function. We study the effect of a bequest motive and the annuity rate used by the insurer on the optimal choice. Several numerical applications are presented to illustrate the optimal annuitisation decision results and the optimal consumption paths.

In this article, we focus upon a family of matrix valued stochastic processes and study the problem of determining the smallest time such that their Laplace transforms become infinite. In particular, we concentrate upon the class of Wishart processes, which have proved to be very useful in different applications by their ability in describing non-trivial dependence. Thanks to this remarkable property we are able to explain the behavior of the explosion times for the Laplace transforms of the Wishart process and its time integral in terms of the relative importance of the involved factors and their correlations., SCOPUS: ar.j, info:eu-repo/semantics/published

This paper studies the pricing of some multivariate European options, namely Exchange options and Quanto options, when the risky assets involved are modelled by Markov-Modulated Lvy Processes (MMLPs). Pricing formulae are based upon the characteristic exponents by using the well known FFT methodology. We study these pricing issues both under a risk neutral martingale measure and the historical measure. The dependence between the assets components is incorporated in the joint characteristic function of the MMLPs. As an example, we concentrate upon a regime-switching version of the model of Ballotta etal. (2015) in which the dependence structure is introduced in a flexible way. Several numerical examples are provided to illustrate our results.

Financial products are priced using risk-neutral expectations justified by hedging portfolios that (as accurate as possible) match the product’s payoff. In insurance, premium calculations are based on a real-world best-estimate value plus a risk premium. The insurance risk premium is typically reduced by pooling of (in the best case) independent contracts. As hybrid life insurance contracts depend on both financial and insurance risks, their valuation requires a hybrid valuation principle that combines the two concepts of financial and actuarial valuation. The aim of this paper is to present a novel three-step projection algorithm to valuate hybrid contracts by decomposing their payoff in three parts: a financial, hedgeable part, a diversifiable actuarial part, and a residual part that is neither hedgeable nor diversifiable. The first two parts of the resulting premium are directly linked to their corresponding hedging and diversification strategies, respectively. The method allows for a separate treatment of unsystematic, diversifiable mortality risk and systematic, aggregate mortality risk related to, for example, epidemics or population-wide improvements in life expectancy. We illustrate our method in the case of CAT bonds and a pure endowment insurance contract with profit and compare the three-step method to alternative valuation operators suggested in the literature.

A way to model the clustering of jumps in asset prices consists in combining a diffusion process with a jump Hawkes process in the dynamics of the asset prices. This article proposes a new alternative model based on regime switching processes, referred to as a self-exciting switching jump diffusion (SESJD) model. In this model, jumps in the asset prices are synchronized with changes of states of a hidden Markov chain. The matrix of transition probabilities of this chain is designed in order to approximate the dynamics of a Hawkes process. This model presents several advantages compared to other jump clustering models. Firstly, the SESJD model is easy to fit to time series since estimation can be performed with an enhanced Hamilton filter. Secondly, the model explains various forms of option volatility smiles. Thirdly, several properties about the hitting times of the SESJD model can be inferred by using a fluid embedding technique, which leads to closed form expressions for some financial derivatives, like perpetual binary options., SCOPUS: ar.j, info:eu-repo/semantics/published

Financial products are priced using risk-neutral expectations justified by hedging portfolios that (as accurate as possible) match the product’s payoff. In insurance, premium calculations are based on a real-world best-estimate value plus a risk premium. The insurance risk premium is typically reduced by pooling of (in the best case) independent contracts. As hybrid life insurance contracts depend on both financial and insurance risks, their valuation requires a hybrid valuation principle that combines the two concepts of financial and actuarial valuation. The aim of this paper is to present a novel three-step projection algorithm to valuate hybrid contracts by decomposing their payoff in three parts: a financial, hedgeable part, a diversifiable actuarial part, and a residual part that is neither hedgeable nor diversifiable. The first two parts of the resulting premium are directly linked to their corresponding hedging and diversification strategies, respectively. The method allows for a separate treatment of unsystematic, diversifiable mortality risk and systematic, aggregate mortality risk related to, for example, epidemics or population-wide improvements in life expectancy. We illustrate our method in the case of CAT bonds and a pure endowment insurance contract with profit and compare the three-step method to alternative valuation operators suggested in the literature.

