Valuación de opciones de tipo de cambio asumiendo distribuciones α-estables

This paper aims to present the valuation of options using the Black-Scholes method assuming α-stable distributions as an alternative option valuation in the Mexican market. The use of α-stable distributions for modelling financial series allows to overcome the classical valuation main weakness which assumes normality, by capturing the presence of heavy tails and asymmetry in financial time series. One of the main results is the price differential between the two models and the effect of alpha and beta parameters on prices; to show the difference valuation is made of a call option and a put option for the peso-dollar exchange rate. Likewise, basic sensitivity measurements of options (delta, gamma, and rho) were made and the effect of the stability parameter (α) was made on the implied volatility of options assuming the α-stable price as the market price.     

Exact inference and optimal invariant estimation for the stability parameter of symmetric α-stable distributions

Hill estimation (Hill, 1975), the most widespread method for estimating tail thickness of heavy-tailed financial data, suffers from two drawbacks. One is that the optimal number of tail observations to use in the estimation is a function of the unknown tail index being estimated, which diminishes the empirical relevance of the Hill estimation. The other is that the hypothesis test of the underlying data lying in the domain of attraction of an α-stable law (α < 2) or of a normal law (α ≥ 2) for finite samples, is performed on the basis of the asymptotic distribution, which can be different from those for finite samples. In this paper, using the Monte Carlo technique, we propose an exact test method for the stability parameter of α-stable distributions which is based on the Hill estimator, yet is able to provide exact confidence intervals for finite samples. Our exact test method automatically includes an estimation procedure which does not need the assumption of a known number of observations on the distributional tail. Empirical applications demonstrate the advantages of our new method in comparison with the Hill estimation.     

Multilevel Monte Carlo for exponential Lévy models

We apply the multilevel Monte Carlo method for option pricing problems using exponential Lévy models with a uniform timestep discretisation. For lookback and barrier options, we derive estimates of the convergence rate of the error introduced by the discrete monitoring of the running supremum of a broad class of Lévy processes. We then use these to obtain upper bounds on the multilevel Monte Carlo variance convergence rate for the variance gamma, NIG and \(\alpha\)-stable processes. We also provide an analysis of a trapezoidal approximation for Asian options. Our method is illustrated by numerical experiments.     

Los procesos α-estables y su relación con el exponente de autosimilitud: paridades de los tipos de cambio dólar estadounidense,dólar canadiense,euro y yen

In this research we analyze the performance of the exchange rates of USA Dollar, Canadian Dollar, Euro and Yen; we estimate the basic statistics, α-stable parameters, we performed tests of goodness fit Kolmogorov-Smirnov, Anderson-Darling and Lilliefors; we estimate self-similarity exponents and we performed t y F tests, ruling that the series of the exchange rates are multi-fractal; we estimate confidence intervals of the exchange rates and we conclude that the estimated α-stable distributions are more efficient than the gaussian distribution to quantify market risks and the series are self-similar; by the ? index we infer the risk of events and we indicate that exchange rates are anti-persistent, have mean reversión, short-term memory, negative correlation and high risk in the short and medium term; the estimation and validation of α-stable distributions and the exponent of self-similarity are important for pricing and the creation of innovative investment instruments by financial engineering, risk management and derivatives pricing.     

A unified approach for the pricing of options relating to averages

In this paper, we consider generalized Asian options and propose a unified approximation method for the pricing of such options when the underlying process is a diffusion. Through numerical examples, we show that our approximation method is accurate enough to be used in practice for the pricing of any type of Asian options that has been treated separately in the literature. Comparisons are made with the existing methods in the literature to support the usefulness of our method.     

Valuación de opciones sobre subyacentes con rendimientos α-estables

In this work, we analyze the log-stable option pricing model, we estimate the parameters of the distribution of the peso-dollar exchange depreciation rate through the methods: 1) maximum likelihood, 2) tabulated quantiles of α-stable distributions and 3) regression on the sample characteristic function; we conducted a qualitative analysis to show the quality of the distribution’s fit and through a quantitative analysis we chose the best α-parameters estimation and we compare the McCulloch (2003) log-stable option pricing model with the Black and Scholes (1973) log-normal model and a MexDer’s prices vector; finally, we show that the log-stable model has advantages over the log-normal model.     

