Black-Scholes Framework Research Articles

A Finite Difference Model for Valuation of Employee Stock Options

Employee stock option grants are a common incentive for employees and are a key remuneration device. These options differ from ordinary options in that they cannot be traded nor hedged. Nevertheless, the work of Carpenter enables one to price these options within a Black-Scholes framework, with one additional parameter calibrated from historical data. We develop a model where the price of the grant obeys the Black-Scholes differential equation with two additional parameters: one which controls the rate at which employees forfeit unvested options, and another which controls the rate at which employees exercise vested options. We implement a finite difference scheme for computation of the option values derived from this model.

Pricing financial claims contingent upon an underlying asset monitored at discrete times

Exotic option contracts typically specify a contingency upon an underlying asset price monitored at a discrete set of times. Yet, techniques used to price such options routinely assume continuous monitoring leading to often substantial price discrepancies. A brief review of relevant option-pricing methods is presented. The pricing problem is transformed into one of Wiener–Hopf type using a z-transform in time and a Fourier transform in the logarithm of asset prices. The Wiener–Hopf technique is used to obtain probabilistic identities for the related random walks killed by an absorbing boundary. An accurate and efficient approximation is obtained using Padé approximants and an approximate inverse z-transform based on the trapezoidal rule. For simplicity, European barrier options in a Gaussian Black–Scholes framework are used to exemplify the technique (for which exact analytic expressions are obtained). Extensions to different option contracts and options driven by other Lévy processes are discussed.

A Closed Form Approach to the Valuation and Hedging of Basket and Spread Option

The lognormal diffusion model for the returns on the underlying asset is a key assumption of Black-Scholes and many other derivatives models. One advantage of the lognormal is that it is impossible for the price to become negative in the model; another is that asset prices exhibit positive skewness, as is apparent in real world securities. However, when an option is written on a spread or on a portfolio containing both long and short positions in assets that individually follow lognormal diffusions, these properties may not be appropriate. In this paper, Borovkova, Permana, and v.d. Weide propose modeling such portfolios using negative and/or shifted lognormal distributions. This allows a much better fit to the empirical distributions for such cases, while still retaining the closed-form solutions of the Black-Scholes framework for option values and Greek letter risks. The effectiveness and simplicity of the technique are demonstrated by Monte Carlo simulations with a variety of basket options. <b>TOPICS:</b>Options, simulations

Trading Basis Part 1

The Credit Default Swap (CDS) market is still fairly young. As a result, there are some areas that have not yet been fully investigated. One of those areas is the behavior of the Cheapest to Deliver Option (CTD) embedded within the CDS contract. Fluctuations in the CTD translate into changes in the ?Basis?, the visible difference between the credit spread of a bond and its CDS. The aim of this paper is to develop a trading strategy that profits from fluctuations in the value of the CTD. To do this, we try to shed some light on the factors affecting the CTD within Emerging Markets (EM) Sovereigns Bonds. We propose a stylized closed-form Black-Scholes framework that identifies those factors which are likely to be most important. We run statistical tests to verify the validity of this approach and develop a Monte Carlo simulation model to put a value on the CTD. We find that the statistical tests give credence to the Black-Scholes formulation and are corroborated by the implications of the Monte Carlo model. The resulting trading strategy is highly profitable. We also discuss some potentially important ancillary results, relating to a method for obtaining the implied market values for both bond default probabilities and recovery rates.

Trading Basis Part 2

The Credit Default Swap (CDS) market is still fairly young. As a result, there are some areas that have not yet been fully investigated. One of those areas is the behavior of the Cheapest to Deliver Option (CTD) embedded within the CDS contract. Fluctuations in the CTD translate into changes in the ?Basis?, the visible difference between the credit spread of a bond and its CDS. The aim of this paper is to develop a trading strategy that profits from fluctuations in the value of the CTD. To do this, we try to shed some light on the factors affecting the CTD within Emerging Markets (EM) Sovereigns Bonds. We propose a stylized closed-form Black-Scholes framework that identifies those factors which are likely to be most important. We run statistical tests to verify the validity of this approach and develop a Monte Carlo simulation model to put a value on the CTD. We find that the statistical tests give credence to the Black-Scholes formulation and are corroborated by the implications of the Monte Carlo model. The resulting trading strategy is highly profitable. We also discuss some potentially important ancillary results, relating to a method for obtaining the implied market values for both bond default probabilities and recovery rates.

