A Closer Look at Barrier Exchange Options

A barrier exchange option is an exchange option that is knocked out the first time the prices of two underlying assets become equal. Lindset, S., & Persson, S.‐A. (2006) present a simple dynamic replication argument to show that, in the absence of arbitrage, the current value of the barrier exchange option is equal to the difference in the current prices of the underlying assets and that this pricing formula applies irrespective of whether the option is European or American. In this study, we take a closer look at barrier exchange options and show, despite the simplicity of the pricing formula presented by Lindset, S., & Persson, S.‐A. (2006), that the barrier exchange option in fact involves a surprising array of key concepts associated with the pricing of derivative securities including: put–call parity, barrier in–out parity, static vs. dynamic replication, martingale pricing, continuous vs. discontinuous price processes, and numeraires. We provide valuable intuition behind the pricing formula which explains its apparent simplicity. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 33:29–43, 2013     

THE WIENER–HOPF TECHNIQUE AND DISCRETELY MONITORED PATH‐DEPENDENT OPTION PRICING

Fusai, Abrahams, and Sgarra (2006) employed the Wiener–Hopf technique to obtain an exact analytic expression for discretely monitored barrier option prices as the solution to the Black–Scholes partial differential equation. The present work reformulates this in the language of random walks and extends it to price a variety of other discretely monitored path‐dependent options. Analytic arguments familiar in the applied mathematics literature are used to obtain fluctuation identities. This includes casting the famous identities of Baxter and Spitzer in a form convenient to price barrier, first‐touch, and hindsight options. Analyzing random walks killed by two absorbing barriers with a modified Wiener–Hopf technique yields a novel formula for double‐barrier option prices. Continuum limits and continuity correction approximations are considered. Numerically, efficient results are obtained by implementing Padé approximation. A Gaussian Black–Scholes framework is used as a simple model to exemplify the techniques, but the analysis applies to Lévy processes generally.     

Piecewise linear double barrier options

A piecewise linear double barrier option generalizes classical double barrier options because of its versatility in designing various double boundaries. This paper discusses how to price piecewise linear double barrier options. To this purpose, we derive the probability that an underlying process does not cross a given piecewise linear double barrier, where the underlying process follows the Brownian motion of piecewise constant drift. Using the established non-crossing probability, we provide the explicit pricing formulas of piecewise linear double barrier options and show how the shape of a double barrier affects the option prices through extensive numerical experiments.     

Option pricing with orthogonal polynomial expansions

We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein–Stein, and Hull–White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accurately as the Fourier‐transform‐based method in the nested affine cases. We also derive and numerically validate series representations for option Greeks. We depict an extension of our approach to exotic options whose payoffs depend on a finite number of prices.     

SELF-DECOMPOSABILITY AND OPTION PRICING

The risk-neutral process is modeled by a four parameter self-similar process of independent increments with a self-decomposable law for its unit time distribution. Six different processes in this general class are theoretically formulated and empirically investigated. We show that all six models are capable of adequately synthesizing European option prices across the spectrum of strikes and maturities at a point of time. Considerations of parameter stability over time suggest a preference for two of these models. Currently, there are several option pricing models with 6–10 free parameters that deliver a comparable level of performance in synthesizing option prices. The dimension reduction attained here should prove useful in studying the variation over time of option prices.     

HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK–MERTON–SCHOLES?

We compare the option pricing formulas of Louis Bachelier and Black–Merton–Scholes and observe—theoretically as well as for Bachelier's original data—that the prices coincide very well. We illustrate Louis Bachelier's efforts to obtain applicable formulas for option pricing in pre-computer time. Furthermore we explain—by simple methods from chaos expansion—why Bachelier's model yields good short-time approximations of prices and volatilities.     

THE BLACK-SCHOLES EQUATION REVISITED: ASYMPTOTIC EXPANSIONS AND SINGULAR PERTURBATIONS

In this paper, novel singular perturbation techniques are applied to price European, American, and barrier options. Employment of these methods leads to a significant simplification of the problem in all cases, by reducing the number of parameters. For American options, the valuation problem is reduced to a procedure that may be performed on a rudimentary handheld calculator. The method also sheds light on the evolution of option prices for all of the cases considered, the results being particularly illuminating for American and barrier options.     

Why do expiring futures and cash prices diverge for grain markets?

