From Discrete to Continuous Financial Models: New Convergence Results For Option Pricing

In this paper we develop a new notion of convergence for discussing the relationship between discrete and continuous financial models, D ²-convergence. This is stronger than weak convergence, the commonly used mode of convergence in the finance literature. We show that D ²-convergence, unlike weak convergence, yields a number of important convergence preservation results, including the convergence of contingent claims, derivative asset prices and hedge portfolios in the discrete Cox-Ross-Rubinstein option pricing models to their continuous counterparts in the Black-Scholes model. Our results show that D ²-convergence is characterized by a natural lifting condition from nonstandard analysis (NSA), and we demonstrate how this condition can be reformulated in standard terms, i.e., in language that only involves notions from standard analysis. From a practical point of view, our approach suggests procedures for constructing good (i.e., convergent) approximate discrete claims, prices, hedge portfolios, etc. This paper builds on earlier work by the authors, who introduced methods from NSA to study problems arising in the theory of option pricing.     

运用期权定价模型评估公司价值

本着把期权思想应用于公司价值评估,分别运用二叉树模型和Black-Scholes模型计算公司价值,并对这一方法的应用价值及局限性进行一定的探讨。     

Richardson extrapolation techniques for the pricing of American‐style options

In this article, the authors reexamine the American‐style option pricing formula of R. Geske and H.E. Johnson (1984), and extend the analysis by deriving a modified formula that can overcome the possibility of nonuniform convergence (which is likely to occur for nonstandard American options whose exercise boundary is discontinuous) encountered in the original Geske–Johnson methodology. Furthermore, they propose a numerical method, the Repeated‐Richardson extrapolation, which allows the estimation of the interval of true option values and the determination of the number of options needed for an approximation to achieve a given desired accuracy. Using simulation results, our modified Geske–Johnson formula is shown to be more accurate than the original Geske–Johnson formula for pricing American options, especially for nonstandard American options. This study also illustrates that the Repeated‐Richardson extrapolation approach can estimate the interval of true American option values extremely well. Finally, the authors investigate the possibility of combining the binomial Black–Scholes method proposed by M. Broadie and J.B. Detemple (1996) with the Repeated‐Richardson extrapolation technique. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:791–817, 2007     

A Dynamic Investment Model with Control on the Portfolio's Worst Case Outcome

This paper considers a portfolio problem with control on downside losses. Incorporating the worst-case portfolio outcome in the objective function, the optimal policy is equivalent to the hedging portfolio of a European option on a dynamic mutual fund that can be replicated by market primary assets. Applying the Black-Scholes formula, a closed-form solution is obtained when the utility function is HARA and asset prices follow a multivariate geometric Brownian motion. The analysis provides a useful method of converting an investment problem to an option pricing model.     

Two‐State Option Pricing: Binomial Models Revisited

This article revisits the topic of two‐state option pricing. It examines the models developed by Cox, Ross, and Rubinstein (1979), Rendleman and Bartter (1979), and Trigeorgis (1991) and presents two alternative binomial models based on the continuous‐time and discrete‐time geometric Brownian motion processes, respectively. This work generalizes the standard binomial approach, incorporating the main existing models as particular cases. The proposed models are straightforward and flexible, accommodate any drift condition, and afford additional insights into binomial trees and lattice models in general. Furthermore, the alternative parameterizations are free of the negative aspects associated with the Cox, Ross, and Rubinstein model. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:987–1001, 2001     

THE GARCH OPTION PRICING MODEL

This article develops an option pricing model and its corresponding delta formula in the context of the generalized autoregressive conditional heteroskedastic (GARCH) asset return process. the development utilizes the locally risk-neutral valuation relationship (LRNVR). the LRNVR is shown to hold under certain combinations of preference and distribution assumptions. the GARCH option pricing model is capable of reflecting the changes in the conditional volatility of the underlying asset in a parsimonious manner. Numerical analyses suggest that the GARCH model may be able to explain some well-documented systematic biases associated with the Black-Scholes model.     