We propose a new approach for bivariate financial time series modelling which allows for mutual excitation between shocks. Jumps are triggered by changes of regime of a hidden Markov chain whose matrix of transition probabilities is constructed in order to approximate a bivariate Hawkes process. This model, called the Bivariate Mutually-Excited Switching Jump Diffusion (BMESJD) presents several interesting features. Firstly, compared to alternative approaches for modelling the contagion between jumps, the calibration is easier and performed with a modified Hamilton’s filter. Secondly, the BMESJD allows for simultaneous jumps when markets are highly stressed. Thirdly, a family of equivalent probability measures under which the BMESJD dynamics are preserved, is well identified. Finally, the BMESJD is a continuous time process that is well adapted for pricing options with two underlying assets., info:eu-repo/semantics/published

In this paper, we address the issue of determining the optimal contribution rate of a defined benefit pension fund. The affiliate's mortality is modelled by a jump process and the benefits paid at retirement are function of the evolution of future salaries. Assets of the fund are invested in cash, stocks, and a rolling bond. Interest rates are driven by a Vasicek model. The objective is to minimize both the quadratic spread between the contribution rate and the normal cost, and the quadratic spread between the terminal wealth and the mathematical reserve required to cover benefits. The optimization is done under a budget constraint that guarantees the actuarial equilibrium between the current asset and future contributions and benefits. The method of resolution is based on the Cox–Huang approach and on dynamic programming.

In this paper we propose some moment matching pricing methods for European-style discrete arithmetic Asian basket options in a Black & Scholes framework. We generalize the approach of [M. Curran, Valuing Asian and portfolio by conditioning on the geometric mean price, Management Science 40 (1994) 1705-1711] and of [G. Deelstra, J. Liinev, M. Vanmaele, Pricing of arithmetic basket options by conditioning, Insurance: Mathematics & Economics 34 (2004) 55-57] in several ways. We create a framework that allows for a whole class of conditioning random variables which are normally distributed. We moment match not only with a lognormal random variable but also with a log-extended-skew-normal random variable. We also improve the bounds of [G. Deelstra, I. Diallo, M. Vanmaele, Bounds for Asian basket options, Journal of Computational and Applied Mathematics 218 (2008) 215-228]. Numerical results are included and on the basis of our numerical tests, we explain which method we recommend depending on moneyness and time-to-maturity.

In this paper, we investigate static super-replicating strategies for European-type call options written on a weighted sum of asset prices. This class of exotic options includes Asian options and basket options among others. We assume that there exists a market where the plain vanilla options on the different assets are traded and hence their prices can be observed in the market. Both the infinite market case (where prices of the plain vanilla options are available for all strikes) and the finite market case (where only a finite number of plain vanilla option prices are observed) are considered. We prove that the finite market case converges to the infinite market case when the number of observed plain vanilla option prices tends to infinity. We show how to construct a portfolio consisting of the plain vanilla options on the different assets, whose pay-off super-replicates the pay-off of the exotic option. As a consequence, the price of the super-replicating portfolio is an upper bound for the price of the exotic option. The super-hedging strategy is model-free in the sense that it is expressed in terms of the observed option prices on the individual assets, which can be e.g. dividend paying stocks with no explicit dividend process known. This paper is a generalization of the work of Simon et al. [Simon, S., Goovaerts, M., Dhaene, J., 2000. An easy computable upper bound for the price of an arithmetic Asian option. Insurance Math. Econom. 26 (2-3), 175-184] who considered this problem for Asian options in the infinite market case. Laurence and Wang [Laurence, P., Wang, T.H., 2004. What's a basket worth? Risk Mag. 17, 73-77] and Hobson et al. [Hobson, D., Laurence, P., Wang, T.H., 2005. Static-arbitrage upper bounds for the prices of basket options. Quant. Fin. 5 (4), 329-342] considered this problem for basket options, in the infinite as well as in the finite market case. As opposed to Hobson et al. [Hobson, D., Laurence, P., Wang, T.H., 2005. Static-arbitrage upper bounds for the prices of basket options. Quant. Fin. 5 (4), 329-342] who use Lagrange optimization techniques, the proofs in this paper are based on the theory of integral stochastic orders and on the theory of comonotonic risks. © 2008 Elsevier B.V. All rights reserved. ispartof: Insurance Mathematics & Economics vol:42 issue:3 pages:1067-1085 ispartof: location:Samos, Karlovassi, Greece status: published

In this paper we investigate the consequences on the pricing of insurance contingent claims when we relax the typical independence assumption made in the actuarial literature between mortality risk and interest rate risk. Starting from the Gaussian approach of Liu et al. (2014), we consider some multifactor models for the mortality and interest rates based on more general affine models which remain positive and we derive pricing formulas for insurance contracts like Guaranteed Annuity Options (GAOs). In a Wishart affine model, which allows for a non-trivial dependence between the mortality and the interest rates, we go far beyond the results found in the Gaussian case by Liu et al. (2014), where the value of these insurance contracts can be explained only in terms of the initial pairwise linear correlation.