Partially exact and bounded approximations for arithmetic Asian options

This paper considers the pricing of European Asian options in the Black-Scholes framework. All approaches we consider are readily extendable to the case of an Asian basket option. We consider three methods for evaluating the price of an Asian option, and contribute to all three. Firstly, we show the link between the approaches of Rogers and Shi [1995], Andreasen [1999], Hoogland and Neumann [2000] and Veceř [2001]. For the latter formulation we propose two reductions, which increase the numerical stability and reduce the calculation time. Secondly, we show how a closed-form expression can be derived for Curran’s and Rogers and Shi’s lower bound for the general case of multiple underlyings. Thirdly, we considerably sharpen Thompson’s [1999a,b] upper bound such that it is tighter than all known upper bounds. Finally, we consider analytical approximations and combine the traditional moment matching approximations with Curran’s conditioning approach. The resulting class of partially exact and bounded approximations can be proven to lie between a sharp lower and upper bound. In numerical examples we demonstrate that they outperform all current state-of-the-art bounds and approximations.

OPTIMAL CONSTANT-REBALANCED PORTFOLIO INVESTMENT STRATEGIES FOR DYNAMIC PORTFOLIO SELECTION

In this paper we propose a variant of the continuous-time Markowitz mean-variance model by incorporating the Earnings-at-Risk measure in the portfolio optimization problem. Under the Black-Scholes framework, we obtain closed-form expressions for the optimal constant-rebalanced portfolio (CRP) investment strategy. We also derive explicitly the corresponding mean-EaR efficient portfolio frontier, which is a generalization of the Markowitz mean-variance efficient frontier.

Numerical solution of variational inequalities for pricing Asian options by higher order Lagrange–Galerkin methods

Asian options prices can be modelled in the Black–Scholes framework leading to two-factor models depending on the asset price, the average of the asset price and the time. They can also involve inequality constraints, as in the case of Amerasian options, leading to variational inequalities (VI). In the first section, we completely describe the pricing model for fixed-strike Eurasian and Amerasian options and list some properties satisfied by the option value function. Then, since no solutions in closed form are known, we deal with the numerical solution of the above problems proposing a general methodology: an iterative algorithm for the VI, combined with higher order Lagrange–Galerkin methods for partial differential equations. Finally, numerical results are shown.

An exact analytical solution for discrete barrier options

In the present paper we provide an analytical solution for pricing discrete barrier options in the Black-Scholes framework. We reduce the valuation problem to a Wiener-Hopf equation that can be solved analytically. We are able to give explicit expressions for the Greeks of the contract. The results from our formulae are compared with those from other numerical methods available in the literature. Very good agreement is obtained, although evaluation using the present method is substantially quicker than the alternative methods presented.

Double-Exponential Fast Gauss Transform Algorithms for Pricing Discrete Lookback Options

This paper presents fast and accurate algorithms for computing the prices of discretely sampled lookback options. Under the Black–Scholes framework, the pricing of a discrete lookback option can be reduced to a series of convolutions of a function with the Gaussian distribution. Using this fact, an effcient algorithm, which computes these convolutions by a combination of the double-exponential integration formula and the fast Gauss transform, has been proposed recently. We extend this algorithm to lookback options under Merton’s jump-diffusion model and American lookback options. Numerical experiments show that our method is much faster and more accurate than conventional methods for lookback options under Merton’s model. For American lookback options, our method outperforms conventional methods when required accuracy is relatively high.

A Double-Exponential Fast Gauss Transform Algorithm for Pricing Discrete Path-Dependent Options

This paper develops algorithms for the pricing of discretely sampled barrier, lookback, and hindsight options and discretely exercisable American options. Under the Black-Scholes framework, the pricing of these options can be reduced to evaluation of a series of convolutions of the Gaussian distribution and a known function. We compute these convolutions efficiently using the double-exponential integration formula and the fast Gauss transform. The resulting algorithms have computational complexity of O(nN), where the number of monitoring/exercise dates is n and the number of sample points at each date is N, and our results show the error decreases exponentially with N. We also extend the approach and provide results for Merton’s lognormal jump-diffusion model.

