A BLACK-SCHOLES FORMULA FOR OPTION PRICING WITH DIVIDENDS

Abstract. We obtain a Black-Scholes formula for the arbitrage-free pricing of Eu-ropean Call options with constant coefficients when the underlylng stock generatesdividends. To hedge the Call option, we will always borrow money from bank. We seethe influence of the dividend term on the option pricing via the comparison theoremof BSDE(backward stochastic di~erential equation [5], [7]). We also consider the option pricing problem in terms of the borrowing rate R whichis not equal to the interest rate r. The corresponding Black-Sdxoles formula is given.We notice that it is in fact the borrowing rate that plays the role in the pricing formula.     

On the general theory of exact controllability for skew symmetric operators

We introduce a general formalism for linear evolution equations with skew adjoint operators. We make explicit the controllability operator as an expansion with respect to eigenfunctions. Using the fact that the eigenvalues are purely imaginary, we give sufficient controllability conditions. This approach is convenient for studying the asymptotic behaviour of the optimal control.     

Variational inequalities and the pricing of American options

This paper is devoted to the derivation of some regularity properties of pricing functions for American options and to the discussion of numerical methods, based on the Bensoussan-Lions methods of variational inequalities. In particular, we provide a complete justification of the so-called Brennan-Schwartz algorithm for the valuation of American put options.Research supported in part by a contract from Banque INDOSUEZ.     

Topological inverse problem for oscillating systems and its application

This note is just an introduction to a problem to find a topological type of potentials from given data and a problem to see which data or experiment is necessary to obtain the topological classification of practical use. Here we propose a method of hysteresis on the space of Fourier coefficients, which reduces to the theory of resonance curves in a very special case. The direction of the hysteresis curve is proved to characterize the (first) topological type of potential. In contrast with the usual direction, which is common to the transistor oscillator, the unusual direction is found in EEG experiments (on humans) called photic driving experiments on rhythm.     

On the spectral radius and the spectral norm of Hadamard products of nonnegative matrices

We prove the spectral radius inequality ρ(A₁°A₂°?°A_k)?ρ(A₁A₂?A_k) for nonnegative matrices using the ideas of Horn and Zhang. We obtain the inequality ‖A°B‖?ρ(A^TB) for nonnegative matrices, which improves Schur’s classical inequality ‖A°B‖?‖A‖‖B‖, where ‖·‖ denotes the spectral norm. We also give counterexamples to two conjectures about the Hadamard product.     

Coupling and option price comparisons in a jump-diffusion model

In this paper, we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete, there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for convex payoffs, the option price is increasing in the jump-risk parameter. We apply this result to deduce general inequalities, comparing the prices of contingent claims under various martingale measures, which have been proposed in the literature as candidate pricing measures.

Our proofs are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities.     

Smallest g -supersolution with constraint

In this paper the existence and uniqueness of the smallest g-supersolution for BSDE is discussed in the case without Lipschitz condition imposing on both constraint function and drift coefficient in the different method from the one with Lipschitz condition. Then by considering (ξ, g) as a parameter of BSDE, and (ξ ^α, g ^α) as a class of parameters for BSDE, where α belongs to a set , for every there exists a pair of solution {Y ^a, Z^a} for the BSDE, the properties of which is also a solution for some BSDE is studied. This result may be used to discuss optimal problems with recursive utility. This work was supported by NSFC (79790130)     

Approach theory meets probability theory

In this paper we reconsider the basic topological and metric structures on spaces of probability measures and random variables, such as e.g. the weak topology and the total variation metric, replacing them with more intrinsic and richer approach structures. We comprehensibly investigate the relationships among, and basic facts about these structures, and prove that fundamental results, such as e.g. the portmanteau theorem and Prokhorov?s theorem, can be recaptured in a considerably stronger form in the new setting.     