Improved method for static replication under the CEV model

This paper studies the hedging performance of static replication approach proposed by Derman, Ergener, and Kani (DEK, 1995) for continuous barrier options under the constant elasticity of variance (CEV) model of Cox (1975) and Cox and Ross (1976), and then focuses on how to improve the DEK method. Given the time-varying volatility feature of the CEV model, I show that the DEK static hedging portfolio exhibits serious mismatches of the theta values on the barrier, particularly when one of the component options of the portfolio is around the neighborhood of expiration, which primarily explains why static portfolio values are greater than zero on the barrier except at the matching points. The DEK method (hereafter, the improved DEK method) is improved by re-forming a static replication portfolio consisting of plain vanilla options and cash-or-nothing binary options with different maturities to match both the value-matching condition and the theta-matching condition on the barrier. The numerical analyses indicate that under the CEV model, the improved DEK method significantly reduces replication errors for an up-and-out call option.     

The α-stable processes and their relationship with the exponent of self-similarity: Exchange rates of USA Dollar,Canadian Dollar,Euro and Yen

This research work analyzes the yields of the exchange rate parities of the American dollar, Canadian dollar, Euro, and Yen; estimates the basic statistics and the α-stables; carries out the Kolmogorov–Smirnov, Anderson–Darling, and Lilliefors goodness of fit tests; estimates the self-similar exponents and carries out the t and F tests, ruling out that the series of parities are multifractal. It also estimates the confidence intervals of the exchange rate parities and concludes that the estimated α-stable distributions are more efficient than the Gaussian distribution to quantify the risks of the market, and that the series are self-similar. Through the ? index, we can infer the risk of the events, indicating that the parities are anti-persistent and thus have short-term memory, mean reversion, and a negative correlation with the high risk in the short and medium term. The estimation and validation of the α-stable distributions and the self-similar exponent are important in the evaluation and creation of innovative investment instruments through financial engineering, risk administration, and the evaluation of derived products.     

Calibration to American options: numerical investigation of the de-Americanization method

American options are the reference instruments for the model calibration of a large and important class of single stocks. For this task, a fast and accurate pricing algorithm is indispensable. The literature mainly discusses pricing methods for American options that are based on Monte Carlo, tree and partial differential equation methods. We present an alternative approach that has become popular under the name de-Americanization in the financial industry. The method is easy to implement and enjoys fast run-times (compared to a direct calibration to American options). Since it is based on ad hoc simplifications, however, theoretical results guaranteeing reliability are not available. To quantify the resulting methodological risk, we empirically test the performance of the de-Americanization method for calibration. We classify the scenarios in which de-Americanization performs very well. However, we also identify the cases where de-Americanization oversimplifies and can result in large errors.     

Asymptotic and exact pricing of options on variance

We consider the pricing of derivatives written on the discretely sampled realized variance of an underlying security. In the literature, the realized variance is usually approximated by its continuous-time limit, the quadratic variation of the underlying log-price. Here, we characterize the small-time limits of options on both objects. We find that the difference between them strongly depends on whether or not the stock price process has jumps. Subsequently, we propose two new methods to evaluate the prices of options on the discretely sampled realized variance. One of the methods is approximative; it is based on correcting prices of options on quadratic variation by our asymptotic results. The other method is exact; it uses a novel randomization approach and applies Fourier?CLaplace techniques. We compare the methods and illustrate our results by some numerical examples.     

Static hedging and pricing American options

This paper utilizes the static hedge portfolio (SHP) approach of Derman et al. [Derman, E., Ergener, D., Kani, I., 1995. Static options replication. Journal of Derivatives 2, 78–95] and Carr et al. [Carr, P., Ellis, K., Gupta, V., 1998. Static hedging of exotic options. Journal of Finance 53, 1165–1190] to price and hedge American options under the Black-Scholes (1973) model and the constant elasticity of variance (CEV) model of Cox [Cox, J., 1975. Notes on option pricing I: Constant elasticity of variance diffusion. Working Paper, Stanford University]. The static hedge portfolio of an American option is formulated by applying the value-matching and smooth-pasting conditions on the early exercise boundary. The results indicate that the numerical efficiency of our static hedge portfolio approach is comparable to some recent advanced numerical methods such as Broadie and Detemple [Broadie, M., Detemple, J., 1996. American option valuation: New bounds, approximations, and a comparison of existing methods. Review of Financial Studies 9, 1211–1250] binomial Black-Scholes method with Richardson extrapolation (BBSR). The accuracy of the SHP method for the calculation of deltas and gammas is especially notable. Moreover, when the stock price changes, the recalculation of the prices and hedge ratios of the American options under the SHP method is quick because there is no need to solve the static hedge portfolio again. Finally, our static hedging approach also provides an intuitive derivation of the early exercise boundary near expiration.     