ARBITRAGE IN FRACTAL MODULATED BLACK–SCHOLES MODELS WHEN THE VOLATILITY IS STOCHASTIC

In this paper an arbitrage strategy is constructed for the modified Black–Scholes model driven by fractional Brownian motion or by a time changed fractional Brownian motion, when the volatility is stochastic. This latter property allows the heavy tailedness of the log returns of the stock prices to be also accounted for in addition to the long range dependence introduced by the fractional Brownian motion. Work has been done previously on this problem for the case with constant "volatility" and without a time change; here these results are extended to the case of stochastic volatility models when the modulator is fractional Brownian motion or a time change of it. (Volatility in fractional Black–Scholes models does not carry the same meaning as in the classic Black–Scholes framework, which is made clear in the text.) Since fractional Brownian motion is not a semi-martingale, the Black–Scholes differential equation is not well-defined sense for arbitrary predictable volatility processes. However, it is shown here that any almost surely continuous and adapted process having zero quadratic variation can act as an integrator over functions of the integrator and over the family of continuous adapted semi-martingales. Moreover it is shown that the integral also has zero quadratic variation, and therefore that the integral itself can be an integrator. This property of the integral is crucial in developing the arbitrage strategy. Since fractional Brownian motion and a time change of fractional Brownian motion have zero quadratic variation, these results are applicable to these cases in particular. The appropriateness of fractional Brownian motion as a means of modeling stock price returns is discussed as well.

Partially Exact and Bounded Approximations for Arithmetic Asian Options

This paper considers the pricing of European Asian options in the Black-Scholes framework. All approaches we consider are readily extendable to the case of an Asian basket option. Firstly we consider the partial differential equation approach to the pricing of Asian options. We show the link between the approaches of Rogers and Shi [1995], Andreasen [1999], Hoogland and Neumann [2000] and Vecer [2001]. For the latter two formulations we propose two reductions, which increase the numerical stability and reduce the calculation time. Secondly, we show how a closed-form expression can be derived for Rogers and Shi's lower bound for the general case of multiple underlyings. Thirdly, we sharpen Thompson's [1999a,b] upper bound for the value of an Asian option. This is important for the practically relevant case of options with long maturities. Numerical results show that when the strike price is not extremely high, the resulting upper bound is tighter than recently introduced upper bounds in studies by Nielsen and Sandmann [2003] and Vanmaele et al. [2005]. Finally, we consider analytical approximations for the value of an Asian option. A much heard criticism on moment-matching approaches is that the error in the approximation is not known beforehand. We combine the traditional moment-matching approaches (e.g. Levy [1992]) with the conditioning approaches (e.g. Curran [1994]) and introduce a class of analytical approximations, which can be proven to lie between a sharp lower and upper bound. In numerical examples the accuracy of these new approximations is demonstrated. The approximations are found to outperform all of the current state-of-the-art upper bounds and approximations.

PRICING AUSTRALIAN S&amp;P200 OPTIONS: A BAYESIAN APPROACH BASED ON GENERALIZED DISTRIBUTIONAL FORMS

SummaryThis paper develops a new class of option price models and applies it to options on the Australian S&amp;P200 Index. The class of models generalizes the traditional Black‐Scholes framework by accommodating time‐varying conditional volatility, skewness and excess kurtosis in the underlying returns process. An important property of these more general pricing models is that the computational requirements are essentially the same as those associated with the Black‐Scholes model, with both methods being based on one‐dimensional integrals. Bayesian inferential methods are used to evaluate a range of models nested in the general framework, using observed market option prices. The evaluation is based on posterior parameter distributions, as well as posterior model probabilities. Various fit and predictive measures, plus implied volatility graphs, are also used to rank the alternative models. The empirical results provide evidence that time‐varying volatility, leptokurtosis and a small degree of negative skewness are priced in Australian stock market options.