Improving lattice schemes through bias reduction

Lattice schemes for option pricing, such as tree or grid/partial differential equation (p.d.e.) methods, are usually designed as a discrete version of an underlying continuous model of stock prices. The parameters of such schemes are chosen so that the discrete version “best” matches the continuous one. Only in the limit does the lattice option price model converge to the continuous one. Otherwise, a discretization bias remains. A simple modification of lattice schemes which reduces the discretization bias is proposed. The modification can, in theory, be applied to any lattice scheme. The main idea is to adjust the lattice parameters in such a way that the option price bias, not the stock price bias, is minimized. European options are used, for which the option price bias can be evaluated precisely, as a template to modify and improve American option methods. A numerical study is provided. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:733–757, 2006     

ON THE CONSISTENCY OF THE LUCAS PRICING FORMULA

In order to find the real market value of an asset in an exchange economy, one would typically apply the Lucas formula, developed in a discrete time framework. This theory has also been extended to continuous time models, in which case the same pricing formula has been universally applied.
While the discrete time theory is rather transparent, there has been some confusion regarding the continuous time analogue. In particular, the continuous time pricing formula must contain a certain type of a square covariance term that does not readily follow from the discrete time formulation. As a result, this term has sometimes been missing in situations where it should have been included.
In this paper, we reformulate the discrete time theory in such a way that this covariance term does not come as a mystery in the continuous time version.
In most real life situations dividends are paid out in lump sums, not in rates. This leads to a discontinuous model, and adding a continuous time framework, it appears that our framework is a most natural one in finance.
Finally, the Gordon growth formula is extended from its deterministic origin, to the present model of uncertainty, and it is indicated how this can be used to to possibly shed some light on the volatility puzzle.     

A TEST OF A GENERAL EQUILIBRIUM STOCK OPTION PRICING MODEL

An empirical version of the Cox, Ingersoll, and Ross (1985a) call option pricing model is derived, assuming execution price uncertainty in the options market. the pricing restrictions come in the form of moment conditions in the option pricing error. These can be estimated and tested using a version of the method of simulated moments (MSM). Simulation estimates, obtained by discretely approximating the risk-neutral processes of the underlying stock price and the interest rate, are substituted for analytically unknown call prices. the asymptotics and other aspects of the MSM estimator are discussed. the model is tested on transaction prices at 15-minute intervals. It substantially outperforms the Black-Scholes model. the empirical success of the Cox-Ingersoll-Ross model implies that the continuous-time interest rate implicit in synchronous transaction quotes of 90-day Treasury-bill futures contracts is an-albeit noisy-proxy for the instantaneous volatility on common stock. the process of the instantaneous volatility is found to be close to nonstationary. It is well approximated by a heteroskedastic unit-root process. With this approximation, the Cox-Ingersoll-Ross model only slightly overprices long-maturity options.     

Asymptotics of the price oscillations of a European call option in a tree model

It is well known that the price of a European vanilla option computed in a binomial tree model converges toward the Black-Scholes price when the time step tends to zero. Moreover, it has been observed that this convergence is of order 1/ n in usual models and that it is oscillatory. In this paper, we compute this oscillatory behavior using asymptotics of Laplace integrals, giving explicitly the first terms of the asymptotics. This allows us to show that there is no asymptotic expansion in the usual sense, but that the rate of convergence is indeed of order 1/ n in the case of usual binomial models since the second term (in     ) vanishes. The next term is of type   C ₂( n )/ n   , with   C ₂( n )  some explicit bounded function of n that has no limit when n tends to infinity.     