In this paper we apply the multivariate construction for Levy processes introduced by Ballotta and Bonfiglioli (2014) to propose an integrated model for the joint dynamics of FX exchange rates and asset prices. We show that the proposed construction is consistent in terms of symmetries with respect to inversion and triangulation, and provides an insight into the quanto adjustment showing that this is affected by higher order cumulants of the pure jump part of the systematic risk factor. Using the Esscher transform, we relate Quanto options to vanilla call and put options, which allows for a fast calibration method to the vanilla and the Quanto market. As Quanto products offer significant exposure to the correlation between exchange rates and asset prices, they allow access to a market implied measure of this correlation. By means of a joint calibration exercise to the CME USD denominated Quanto futures on the Nikkei 225 index and both the Nikkei 225 and USDJPY market implied volatilities, we illustrate this approach for the case in which the driving process is assumed to be a Variance Gamma process.

In this paper, we investigate an optimization problem related to super-replicating strategies for European-type call options written on a weighted sum of asset prices, following the initial approach in Chen et al. (2008). Three issues are investigated. The first issue is the (non-)uniqueness of the optimal solution. The second issue is the generalization to an optimization problem where the weights may be random. This theory is then applied to static super-replication strategies for some exotic options in a stochastic interest rate setting. The third issue is the study of the co-existence of the comonotonicity property and the martingale property., SCOPUS: ar.j, info:eu-repo/semantics/published

We extend the Cox-Ingersoll-Ross (1985) model of the short interest rate by assuming a stochastic reversion level, which better reflects the time dependence caused by the cyclical nature of the economy or by expectations concerning the future impact of monetary policies. In this framework, we have studied the convergence of the long-term return by using the theory of generalised Bessel-square processes. We emphasize the applications of the convergence results. A limit theorem proves evidence of the use of a Brownian motion with drift instead of the integral . For practice, however, this approximation turns out to be only appropriate when there are no explicit formulae and calculations are very time-consuming.

For applications in finance, we study the stochastic differential equation dXs = (2βXs + δs) ds + g(Xs) dBs with β a negative real number, g a continuous function vanishing at zero which satisfies a Holder condition and δ a measurable and adapted stochastic process such that ∫t0 δu du < ∞ a.e. for all t ∈ ℝ+ and which may have a random correlation with the process X itself. In this paper, we concentrate on the Euler discretization scheme for such processes and we study the convergence in L1-supnorm and in ℋ1-norm towards the solution of the stochastic differential equation with stochastic drift term. We also check the order of strong convergence. © 1998 John Wiley & Sons, Ltd.

This chapter first reminds the reader of insurance premium calculation principles and of mathematical tools enabling portfolios to be ordered according to their risk levels—“orders” on risks. Next, it presents the most prevalent model of the process of total claim amounts generated by a portfolio, namely “the collective model”. This first chapter ends with the main results of ruin theory.

We consider non mean-reverting Wishart processes and we study the problem of determining the smallest time such that the Laplace transforms of the process and its integral become infinite. Thanks to the remarkable property of (affine) Wishart processes to reproduce non-trivial dependence among the positive factors, we are able to explain the behavior of the explosion times in terms of the relative importance of the involved factors and their correlations. In this way, we go far beyond the known results that can be recovered in the classical affine framework.

Optimal timing for annuitization is developed along three approaches. Firstly, the mutual fund in which the individual invests before annuitization is modeled by a jump diffusion process. Secondly, instead of maximizing an economic utility, the stopping time is used to maximize the market value of future cash-flows. Thirdly, a solution is proposed in terms of Expected Present Value operators: this shows that the non-annuitization (or continuation) region is either delimited by a lower or upper boundary, in the domain time-assets return. The necessary conditions are given under which these mutually exclusive boundaries exist. Further, a method is proposed to compute the probability of annuitization. Finally, a case study is presented where the mutual fund is fitted to the S&P500 and mortality is modeled by a Gompertz Makeham law with several real scenarios being discussed. © 2014 Elsevier B.V., SCOPUS: ar.j, info:eu-repo/semantics/published

