A subject often recurring in financial papers, is the pricing of stocks and securities when the r... more A subject often recurring in financial papers, is the pricing of stocks and securities when the rate of return is stochastic. In most cases, the stocks considered are assumed not to pay out any dividend. In the present contribution we want to show how it is possible to obtain upper and lower bounds for the (distribution of the) accumulated value of a dividend paying security at a future time t, when the logarithm of the stock price is modelled by means of a Wiener process.
In most practical cases, it is impossible to find an explicit expression for the distribution fun... more In most practical cases, it is impossible to find an explicit expression for the distribution function of the present value of a sequence of cash flows that are discounted using some given stochastic return process. In this paper, we present an easy computable approximation for this distribution function. The approximation is a distribution function which is, in the sense of convex order, an upper bound for the original distribution function.
This paper starts from the GARCH(1,1)-M model of Bollerslev [Generalized autoregressive condition... more This paper starts from the GARCH(1,1)-M model of Bollerslev [Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31 (1986) 307-327], and investigates the limit diffusion form as it is presented in Nelson [ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7-38]. The distribution for the conditional variance process is derived, and in the limit for t going to infinity is shown to coincide with the stationary distribution given in Nelson [ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7-38]. In addition it is shown how the distribution for the complete model can be arrived at; explicit calculations are given in case the conditional variance is a martingale.
Abstract By means of Wiener processes, randomness in interest rates for annuities can be modelled... more Abstract By means of Wiener processes, randomness in interest rates for annuities can be modelled. This paper wants to give an expression for the Laplace transform of annuities certain, when time is exponentially distributed.
... A. DE SCHEPPER, MJ GOOVAERTS and R. KAAS De Schepper A, Goovaerts MJ, Kaas R. A recursive sch... more ... A. DE SCHEPPER, MJ GOOVAERTS and R. KAAS De Schepper A, Goovaerts MJ, Kaas R. A recursive scheme for perpetu-ities with random positive interest rates. Part I. Analytical results. Scand. ... 10, 275-287. De Schepper, A., De Vylder, F., Goovaerts. M. & Kaas, R. (l992a). ...
This paper intends to evaluate the present value of IBNR reserves, when future interest rates are... more This paper intends to evaluate the present value of IBNR reserves, when future interest rates are unknown. We first derive a result for the Laplace transform ofthe present value, when it is assumed that the interest rates are stochastic and can be modelled by means of a stochastic process which is similar to the model of Cox, lngersoll and Ross (1985). Starting from this Laplace transform, it is shown how the probability distribution for the quantity under investigation can be found. The results are illustrated numerically.
In this article, we calculate a market-weighted return index for the  largest stocks listed on ... more In this article, we calculate a market-weighted return index for the  largest stocks listed on the Brussels Stock Exchange over the period -, based on a new, unique and high-quality database. We find that this index captures the most important stylised facts of the value-weighted return of all shares listed on the Brussels Stock Exchange in this period. Our results support the empirical practice of concentrating on just the largest stocks. The indices we construct are based on one of the longest Belgian time series available. The indices take into account the exact dividends, the timing of the dividend cash flows and all capital operations. We are therefore able to decompose total returns into capital gain returns and dividend returns, which is not possible with most historical return series. We show that, to construct a credible return index, it is crucial to fully take into account dividends.
A subject often recurring in financial and actuarial papers is the pricing of stocks and securiti... more A subject often recurring in financial and actuarial papers is the pricing of stocks and securities when the rate of return is stochastic. In most cases, the stocks considered are assumed not to pay out any dividend. In the present contribution we show how it is possible to obtain upper and lower bounds for the (distribution of the) accumulated value of a cash-flow in the presence of dividend barriers at a future time t, when the logarithm of the stock price is modelled by means of a Wiener process.
In this paper, we apply the Jensen's inequality to derive sharp lower and upper bounds for condit... more In this paper, we apply the Jensen's inequality to derive sharp lower and upper bounds for conditional expectations of multiplicative functionals of diffusion processes. We show that the bounds are applicable to the pricing of European-style derivative securities. We also adapt the method to price (double) barrier options and lookback options under local volatility models. As by-product result, we propose a static hedge for lookback options. Numerical illustrations performed for the CEV model demonstrate the high accuracy of the method.
Common interest rate models are faced with the problem of volatilities vanishing for spot rates i... more Common interest rate models are faced with the problem of volatilities vanishing for spot rates in the vicinity of zero. A possible answer to this difficulty can be given by the introduction of a reflecting boundary at zero, at the same time guaranteeing the spot rate to be non-negative, which is needed in order to avoid the possibility of arbitrage. In the present paper, we obtain closed form expressions for transition probabilities and for prices of general interest rate contingent claims by means of path integrals, when the spot rate process is modelled by means of a general diffusion with a reflecting or absorbing boundary. We also show how to derive accurate closed form approximations in case the path integrals are not analytically computable.
In this paper, we apply the results about d and d-function perturbations in order to formulate wi... more In this paper, we apply the results about d and d-function perturbations in order to formulate within the Feynman-Kac integration the solution of the forward Fokker-Planck equation subject to Dirichlet or Neumann boundary conditions. We introduce the concept of convex order to derive upper and lower bounds for path integrals with d and d- functions in the integrand. We suggest the use of bounds as an approximation for the solution.
