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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 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.

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.

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.

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.

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

In literature, lower and upper bounds were obtained for arithmetic Asian and basket options based on comonotonicity results and by conditioning upon one variable. In this paper, we derive analytical expressions for the comonotonic bounds of stop-loss premiums of sums of dependent random variables by conditioning upon two variables. We also use the idea of several conditioning variables to develop an approximation for cases for which it is cumbersome to obtain a comonotonic lower bound. The numerical analysis shows that conditioning on two variables leads to very sharp results.

In a continuous-time framework, we consider the problem of a De…ned Contribution Pension Fund in the presence of a minimum guarantee. The problem of the fund manager is to invest the initial wealth and the (stochastic) contribution ‡ow into the …nancial market, in order to maximize the expected util- ity function of the terminal wealth under the constraint that the terminal wealth must exceed the minimum guarantee. We assume that the stochastic interest rates follow the a¢ne dynamics, including the CIR (Cox, Ingersoll and Ross 1985) model and the Vasiµ model. The optimal investment strategies are obtained by assum- ing the completeness of …nancial markets and a CRRA utility function. Explicit formulae for the optimal investment strategies are included for dierent examples of guarantees and contributions.

In the context of a general multivariate financial market with transaction costs, we consider the problem of maximizing expected utility from terminal wealth. In contrast with the existing literature, where only the liquidation value of the terminal portfolio is relevant, we consider general utility functions which are only required to be consistent with the structure of the transaction costs. An important feature of our analysis is that the utility function is not required to be $C^1$. Such nonsmoothness is suggested by major natural examples. Our main result is an extension of the well-known dual formulation of the utility maximization problem to this context.

We study an optimal investment problem in a continuous-time framework where the interest rates follow Cox-Ingersoll-Ross dynamics. Closed form formulae for the optimal investment strategy are obtained by assuming the completeness of financial markets and the CRRA utility function. In particular, we study the behaviour of the solution when time approaches the terminal date.

We define the concept of conditional dominance and use it for obtaining bounds on the hedging prices of random variables. These bounds depend only on the characteristics of the financial market and the random variables to hedge. Moreover, they are coherent with the equilibrium and tighter than the ones obtained by the classical super-replication approach, significantly in some cases. This approach can be applied in static as well as dynamic frameworks.

In this paper, we observe the convergence of the long-term return, using an extension of the Cox-Ingersoll-Ross [1985] stochastic model of the short interest rate r. Using the theory of Bessel processes, we are able to prove the convergence almost everywhere of \(frac{1}{t}{\int {_{_0}} ^t}{X_\delta }ds\) with the X a generalized Besselsquare process with drift with stochastic reversion level. In order to make some approximations, we observe the convergence in law of the sequence \(left({\sqrt {{{\frac{{ - 2\beta }}{{\delta n}}}^3}} _0}^{nt}\left( {{X_u} + \frac{{{\delta _u}}}{{2\beta }}} \right)du)t \geqslant 0\). By Aldous’ criterion, we are able to prove that this sequence converges in law to a Brownian motion.