In recent years, cash and futures prices have failed to converge at expiration for selected corn, soybean, and wheat commodity contracts. This lack of convergence raises questions about the effectiveness of arbitrage activities, and increases concerns about the usefulness of these contracts for hedging. We describe the delivery process for these contracts, and show that it embeds a valuable real option on the long side—the option to exchange the deliverable for another futures contract. As the relative volatility of cash and futures prices increases, this option increases in value, which disconnects the cash market from the deliverable instrument in a futures contract. Our estimates of this option's value show that it may create significant price divergence. We parameterize an option pricing model using data on these three commodities from 2000 to 2008 and show that the option model fits closely to recent episodes of non‐convergence, which lends support to the importance of real option effects. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark

相似文献   

ANALYTIC APPROXIMATIONS FOR MULTI‐ASSET OPTION PRICING

We derive general analytic approximations for pricing European basket and rainbow options on N assets. The key idea is to express the option’s price as a sum of prices of various compound exchange options, each with different pairs of subordinate multi‐ or single‐asset options. The underlying asset prices are assumed to follow lognormal processes, although our results can be extended to certain other price processes for the underlying. For some multi‐asset options a strong condition holds, whereby each compound exchange option is equivalent to a standard single‐asset option under a modified measure, and in such cases an almost exact analytic price exists. More generally, approximate analytic prices for multi‐asset options are derived using a weak lognormality condition, where the approximation stems from making constant volatility assumptions on the price processes that drive the prices of the subordinate basket options. The analytic formulae for multi‐asset option prices, and their Greeks, are defined in a recursive framework. For instance, the option delta is defined in terms of the delta relative to subordinate multi‐asset options, and the deltas of these subordinate options with respect to the underlying assets. Simulations test the accuracy of our approximations, given some assumed values for the asset volatilities and correlations. Finally, a calibration algorithm is proposed and illustrated.     

PRICING CHAINED OPTIONS WITH CURVED BARRIERS

This paper studies barrier options which are chained together, each with payoff contingent on curved barriers. When the underlying asset price hits a primary curved barrier, a secondary barrier option is given to a primary barrier option holder. Then if the asset price hits another curved barrier, a third barrier option is given, and so on. We provide explicit price formulas for these options when two or more barrier options with exponential barriers are chained together. We then extend the results to the options with general curved barriers.     

Recursive Formula for Arithmetic Asian Option Prices

I derive a recursive formula for arithmetic Asian option prices with finite observation times in semimartingale models. The method is based on the relationship between the risk‐neutral expectation of the quadratic variation of the return process and European option prices. The computation of arithmetic Asian option prices is straightforward whenever European option prices are available. Applications with numerical results under the Black–Scholes framework and the exponential Lévy model are proposed. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 34:220–234, 2014     

Bounds on European Option Prices under Stochastic Volatility

In this paper we consider the range of prices consistent with no arbitrage for European options in a general stochastic volatility model. We give conditions under which the infimum and the supremum of the possible option prices are equal to the intrinsic value of the option and to the current price of the stock, respectively, and show that these conditions are satisfied in most of the stochastic volatility models from the financial literature. We also discuss properties of Black–Scholes hedging strategies in stochastic volatility models where the volatility is bounded.     

ANAYTICAL SOLUTIONS FOR THE PRICING OF AMERICAN BOND AND YIELD OPTIONS1

In this paper we use the Cox, Ingersoll, and Ross (1985b) single-factor, term structure model and extend it to the pricing of American default-free bond puts. We provide a quasi-analytical formula for these option prices based on recently established mathematical results for Bessel bridges, coupled with the optimal stopping time method. We extend our results to another interest rate contingent claim and provide a quasi-analytical solution for American yield option prices which illustrates the flexibility of our framework.     

A Continuity Correction for Discrete Barrier Options

The payoff of a barrier option depends on whether or not a specified asset price, index, or rate reaches a specified level during the life of the option. Most models for pricing barrier options assume continuous monitoring of the barrier; under this assumption, the option can often be priced in closed form. Many (if not most) real contracts with barrier provisions specify discrete monitoring instants; there are essentially no formulas for pricing these options, and even numerical pricing is difficult. We show, however, that discrete barrier options can be priced with remarkable accuracy using continuous barrier formulas by applying a simple continuity correction to the barrier. The correction shifts the barrier away from the underlying by a factor of exp(bet sig sqrt dt), where bet approx 0.5826, sig is the underlying volatility, and dt is the time between monitoring instants. The correction is justified both theoretically and experimentally.     