CONTINGENT CLAIMS VALUED AND HEDGED BY PRICING AND INVESTING IN A BASIS

Contingent claims with payoffs depending on finitely many asset prices are modeled as elements of a separable Hilbert space. Under fairly general conditions, including market completeness, it is shown that one may change measure to a reference measure under which asset prices are Gaussian and for which the family of Hermite polynomials serves as an orthonormal basis. Basis pricing synthesizes claim valuation and basis investment provides static hedging opportunities. For claims written as functions of a single asset price we infer from observed option prices the implicit prices of basis elements and use these to construct the implied equivalent martingale measure density with respect to the reference measure, which in this case is the Black-Scholes geometric Brownian motion model. Data on S & P 500 options from the Wall Street Journal are used to illustrate the calculations involved. On this illustrative data set the equivalent martingale measure deviates from the Black-Scholes model by relatively discounting the larger price movements with a compensating premia placed on the smaller movements.     

HEDGING WITH ENERGY

In the setting of diffusion models for price evolution, we suggest an easily implementable approximate evaluation formula for measuring the errors in option pricing and hedging due to volatility misspecification. The main tool we use in this paper is a (suitably modified) classical inequality for the L ² norm of the solution, and the derivatives of the solution, of a partial differential equation (the so-called "energy" inequality). This result allows us to give bounds on the errors implied by the use of approximate models for option valuation and hedging and can be used to justify formally some "folk" belief about the robustness of the Black and Scholes model. Surprisingly enough, the result can also be applied to improve pricing and hedging with an approximate model. When statistical or a priori information is available on the "true" volatility, the error measure given by the energy inequality can be minimized w.r.t. the parameters of the approximating model. The method suggested in this paper can help in conjugating statistical estimation of the volatility function derived from flexible but computationally cumbersome statistical models, with the use of analytically tractable approximate models calibrated using error estimates.     

Convergence of the Critical Price In the Approximation of American Options

We consider the American put option in the Black-Scholes model. When the value of the option is computed through numerical methods (such as the binomial method and the finite difference method) the approximation yields an approximate critical price. We prove the convergence of this approximate critical price towards the exact critical price.     

Option pricing for the transformed‐binomial class

This article generalizes the seminal Cox‐Ross‐Rubinstein (1979) binomial option pricing model to all members of the class of transformed‐binomial pricing processes. The investigation addresses issues related with asset pricing modeling, hedging strategies, and option pricing. Formulas are derived for (a) replicating or hedging portfolios, (b) risk‐neutral transformed‐binomial probabilities, (c) limiting transformed‐normal distributions, and (d) the value of contingent claims, including limiting analytical option pricing equations. The properties of the transformed‐binomial class of asset pricing processes are also studied. The results of the article are illustrated with several examples. © 2006 Wiley Periodicals, Inc. Jrl. Fut Mark 26:759–787, 2006     

基于实物期权理论的风险投资项目评价

期权理论在金融领域的应用十分广泛,实物期权理论也被用于企业投资项目评价之中。针对风险投资项目的特殊性,以期权理论为基础,阐述风险投资项目的期权特征,进而以实例分析Black—Scholes期权定价模型在风险投资项目评价中的应用,并与NPV法所得出的结果进行对比,从而达到对投资和管理进行决策的目的。     

A MULTINOMIAL APPROXIMATION FOR AMERICAN OPTION PRICES IN LÉVY PROCESS MODELS

This paper gives a tree-based method for pricing American options in models where the stock price follows a general exponential Lévy process. A multinomial model for approximating the stock price process, which can be viewed as generalizing the binomial model of Cox, Ross, and Rubinstein (1979) for geometric Brownian motion, is developed. Under mild conditions, it is proved that the stock price process and the prices of American-type options on the stock, calculated from the multinomial model, converge to the corresponding prices under the continuous time Lévy process model. Explicit illustrations are given for the variance gamma model and the normal inverse Gaussian process when the option is an American put, but the procedure is applicable to a much wider class of derivatives including some path-dependent options. Our approach overcomes some practical difficulties that have previously been encountered when the Lévy process has infinite activity.     