Brealey,Myers, and Allen on Real Options

Real options are valuable sources of flexibility that are either inherent in, or can be built into, corporate assets. The value of such options are generally not captured by the standard discounted cash flow (DCF) approach, but can be estimated using a variant of financial option pricing techniques. This article provides an overview of the basics of real option valuation by examining four important kinds of real options:

• 1 The option to make follow‐on investments. Companies often cite “strategic” value when taking on negative‐NPV projects. A close look at the payoffs from such projects reveals call options on follow‐on projects in addition to the immediate cash flows from the projects. Today's investments can generate tomorrow's opportunities.
• 2 The option to wait (and learn) before investing. This is equivalent to owning a call option on the investment project. The call is exercised when the firm commits to the project. But often it's better to defer a positive‐NPV project in order to keep the call alive. Deferral is most attractive when uncertainty is great and immediate project cash flows—which are lost or postponed by waiting—are small.
• 3 The option to abandon. The option to abandon a project provides partial insurance against failure. This is a put option; the put's exercise price is the value of the project's assets if sold or shifted to a more valuable use.
• 4 The option to vary the firm's output or its production methods. Companies often build flexibility into their production facilities so that they can use the cheapest raw materials or produce the most valuable set of outputs. In this case they effectively acquire the option to exchange one asset for another.

The authors also make the point that, in most applications, real‐option valuation methods are a complement to, not a substitute for, the DCF method. Indeed DCF, which is best suited to and usually sufficient for safe investments and “cash cow” assets, is typically the starting point for real‐option analyses. In such cases, DCF is used to generate the values of the “underlying assets”—that is, the projects when viewed without their options or sources of flexibility.     

Improved lower and upper bound algorithms for pricing American options by simulation

This paper introduces new variance reduction techniques and computational improvements to Monte Carlo methods for pricing American-style options. For simulation algorithms that compute lower bounds of American option values, we apply martingale control variates and introduce the local policy enhancement, which adopts a local simulation to improve the exercise policy. For duality-based upper bound methods, specifically the primal–dual simulation algorithm, we have developed two improvements. One is sub-optimality checking, which saves unnecessary computation when it is sub-optimal to exercise the option along the sample path; the second is boundary distance grouping, which reduces computational time by skipping computation on selected sample paths based on the distance to the exercise boundary. Numerical results are given for single asset Bermudan options, moving window Asian options and Bermudan max options. In some examples the computational time is reduced by a factor of several hundred, while the confidence interval of the true option value is considerably tighter than before the improvements.     

The Credit Risk and Pricing of OTC Options

In the over-the-counter (OTC) markets, the options traded are always subject to credit risk. Therefore the counterparty’s credit risk is a striking factor when pricing options, whereas it is not considered in the classic Black-Scholes models. Based on the first passage time models, this paper develops the credit risk and valuation model for the European options in the OTC markets, incorporating a practical default trigger mechanism. The default probability and the pricing formulae of the OTC options are obtained by using partial differential equation (PDE) techniques, especially Green’s function.     

Does vega-neutral options trading contain information?

This study suggests a novel approach for decomposing net options demands into the options order imbalances with and without volatility risk. By analyzing a high-frequency index futures and options dataset, we examine the information content of (i) the direction-motivated order imbalance induced by a single option type, which is exposed to volatility risk, and (ii) that constructed by both calls and puts, which is vega-neutral. The aggregate options order imbalance does not convey information after controlling for futures market trading. However, the intraday options order imbalance by trading without volatility risk significantly predicts spot index returns, though its longer-horizon forecasting ability is relatively weak because of a possible cross-market hedging effect. The predictive abilities of informed foreigners’ trades and out-of-the-money options trading are prominent. Our empirical results suggest that the vega-neutral options trading conveys additional information distinct from the futures order imbalance.     