Dividend‐Rollover Effect and the Ad Hoc Black‐Scholes Model

One of the most widely used option‐valuation models among practitioners is the ad hoc Black‐Scholes (AHBS) model. The main contribution of this study is methodological. We carefully consider three dividend strategies (No dividend, Implied‐forward dividend, and Actual dividend) for the AHBS model to investigate their effect on pricing errors. We suggest a new dividend strategy, implied‐forward dividend, which incorporates expectational information on dividends embedded in option prices. We demonstrate that our implied‐forward dividend strategy produces more consistent estimates between in‐sample market and model option prices. More importantly our new implied‐forward dividend strategy makes more accurate out‐of‐sample forecasts for one‐day or one‐week ahead prices. Second, we document that both a “Return‐volatility” Smile and a “Return‐pricing Error” Smile exist. From these return characteristics, we make two conclusions: (1) the return dependency of implied volatility is an important explanatory variable and should be controlled to reduce the pricing error of an AHBS model, and (2) it is important for the hedging horizon to be based on return size, that is, the larger the contemporaneous return, the more frequent an option issuer must rebalance the option's hedge. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 32:742‐772, 2012     

PROPERTIES OF OPTION PRICES IN MODELS WITH JUMPS

We study convexity and monotonicity properties of option prices in a model with jumps using the fact that these prices satisfy certain parabolic integro–differential equations. Conditions are provided under which preservation of convexity holds, i.e., under which the value, calculated under a chosen martingale measure, of an option with a convex contract function is convex as a function of the underlying stock price. The preservation of convexity is then used to derive monotonicity properties of the option value with respect to the different parameters of the model, such as the volatility, the jump size, and the jump intensity.     

MCMC ESTIMATION OF LÉVY JUMP MODELS USING STOCK AND OPTION PRICES

We examine the performances of several popular Lévy jump models and some of the most sophisticated affine jump‐diffusion models in capturing the joint dynamics of stock and option prices. We develop efficient Markov chain Monte Carlo methods for estimating parameters and latent volatility/jump variables of the Lévy jump models using stock and option prices. We show that models with infinite‐activity Lévy jumps in returns significantly outperform affine jump‐diffusion models with compound Poisson jumps in returns and volatility in capturing both the physical and risk‐neutral dynamics of the S&P 500 index. We also find that the variance gamma model of Madan, Carr, and Chang with stochastic volatility has the best performance among all the models we consider.     

Arbitrage Values Generally Depend On A Parametric Rate of Return

Let X denote a positive Markov stochastic integral, and let S ( t , μ) = exp(μ t ) X ( t ) represent the price of a security at time t with infinitesimal rate of return μ. Contingent claim (option) pricing formulas typically do not depend on μ. We show that if a contingent claim is not equivalent to a call option having exercise price equal to zero, then security prices having this property—option prices do not depend on μ—must satisfy: for some V (0, T ), In( S ( t , μ) X ( V )) is Gaussian on a time interval [ V, T ], and hence S ( t , μ) has independent observed returns. With more assumptions, V = 0, and there exist equivalent martingale measures.     

The Chinese warrant bubble: A fundamental analysis

We investigate the information content in Chinese warrant prices based on an option pricing framework that incorporates short‐selling and margin‐trading constraints in the underlying stock market. We show that Chinese warrant prices can be explained under this pricing framework. On the basis of this new model, we develop a price deviation measure to quantify stock market investors' unobserved demand for short selling or margin trading due to market constraints. We find that warrant‐price deviations are driven by underlying stock valuation to a great extent. Chinese warrant prices, save for the time around expiration dates, are better characterized as derivatives than as pure bubbles.     

Lévy betas: Static hedging with index futures

This study considers calibration to forward‐looking betas by extracting information on equity and index options from prices using Lévy models. The resulting calibrated betas are called Lévy betas. The objective of the proposed approach is to capture market expectations for future betas through option prices, as betas estimated from historical data may fail to reflect structural change in the market. By assuming a continuous‐time capital asset pricing model (CAPM) with Lévy processes, we derive an analytical solution to index and stock options, thus permitting the betas to be implied from observed option prices. One application of Lévy betas is to construct a static hedging strategy using index futures. Employing Hong Kong equity and index option data from September 16, 2008 to October 15, 2009, we show empirically that the Lévy betas during the sub‐prime mortgage crisis period were much more volatile than those during the recovery period. We also find evidence to suggest that the Lévy betas improve static hedging performance relative to historical betas and the forward‐looking betas implied by a stochastic volatility model.