On rates of convergence and asymptotic normality in the multiknapsack problem

In Meanti et al. (1990) an almost sure asymptotic characterization has been derived for the optimal solution value as function of the knapsack capacities, when the profit and requirement coefficients of items to be selected from are random variables. In this paper we establish a rate of convergence for this process using results from the theory of empirical processes.     

Some recent advances in theory and simulation of fractional diffusion processes

To offer an insight into the rapidly developing theory of fractional diffusion processes, we describe in some detail three topics of current interest: (i) the well-scaled passage to the limit from continuous time random walk under power law assumptions to space-time fractional diffusion, (ii) the asymptotic universality of the Mittag–Leffler waiting time law in time-fractional processes, (iii) our method of parametric subordination for generating particle trajectories.     

An irregular grid approach for pricing high-dimensional American options

We propose and test a new method for pricing American options in a high-dimensional setting. The method is centered around the approximation of the associated complementarity problem on an irregular grid. We approximate the partial differential operator on this grid by appealing to the SDE representation of the underlying process and computing the root of the transition probability matrix of an approximating Markov chain. Experimental results in five-dimensions are presented for four different payoff functions.     

Explicit characterization of the super-replication strategy in financial markets with partial transaction costs

We consider a continuous time multivariate financial market with proportional transaction costs and study the problem of finding the minimal initial capital needed to hedge, without risk, European-type contingent claims. The model is similar to the one considered in Bouchard and Touzi [B. Bouchard, N. Touzi, Explicit solution of the multivariate super-replication problem under transaction costs, The Annals of Applied Probability 10 (3) (2000) 685–708] except that some of the assets can be exchanged freely, i.e. without paying transaction costs. In this context, we generalize the result of the above paper and prove that the super-replication price is given by the cost of the cheapest hedging strategy in which the number of non-freely exchangeable assets is kept constant over time. Our proof relies on the introduction of a new auxiliary control problem whose value function can be interpreted as the super-hedging price in a model with unbounded stochastic volatility (in the directions where transaction costs are non-zero). In particular, it confirms the usual intuition that transaction costs play a similar role to stochastic volatility.     

Mathematical recreations of Dénes König and his work on graph theory

Dénes König (1884–1944) is a Hungarian mathematician well known for his treatise on graph theory (König, 1936). When he was a student, he published two books on mathematical recreations ( and ). Does his work on mathematical recreations have any relation to his work on graph theory? If yes, how are they connected? To answer these questions, we will examine his books of 1902, 1905 and 1936, and compare them with each other. We will see that the books of 1905 and 1936 include many common topics, and that the treatment of these topics is different between 1905 and 1936.     

On the ergodicity and mixing of max-stable processes

Max-stable processes arise in the limit of component-wise maxima of independent processes, under appropriate centering and normalization. In this paper, we establish necessary and sufficient conditions for the ergodicity and mixing of stationary max-stable processes. We do so in terms of their spectral representations by using extremal integrals.     

On the pricing of longevity-linked securities

For annuity providers, longevity risk, i.e. the risk that future mortality trends differ from those anticipated, constitutes an important risk factor. In order to manage this risk, new financial products, so-called longevity derivatives, may be needed, even though a first attempt to issue a longevity bond in 2004 was not successful.While different methods of how to price such securities have been proposed in recent literature, no consensus has been reached. This paper reviews, compares and comments on these different approaches. In particular, we use data from the United Kingdom to derive prices for the proposed first longevity bond and an alternative security design based on the different methods.     

可解内生生育率的C-K模型

Abstract. In this paper,a C-K model with solvable endogenous fertility under the strongly addi-tive utility function is presented. The discrimination conditions of the existence of the nonzerosteady states are given. Under a kind of utility function and production function,we prove thatthese conditions are satisfied and the economy at least has an optimal growth path. The position-al relationship of the multiple steady states on the plane is discussed when multiple steady statesand multiple growth paths exist. By numerical analysis ,the fertility decreses with the per capitacapital and per capita consumption increasing and increases with the per capita capital and percapita consumption decreasing on the economic growth path are obtained.