Optimal No‐Arbitrage Bounds on S&P 500 Index Options and the Volatility Smile

This article shows that the volatility smile is not necessarily inconsistent with the Black–Scholes analysis. Specifically, when transaction costs are present, the absence of arbitrage opportunities does not dictate that there exists a unique price for an option. Rather, there exists a range of prices within which the option's price may fall and still be consistent with the Black–Scholes arbitrage pricing argument. This article uses a linear program (LP) cast in a binomial framework to determine the smallest possible range of prices for Standard & Poor's 500 Index options that are consistent with no arbitrage in the presence of transaction costs. The LP method employs dynamic trading in the underlying and risk‐free assets as well as fixed positions in other options that trade on the same underlying security. One‐way transaction‐cost levels on the index, inclusive of the bid–ask spread, would have to be below six basis points for deviations from Black–Scholes pricing to present an arbitrage opportunity. Monte Carlo simulations are employed to assess the hedging error induced with a 12‐period binomial model to approximate a continuous‐time geometric Brownian motion. Once the risk caused by the hedging error is accounted for, transaction costs have to be well below three basis points for the arbitrage opportunity to be profitable two times out of five. This analysis indicates that market prices that deviate from those given by a constant‐volatility option model, such as the Black–Scholes model, can be consistent with the absence of arbitrage in the presence of transaction costs. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:1151–1179, 2001     

Dynamically consistent alpha‐maxmin expected utility

The alpha‐maxmin model is a prominent example of preferences under Knightian uncertainty as it allows to distinguish ambiguity and ambiguity attitude. These preferences are dynamically inconsistent for nontrivial versions of alpha. In this paper, we derive a recursive, dynamically consistent version of the alpha‐maxmin model. In the continuous‐time limit, the resulting dynamic utility function can be represented as a convex mixture between worst and best case, but now at the local, infinitesimal level. We study the properties of the utility function and provide an Arrow–Pratt approximation of the static and dynamic certainty equivalent. We then derive a consumption‐based capital asset pricing formula and study the implications for derivative valuation under indifference pricing.     

LIQUIDITY IN A BINOMIAL MARKET

We study the binomial version of the illiquid market model introduced by Çetin, Jarrow, and Protter for continuous time and develop efficient numerical methods for its analysis. In particular, we characterize the liquidity premium that results from the model. In Çetin, Jarrow, and Protter, the arbitrage free price of a European option traded in this illiquid market is equal to the classical value. However, the corresponding hedge does not exist and the price is obtained only in L² ‐approximating sense. Çetin, Soner, and Touzi investigated the super‐replication problem using the same supply curve model but under some restrictions on the trading strategies. They showed that the super‐replicating cost differs from the Black–Scholes value of the claim, thus proving the existence of liquidity premium. In this paper, we study the super‐replication problem in discrete time but with no assumptions on the portfolio process. We recover the same liquidity premium as in the continuous‐time limit. This is an independent justification of the restrictions introduced in Çetin, Soner, and Touzi. Moreover, we also propose an algorithm to calculate the option’s price for a binomial market.     

Equilibrium Pricing in the Presence of Cumulative Dividends Following a Diffusion

The paper presents some security market pricing results in the setting of a security market equilibrium in continuous time. The theme of the paper is financial valuation theory when the primitive assets pay out real dividends represented by processes of unbounded variation. In continuous time, when the models are also continuous, this is the most general representation of real dividends, and it can be of practical interest to analyze such models.
Taking as the starting point an extension to continuous time of the Lucas consumption-based model, we derive the equilibrium short-term interest rate, present a new derivation of the consumption-based capital asset pricing model, demonstrate how equilibrium forward and futures prices can be derived, including several examples, and finally we derive the equilibrium price of a European call option in a situation where the underlying asset pays dividends according to an Itô process of unbounded variation. In the latter case we demonstrate how this pricing formula simplifies to known results in special cases, among them the famous Black–Scholes formula and the Merton formula for a special dividend rate process.