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Standard derivative pricing theory is based on the assumption of agents acting as price takers on the market for the underlying asset. We relax this hypothesis and study if and how a large agent whose trades move prices can replicate the payoff of a derivative security. Our analysis extends prior work of Jarrow to economies with continuous security trading. We characterize the solution to the hedge problem in terms of a nonlinear partial differential equation and provide results on existence and uniqueness of this equation. Simulations are used to compare the hedging strategies in our model to standard Black-Scholes strategies.  相似文献   

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We discuss here an alternative interpretation of the familiar binomial lattice approach to option pricing, illustrating it with reference to pricing of barrier options, one- and two-sided, with fixed, moving or partial barriers, and also the pricing of American put options. It has often been observed that if one tries to price a barrier option using a binomial lattice, then one can find slow convergence to the true price unless care is taken over the placing of the grid points in the lattice; see, for example, the work of Boyle & Lau [2]. The placing of grid points is critical whether one uses a dynamic programming approach, or a Monte Carlo approach, and this can make it difficult to compute hedge ratios, for example. The problems arise from translating a crossing of the barrier for the continuous diffusion process into an event for the binomial approximation. In this article, we show that it is not necessary to make clever choices of the grid positioning, and by interpreting the nature of the binomial approximation appropriately, we are able to derive very quick and accurate pricings of barrier options. The interpretation we give here is applicable much more widely, and helps to smooth out the ‘odd-even’ ripples in the option price as a function of time-to-go which are a common feature of binomial lattice pricing.  相似文献   

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We compute the limiting hedging error of the Leland strategy for the approximate pricing of the European call option in a market with transactions costs. It is not equal to zero in the case when the level of transactions costs is a constant, in contradiction with the claim in Leland (1985).  相似文献   

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For a Markov process , the forward measure over the time interval is defined by the Radon-Nikodym derivative , where is a given non-negative function and is the normalizing constant. In this paper, the law of under the forward measure is identified when is a diffusion process or, more generally, a continuous-path Markov process.  相似文献   

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Dynamic programming and mean-variance hedging   总被引:4,自引:0,他引:4  
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Mean-variance hedging for continuous processes: New proofs and examples   总被引:4,自引:0,他引:4  
Let be a special semimartingale of the form and denote by the mean-variance tradeoff process of . Let be the space of predictable processes for which the stochastic integral is a square-integrable semimartingale. For a given constant and a given square-integrable random variable , the mean-variance optimal hedging strategy by definition minimizes the distance in between and the space . In financial terms, provides an approximation of the contingent claim by means of a self-financing trading strategy with minimal global risk. Assuming that is bounded and continuous, we first give a simple new proof of the closedness of in and of the existence of the F?llmer-Schweizer decomposition. If moreover is continuous and satisfies an additional condition, we can describe the mean-variance optimal strategy in feedback form, and we provide several examples where it can be computed explicitly. The additional condition states that the minimal and the variance-optimal martingale measures for should coincide. We provide examples where this assumption is satisfied, but we also show that it will typically fail if is not deterministic and includes exogenous randomness which is not induced by .  相似文献   

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A recent article of Flesaker and Hughston introduces a one factor interest rate model called the rational lognormal model. This model has a lot to recommend it including guaranteed finite positive interest rates and analytic tractability. Consequently, it has received a lot of attention among practioners and academics alike. However, it turns out to have the undesirable feature of predicting that the asymptotic value of the short rate volatility is zero. This theoretical result is proved rigorously in this article. The outcome of an empirical study complementing the theoretical result is discussed at the end of the article. European call options are valued with the rational lognormal model and a comparably calibrated mean reverting Gaussian model. unsurprisingly, rational lognormal option values are considerably lower than the analogous mean reverting Gaussian option values. In other words, the volatility in the rational lognormal model declines so quickly that options are severely undervalued.  相似文献   

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Optimal stopping for a diffusion with jumps   总被引:3,自引:0,他引:3  
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We propose a general one-factor model for the term structure of interest rates which based upon a model for the short rate. The dynamics of the short rate is described by an appropriate function of a time-changed Wiener process. The model allows for perfect fitting of given term structure of interest rates and volatilities, as well as for mean reversion. Moreover, every type of distribution of the short rate can be achieved, in particular, the distribution can be concentrated on an interval. The model includes several popular models such as the generalized Vasicek (or Hull-White) model, the Black-Derman-Toy, Black-Karasinski model, and others. There is a unified numerical approach to the general model based on a simple lattice approximation which, in particular, can be chosen as a binomial or -nomial lattice with branching probabilities .  相似文献   

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Weighted norm inequalities and hedging in incomplete markets   总被引:1,自引:0,他引:1  
Let be an -valued special semimartingale on a probability space with canonical decomposition . Denote by the space of all random variables , where is a predictable -integrable process such that the stochastic integral is in the space of semimartingales. We investigate under which conditions on the semimartingale the space is closed in , a question which arises naturally in the applications to financial mathematics. Our main results give necessary and/or sufficient conditions for the closedness of in . Most of these conditions deal with BMO-martingales and reverse H?lder inequalities which are equivalent to weighted norm inequalities. By means of these last inequalities, we also extend previous results on the F?llmer-Schweizer decomposition.  相似文献   

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In this paper we consider the valuation of an option with time to expiration and pay-off function which is a convex function (as is a European call option), and constant interest rate , in the case where the underlying model for stock prices is a purely discontinuous process (hence typically the model is incomplete). The main result is that, for “most” such models, the range of the values of the option, using all possible equivalent martingale measures for the valuation, is the interval , this interval being the biggest interval in which the values must lie, whatever model is used.  相似文献   

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