Albrecheret al. (Albrecher, H., Mayer Ph., Schoutens, W. (2008) General lower bounds for arithmetic Asian option prices.Applied Mathematical Finance,15, 123–149) have proposed model-independent lower bounds for arithmetic Asian options. In this paper we provide an alternative and more elementary derivation of their results. We use the bounds as control variates to develop a simple Monte Carlo method for pricing contracts with Asian-style features. The conditioning idea that is inherent in our approach also inspires us to propose a new semi-analytic pricing approach. We compare both approaches and conclude that these both have their merits and are useful in practice. In particular, we point out that our newly proposed Monte Carlo method allows to deal with Asian-style products that appear in insurance (e.g., unit-linked contracts) in a very efficient way, and outperforms other known Monte Carlo methods that are based on control variates.

This chapter describes the institutional context of reinsurance. It first strives to clarify the legal nature of reinsurance transactions. It next describes the structure of the reinsurance market, and then the different legal and technical features of reinsurance contracts, called reinsurance “treaties” by practitioners. Indeed, the business of reinsurance, though only lightly regulated, takes place within a set of customary rules, making it thereby easier to describe and understand. In particular, traditional reinsurance treaties fall into a limited number of broad categories.

Reinsurance is an important production factor of non-life insurance. The efficiency and the capacity of the reinsurance market directly regulate those of insurance markets. The purpose of this book is to provide a concise introduction to risk theory, as well as to its main application procedures to reinsurance. The first part of the book covers risk theory. It presents the most prevalent model of ruin theory, as well as a discussion on insurance premium calculation principles and the mathematical tools that enable portfolios to be ordered according to their risk levels. The second part describes the institutional context of reinsurance. It first strives to clarify the legal nature of reinsurance transactions. It describes the structure of the reinsurance market and then the different legal and technical features of reinsurance contracts, known as reinsurance ‘treaties’ by practitioners. The third part creates a link between the theories presented in the first part and the practice described in the second one. Indeed, it sets out, mostly through examples, some methods for pricing and optimizing reinsurance. The authors aim is to apply the formalism presented in the first part to the institutional framework given in the second part. It is reassuring to find such a relationship between approaches seemingly abstract and solutions adopted by practitioners. Risk Theory and Reinsurance is mainly aimed at master's students in actuarial science but will also be useful for practitioners wishing to revive their knowledge of risk theory or to quickly learn about the main mechanisms of reinsurance. (Publisher's abstract)

We study the local volatility function in the Foreign Exchange market where both domestic and foreign interest rates are stochastic. This model is suitable to price long-dated FX derivatives. We derive the local volatility function and obtain several results that can be used for the calibration of this local volatility on the FX option's market. Then, we study two different extensions which allow the volatility of the spot FX rate to have stochastic behavior. First, we introduce a stochastic structure on the local volatility surface and show that local volatilities are risk-adjusted expectations of future instantaneous volatilities. The second extension is obtained by multiplying the FX spot local volatility with a stochastic volatility. Thanks to the Gyongy's mimicking property, we obtain a calibration method for the local volatility associated to this model.

Using an extension of the Cox-Ingersoll-Ross [1] stochastic model of the short interest rate r, we study the convergence in law of the long-term return in order to make some approximations. We use the theory of Bessel processes and observe the convergence in law of the sequence with X a generalized Besselsquare process with drift with stochastic reversion level. By Aldous' criterion, we are able to prove that this sequence converges in law to a Brownian motion

In this paper, we study the price of Variable Annuity Guarantees, particularly those of Guaranteed Annuity Options (GAO) and Guaranteed Minimum Income Benefit (GMIB), in the settings of a derivative pricing model where the underlying spot (the fund) is locally governed by a geometric Brownian motion with local volatility, while interest rates follow a Hull–White one-factor Gaussian model. Notwithstanding the fact that in this framework, the local volatility depends on a particularly complex expectation where no closed-form expression exists and it is neither directly related to European call prices or other liquid products, we present in this contribution a method based on Monte Carlo Simulations to calibrate the local volatility model. We further compare the Variable Annuity Guarantee prices obtained in three different settings, namely the local volatility, the stochastic volatility and the constant volatility models all combined with stochastic interest rates and show that an appropriate volatility modeling is important for these long-dated derivatives. More precisely, we compare the prices of GAO, GMIB Rider and barrier types GAO obtained by using the local volatility, stochastic volatility and constant volatility models.