Volatility Risk For Regime-Switching Models

Regime-switching models have proven to be well-suited for capturing the time series behavior of many financial variables. In particular, they have become a popular framework for pricing equity-linked insurance products. The success of these models demonstrates that realistic modeling of financial time series must allow for random changes in volatility. In the context of valuation of contingent claims, however, random volatility poses additional challenges when compared with the standard Black-Scholes framework. The main reason is the incompleteness of such models, which implies that contingent claims cannot be hedged perfectly and that a unique identification of the correct risk-neutral measure is not possible. The objective of this paper is to provide tools for managing the volatility risk. First we present a formula for the expected value of a shortfall caused by misspecification of the realized cumulative variance. This, in particular, leads to a closed-form expression for the expected shortfall for any strategy a hedger may use to deal with the stochastic volatility. Next we identify a method of selection of the initial volatility that minimizes the expected shortfall. This strategy is the same as delta hedging based on the cumulative volatility that matches the Black-Scholes model with the stochastic volatility model. We also discuss methods of managing the volatility risk under model uncertainty. In these cases, super-hedging is a possible strategy but it is expensive. The results presented enable a more accurate analysis of the trade-off between the initial cost and the risk of a shortfall.

A NEW MONTE CARLO METHOD FOR AMERICAN OPTIONS

We introduce a new Monte Carlo method for constructing the exercise boundary of an American option in a generalized Black–Scholes framework. Based on a known exercise boundary, it is shown how to price and hedge the American option by Monte Carlo simulation of suitable probabilistic representations in connection with the respective parabolic boundary value problem. The method presented is supported by numerical experiments.

Valuing Exotic Derivatives with Jump Diffusions: The Case of Basket Options

Exotic options are complicated derivatives instruments whose structure does not allow, in general, for closed-form analytic solutions, thus, making their pricing and hedging a difficult task. To overcome additional complexity such products are, as a rule, priced within a Black-Scholes framework, assuming that the underlying asset follows a Geometric Brownian Motion (GBM) stochastic process. This paper develops a more realistic framework for the pricing of exotic derivatives; and derives closed-form analytic solutions for the pricing and hedging of basket options. We relax the simplistic assumption of the GBM, by introducing the Bernoulli Jump Diffusion process (BJD) and approximate the terminal distribution of the underlying asset with a log-normal distribution. Potential extension of the model with the use of the Edgeworth Series Expansion (ESE) is also discussed. Monte Carlo simulation confirms the validity of the proposed BJD model.

Pricing of American-Style Fixed Strike Asian Options with Continuous Arithmetic Average

This article explores the analytic valuation of American-style fixed strike Asian option or average rate option based on the continuous arithmetic average in the Black-Scholes framework. Because there is no closed-form exact valuation formula for the average rate option with the arithmetic average, a very well-approximated arithmetic average density function is used for the valuation. The optimal exercise boundary and the values of American average rate options are compared with those of American plain vanilla options. Especially, this article shows that American average rate option can have two different optimal exercise boundaries depending on the parameters. Numerical experiments are also performed to demonstrate the influence of the component factors on the values of American average rate options and to illustrate the accuracy and efficiency of the valuation formula.

Résolution Numérique De Problèmes De Complémentarité Linéaire Et Évaluation D'Options Américaines

In the Black-Scholes framework, the American option pricing problem can be discretized into a finite-dimensional linear complementarity problem. We compare the numerical performances of three algorithms for the linear complementarity problem: the pivotal algorithms of Lemke and of Boriçi and Lüthi, and the iterative approach “Projected Successive Over-Relaxation”. We conclude that a special-purpose algorithm can provide performances vastly superior to those of generic algorithms.

Option Pricing and Hedging on Correlated Stocks

We develop a theory for option pricing with perfect hedging in an inefficient market model where the underlying price variations are autocorrelated over a time. This is accomplished by assuming that the underlying noise in the system is derived by an Ornstein-Uhlenbeck, rather than from a Wiener process. After obtaining an effective one-dimensional market model, we achieve a closed expression for the European call price within the Black-Scholes framework and find that our price is always lower than the Black-Scholes price. We obtain the same price if we start from a modified portfolio although now we get a different hedging strategy than that of Black-Scholes. We compare these strategies and study the sensitivity of the call price to several parameters where the correlation effects are also observed.