Incorporating real-options analysis into the accounting curriculum

In this paper, we argue that accounting curricula should be expanded to cover the topic of real options. Our argument relies on reference to the [American Institute of Certified Public Accountants (AICPA) (1999) (Core Competency Framework, New York, NY: AICPA <http://ceae.aicpa.org/Resources/Education+and+Curriculum+Development/Core+Competency+Framework+and+Educational+Competency+Assessment+Web+Site/> Accessed 21.08.08], the framework for curriculum change espoused by [Arya, A., Fellingham, J. C., & Schroeder, D. A. (2003). An academic curriculum proposal. Issues in Accounting Education, 18(1) 29–35], a global study of core competencies for management accountants [International Federation of Accountants (IFAC), (2002). Competency profiles for management accounting practice and practitioners. New York, NY: International Federation of Accountants], a global capital-budgeting “best practices statement” [International Federation of Accountants (IFAC), (2008). International good practice guidance: Project appraisal using discounted cash flow. New York, NY: International Federation of Accountants], current specifications of the CMA exam [Institute of Management Accountants (IMA), (2008). Certified management accountant (CMA) learning outcome statements (effective 07/01/04), updated 07/2008. <http://www.imanet.org/pdf/CMA%20%20LOS.pdf> Accessed 29.10.08.], and elements of the Albrecht and Sack report [Albrecht, W. S., & Sack, R. J. (2000). Accounting education: Charting the course through a perilous future. Accounting education series, Vol. 16. Sarasota, FL: American Accounting Association]. We make special reference to the linkage of the topic of real options to two broad educational goals: decision-modeling and risk analysis. Existing resources that accounting faculty can use to incorporate real options into the curriculum are limited. As a response, we provide an extended example that accounting educators can use to cover the topic of real options. This example uses a set of binomial trees (one for cash inflows and another for cash outflows). The step-by-step approach presented in this paper allows students without a technical/mathematical background to extend discounted-cash-flow (DCF) decision models (e.g., NPV) to incorporate real options that are embedded in proposed investment projects.     

Stock option use by Australian IPOs

This paper examines the use of stock options by Australian IPOs. Stock options in this setting exhibit heterogeneity of ‘in the moneyness’ attributes. We can therefore identify incentive ‘out of the money’ options and reward ‘at or in the money’ options and study their usage. The setting also features options granted to employees (CEO, executives, non-executive directors, and managers) and external parties (e.g., underwriters, lenders, seed capitalists, and promoters). We examine the motives for granting stock options with different ‘in the moneyness’ attributes to employees and external parties, as well as the performance outcomes. Our results suggest ‘in the moneyness’ attributes of options vary according to uncertainties about growth options, general operating risks, and/or agency risks relating to the use of IPO proceeds; however, these relations do not translate into robust links from the option grants to future performance.     

Options on realized variance by transform methods: a non-affine stochastic volatility model

In this paper we study the pricing and hedging of options on realized variance in the 3/2 non-affine stochastic volatility model by developing efficient transform-based pricing methods. This non-affine model gives prices of options on realized variance that allow upward-sloping implied volatility of variance smiles. Heston's model [Rev. Financial Stud., 1993, 6, 327–343], the benchmark affine stochastic volatility model, leads to downward-sloping volatility of variance smiles—in disagreement with variance markets in practice. Using control variates, we propose a robust method to express the Laplace transform of the variance call function in terms of the Laplace transform of the realized variance. The proposed method works in any model where the Laplace transform of realized variance is available in closed form. Additionally, we apply a new numerical Laplace inversion algorithm that gives fast and accurate prices for options on realized variance, simultaneously at a sequence of variance strikes. The method is also used to derive hedge ratios for options on variance with respect to variance swaps.     

Pricing and static hedging of American-style options under the jump to default extended CEV model

This paper prices (and hedges) American-style options through the static hedge approach (SHP) proposed by Chung and Shih (2009) and extends the literature in two directions. First, the SHP approach is generalized to the jump to default extended CEV (JDCEV) model of Carr and Linetsky (2006), and plain-vanilla American-style options on defaultable equity are priced. The robustness and efficiency of the proposed pricing solutions are compared with the optimal stopping approach offered by Nunes (2009), under both the JDCEV framework and the nested constant elasticity of variance (CEV) model of Cox (1975), using different elasticity parameter values. Second, the early exercise boundary near expiration is derived under the JDCEV model.