This paper studies the valuation of credit risk for firms that own several subsidiaries or business lines. We provide simple analytical approximating expressions for probabilities of default, and for equity-debt market values, both in the case when the information is available in continuous time as well as in the case that it is not instantaneously available. The total firm's asset value being modeled as a sum of lognormal random variables, we use convex upper and lower approximations to infer these analytical approximating expressions. We extend the model to firms financed by multiple stochastic liabilities and conclude by numerical illustrations.

In Scheike (1990) a general boundary crossing result for the Brownian motion is obtained. Using path integrals, M. Teunen and M. Goovaerts obtained this result and some generalisations by the methodology of Kac. A Brownian motion process for the surplus of an insurance portfolio is considered which may not cross a given upper boundary. This boundary can be a piecewise linear one consisting of one or more lines.

Over the last decade, it has been shown that the concept of comonotonicity is a helpful tool for solving several research and practical problems in the domain of finance and insurance. In this chapter, we give an extensive bibliographic overview --- without claiming to be complete --- of the developments of the theory of comonotonicity and its applications, with an emphasis on the achievements in the period 2004-2010. These applications range from pricing and hedging of derivatives over risk management to life insurance.

FLWIN, info:eu-repo/semantics/published

We study Vanna-Volga methods which are used to price first generation exotic options in the Foreign Exchange market. They are based on a rescaling of the correction to the Black–Scholes price through the so-called "probability of survival" and the "expected first exit time". Since the methods rely heavily on the appropriate treatment of market data we also provide a summary of the relevant conventions. We offer a justification of the core technique for the case of vanilla options and show how to adapt it to the pricing of exotic options. Our results are compared to a large collection of indicative market prices and to more sophisticated models. Finally we propose a simple calibration method based on one-touch prices that allows the Vanna-Volga results to be in line with our pool of market data.

In ‘A stochastic approach to insurance cycles’, Goovaerts use path integrals. Our paper presents some remarks on this methodology. We will show how the path integral models are related to stochastic differential equations in order to outline a method to evaluate the distribution of a random variable Z ( n , ( x t ) 0≤ t ≤ n ) for a fixed time n , where ( x t ) t ≥0 denotes a Gaussian stochastic process. An explicit expression for the distribution of the discount function exp — n 0 dt(δ + x t ) follows immediately from the theory of Gaussian processes and its relation with stochastic differential equations.

We investigate lower and upper bounds for right tails (stop-loss premiums) of deterministic and stochastic sums of nonindependent random variables. The bounds are derived using the concepts of comonotonicity, convex order, and conditioning. The performance of the presented approximations is investigated numerically for individual life annuity contracts as well as for life annuity portfolios, where mortality is modeled by Makeham's law, whereas investment returns are modeled by a Brownian motion process. ispartof: The Journal of Risk and Insurance vol:76 issue:4 pages:847-866 status: published

In this paper we propose pricing bounds for European-style discrete arithmetic Asian basket options in a Black and Scholes framework. We start from methods used for basket options and Asian options. First, we use the general approach for deriving upper and lower bounds for stop-loss premia of sums of non-independent random variables as in Kaas et al. [Upper and lower bounds for sums of random variables, Insurance Math. Econom. 27 (2000) 151–168] or Dhaene et al. [The concept of comonotonicity in actuarial science and finance: theory, Insurance Math. Econom. 31(1) (2002) 3–33]. We generalize the methods in Deelstra et al. [Pricing of arithmetic basket options by conditioning, Insurance Math. Econom. 34 (2004) 55–57] and Vanmaele et al. [Bounds for the price of discrete sampled arithmetic Asian options, J. Comput. Appl. Math. 185(1) (2006) 51–90]. Afterwards we show how to derive an analytical closed-form expression for a lower bound in the non-comonotonic case. Finally, we derive upper bounds for Asian basket options by applying techniques as in Thompson [Fast narrow bounds on the value of Asian options, Working Paper, University of Cambridge, 1999] and Lord [Partially exact and bounded approximations for arithmetic Asian options, J. Comput. Finance 10 (2) (2006) 1–52]. Numerical results are included and on the basis of our numerical tests, we explain which method we recommend depending on moneyness and time-to-maturity.

