Abstract Binomial models, which describe the asset price dynamics of the continuous{time model in the limit, serve for approximate valuation of options, especially where formulas cannot be derived analytically due to properties of the considered option type. To evaluate results, one inevitably must understand the convergence properties. In the literature we nd various contributions proving convergence of option prices. Di erently, we examine convergence behavior and convergence speed. Unfortunately, even in the case of European call options distorted results occur calculating prices along iteration of tree re nements. These convergence patterns are examined and order of convergence one is proven for the Cox{Ross{Rubinstein[79] model as well as for the tree parameter selections of Jarrow and Rudd[83], and Tian[93]. Furthermore, we de ne new binomial models, where the calculated option prices converge smoothly to the Black{Scholes solution and we achieve order of convergence two with much smaller initial error. Notably, solely the formulas to determine the up{ and down{factors change. Finally, following the approach of Broadie and Detemple[96], all tree approaches are compared with respect to speed and accuracy calculating relative root{mean{squared error of approximate option values for a sample of randomly selected parameters across a set of re nements. Here, on the average the same degree of accuracy is achieved 1400 times faster with the new binomial models. We also give some insights into the peculiarities with the valuation of the American put option. Inspecting the numerical results, the approximation of American type options with the new models exhibits order of convergence one but smaller initial error than with previously existing binomial models, giving 10{times faster the same accuracy as previous binomial methods on the average
Abstract Despite its success, the Black-Scholes formula has become increasingly unreliable over time in the very markets where one would expect it to be most accurate. In addition, attempts by financial economists to extract probabilistic information from option prices have been puny in comparison to what is clearly possible. This paper develops a new method for inferring risk-neutral probabilities (or state-contingent prices) from the simultaneously observed prices of European options. These probabilities are then used to infer a unique fully specified recombining binomial tree that is consistent with these probabilities (and hence consistent with all the observed option prices). If specified exogenously, the model can also accommodate local interest rates and underlying asset payout rates that are general functions of the concurrent underlying asset price and time. One byproduct is a map of the local and risk-neutral global volatility structure of the underlying asset return over future dates and states. In a 200 step lattice, for example, there are a total of 60,301 unknowns: 40,200 potentially different move sizes, 20,100 potentially different move probabilities, and 1 interest rate to be determined from 60,301 independent equations, many of which are non-linear in the unknowns. Despite this, a 3-step backwards recursive solution procedure exists which is only slightly more time-consuming than for a standard binomial tree with given constant move sizes and move probabilities. Moreover, closed-form expressions exist for the values and hedging parameters of European options maturing with or before the end of tree. The tree can also be used to value and hedge American and several types of exotic options. From the standpoint of the standard binomial option pricing model which implies a limiting risk-neutral lognormal distribution for the underlying asset, the approach here provides the natural (and probably the simplest) way to generalize to arbitrary ending risk-neutral probability distributions. Interpreted in terms of continuous-time diffusion processes, the model here assumes that the drift and local volatility are at most functions of the underlying asset price and time. But instead of beginning with a parameterization of these functions (as in previous research), the model derives these functions endogenously to fit current option prices. As a result, it can be thought of as an attempt to exhaust the potential for single state-variable path-independent diffusion processes to rectify problems with the Black- Scholes formula that arise in practice.
Description:
Christian Wallner Uwe Wystup Abstract: No front-office software can survive without providing derivatives of option prices with respect to underlying market or model parameters, the so called Greeks. If a closed form solution for an option exists, Greeks can be computed analytically and they are numerically stable. However, for American style options, there is no closed-form solution. The price is computed by binomial trees, finite difference methods or an analytic approximation. Taking derivatives of these prices leads to instable numerics or misleading results, specially for Greeks of higher order. We compare the computation of the Greeks in various pricing methods and conclude with the recommendation to use Leisen-Reimer trees. Discuss this paper
Abstract A new family of binomial trees as approximations to the Black-Scholes model is introduced. For this class of trees, the existence of complete asymptotic expansions for the prices of vanilla European options is demonstrated and the first three terms are explicitly computed. As special cases, a tree with third order convergence is constructed and the conjecture of Leisen and Reimer that their tree has second order convergence is proven.
)
Date (
Visit Link 
(0 Votes) 
(1 Vote) 
(2 Votes) 