A subject often recurring in financial papers, is the pricing of stocks and securities when the r... more A subject often recurring in financial papers, is the pricing of stocks and securities when the rate of return is stochastic. In most cases, the stocks considered are assumed not to pay out any dividend. In the present contribution we want to show how it is possible to obtain upper and lower bounds for the (distribution of the) accumulated value of a dividend paying security at a future time t, when the logarithm of the stock price is modelled by means of a Wiener process.
In most practical cases, it is impossible to find an explicit expression for the distribution fun... more In most practical cases, it is impossible to find an explicit expression for the distribution function of the present value of a sequence of cash flows that are discounted using some given stochastic return process. In this paper, we present an easy computable approximation for this distribution function. The approximation is a distribution function which is, in the sense of convex order, an upper bound for the original distribution function.
This paper starts from the GARCH(1,1)-M model of Bollerslev [Generalized autoregressive condition... more This paper starts from the GARCH(1,1)-M model of Bollerslev [Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31 (1986) 307-327], and investigates the limit diffusion form as it is presented in Nelson [ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7-38]. The distribution for the conditional variance process is derived, and in the limit for t going to infinity is shown to coincide with the stationary distribution given in Nelson [ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7-38]. In addition it is shown how the distribution for the complete model can be arrived at; explicit calculations are given in case the conditional variance is a martingale.
Abstract By means of Wiener processes, randomness in interest rates for annuities can be modelled... more Abstract By means of Wiener processes, randomness in interest rates for annuities can be modelled. This paper wants to give an expression for the Laplace transform of annuities certain, when time is exponentially distributed.
... A. DE SCHEPPER, MJ GOOVAERTS and R. KAAS De Schepper A, Goovaerts MJ, Kaas R. A recursive sch... more ... A. DE SCHEPPER, MJ GOOVAERTS and R. KAAS De Schepper A, Goovaerts MJ, Kaas R. A recursive scheme for perpetu-ities with random positive interest rates. Part I. Analytical results. Scand. ... 10, 275-287. De Schepper, A., De Vylder, F., Goovaerts. M. & Kaas, R. (l992a). ...
This paper intends to evaluate the present value of IBNR reserves, when future interest rates are... more This paper intends to evaluate the present value of IBNR reserves, when future interest rates are unknown. We first derive a result for the Laplace transform ofthe present value, when it is assumed that the interest rates are stochastic and can be modelled by means of a stochastic process which is similar to the model of Cox, lngersoll and Ross (1985). Starting from this Laplace transform, it is shown how the probability distribution for the quantity under investigation can be found. The results are illustrated numerically.
In this article, we calculate a market-weighted return index for the  largest stocks listed on ... more In this article, we calculate a market-weighted return index for the  largest stocks listed on the Brussels Stock Exchange over the period -, based on a new, unique and high-quality database. We find that this index captures the most important stylised facts of the value-weighted return of all shares listed on the Brussels Stock Exchange in this period. Our results support the empirical practice of concentrating on just the largest stocks. The indices we construct are based on one of the longest Belgian time series available. The indices take into account the exact dividends, the timing of the dividend cash flows and all capital operations. We are therefore able to decompose total returns into capital gain returns and dividend returns, which is not possible with most historical return series. We show that, to construct a credible return index, it is crucial to fully take into account dividends.
A subject often recurring in financial and actuarial papers is the pricing of stocks and securiti... more A subject often recurring in financial and actuarial papers is the pricing of stocks and securities when the rate of return is stochastic. In most cases, the stocks considered are assumed not to pay out any dividend. In the present contribution we show how it is possible to obtain upper and lower bounds for the (distribution of the) accumulated value of a cash-flow in the presence of dividend barriers at a future time t, when the logarithm of the stock price is modelled by means of a Wiener process.
In this paper, we apply the Jensen's inequality to derive sharp lower and upper bounds for condit... more In this paper, we apply the Jensen's inequality to derive sharp lower and upper bounds for conditional expectations of multiplicative functionals of diffusion processes. We show that the bounds are applicable to the pricing of European-style derivative securities. We also adapt the method to price (double) barrier options and lookback options under local volatility models. As by-product result, we propose a static hedge for lookback options. Numerical illustrations performed for the CEV model demonstrate the high accuracy of the method.
Common interest rate models are faced with the problem of volatilities vanishing for spot rates i... more Common interest rate models are faced with the problem of volatilities vanishing for spot rates in the vicinity of zero. A possible answer to this difficulty can be given by the introduction of a reflecting boundary at zero, at the same time guaranteeing the spot rate to be non-negative, which is needed in order to avoid the possibility of arbitrage. In the present paper, we obtain closed form expressions for transition probabilities and for prices of general interest rate contingent claims by means of path integrals, when the spot rate process is modelled by means of a general diffusion with a reflecting or absorbing boundary. We also show how to derive accurate closed form approximations in case the path integrals are not analytically computable.
In this paper, we apply the results about d and d-function perturbations in order to formulate wi... more In this paper, we apply the results about d and d-function perturbations in order to formulate within the Feynman-Kac integration the solution of the forward Fokker-Planck equation subject to Dirichlet or Neumann boundary conditions. We introduce the concept of convex order to derive upper and lower bounds for path integrals with d and d- functions in the integrand. We suggest the use of bounds as an approximation for the solution.
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