In this paper, we elaborate a formula for determining the optimal strike price for a bond put option, used to hedge a position in a bond. This strike price is optimal in the sense that it minimizes, for a given budget, either Value-at-Risk or Tail Value-at-Risk. Formulas are derived for both zero-coupon and coupon bonds, which can also be understood as a portfolio of bonds. These formulas are valid for any short rate model that implies an affine term structure model and in particular that implies a lognormal distribution of future zero-coupon bond prices. As an application, we focus on the Hull–White one-factor model, which is calibrated to a set of cap prices. We illustrate our procedure by hedging a Belgian government bond, and take into account the possibility of divergence between theoretical option prices and real option prices. This paper can be seen as an extension of the work of Ahn and co-workers [Ahn, D., Boudoukh, J., Richardson, M., Whitelaw, R., 1999. Optimal risk management using options. J. Financ. 54, 359–375], who consider the same problem for an investment in a share.

This paper studies a strategy that minimizes the Value-at-Risk (VaR) of a position in a zero-coupon bond by buying a percentage of a put option, subject to a fixed budget available for hedging. We elaborate a formula for determining the optimal strike price for this put option in case of a Vasicek stochastic interest rate model. We demonstrate the relevance of searching the optimal strike price, since moving away from the optimum implies a loss, either due to an increased VaR or due to an increased hedging expenditure. In this way, we extend the results of [Ahn, Boudoukh, Richardson, and Whitelaw (1999). Journal of Finance, 54, 359---375] who minimize VaR for a position in a share. In addition, we look at the alternative risk measure Tail Value-at-Risk.

In this paper we present in a general setting lower and upper bounds for the stop-loss premium of a (stochastic) sum of dependent random variables. Therefore, use is made of the methodology of comonotonic variables and the convex ordering of risks, introduced by Kaas et al. (2000) and Dhaene et al. (2002a, 2002b), combined with actuarial conditioning. The lower bound approximates very accurate the real value of the stop-loss premium. However, the comonotonic upper bounds perform rather badly for some retentions. Therefore, we construct sharper upper bounds based upon the traditional comonotonic bounds. Making use of the ideas of Rogers and Shi (1995), the first upper bound is obtained as the comonotonic lower bound plus an error term. Next this bound is refined by making the error term dependent on the retention in the stop-loss premium. Further, we study the case that the stop-loss premium can be decomposed into two parts. One part which can be evaluated exactly and another part to which comonotonic bounds are applied. As an application we study the bounds for the stop-loss premium of a random variable representing the stochastically discounted value of a series of cash flows with a fixed and stochastic time horizon. The paper ends with numerical examples illustrating the presented approximations. We apply the proposed bounds for life annuities, using Makeham's law, when also the stochastic nature of interest rates is taken into account by means of a Brownian motion.

Inspired by the ideas of Rogers and Shi [J. Appl. Prob. 32 (1995) 1077], Chalasani et al. [J. Comput. Finance 1(4) (1998) 11] derived accurate lower and upper bounds for the price of a European-style Asian option with continuous averaging over the full lifetime of the option, using a discrete-time binary tree model. In this paper, we consider arithmetic Asian options with discrete sampling and we generalize their method to the case of forward starting Asian options. In this case with daily time steps, the method of Chalasani et al. is still very accurate but the computation can take a very long time on a PC when the number of steps in the binomial tree is high. We derive analytical lower and upper bounds based on the approach of Kaas et al. [Insurance: Math. Econ. 27 (2000) 151] for bounds for stop-loss premiums of sums of dependent random variables, and by conditioning on the value of underlying asset at the exercise date. The comonotonic upper bound corresponds to an optimal superhedging strategy. By putting in less information than Chalasani et al. the bounds lose some accuracy but are still very good and they are easily computable and moreover the computation on a PC is fast. We illustrate our results by different numerical experiments and compare with bounds for the Black and Scholes model [J. Pol. Econ. 7 (1973) 637] found in another paper [Bounds for the price of discretely sampled arithmetic Asian options, Working paper, Ghent University, 2002]. We notice that the intervals of Chalasani et al. do not always lie within the Black and Scholes intervals. We have proved that our bounds converge to the corresponding bounds in the Black and Scholes model. Our numerical illustrations also show that the hedging error is small if the Asian option is in the money. If the option is out of the money, the price of the superhedging strategy is not as adequate, but still lower than the straightforward hedge of buying one European option with the same exercise price. Keywords: Comonotonicity; Asian options; Superhedging strategy

Determining the price of a basket option is not a trivial task, because there is no explicit analytical expression available for the distribution of the weighted sum of prices of the assets in the basket. However, by using a conditioning variable, this price can be decomposed in two parts, one of which can be computed exactly. For the remaining part we first derive a lower and an upper bound based on comonotonicity, and another upper bound equal to that lower bound plus an error term. Secondly, we derive an approximation by applying some moment matching method., info:eu-repo/semantics/published