Black-Scholes Formula Research Articles

A Fuzzy LP Approach to Option Portfolio

Abstract: Owing to the fluctuation of financial market from time to time,the volatility and stock price may occur imprecisely in the real world. Therefore, it is natural to consider the fuzzy volatility and fuzzy stock price in the financial market. Under these assumptions,the theoretical price deduced by Black–Scholes formula are will turn into the fuzzy numbers, and the derivatives, called the Greek parameters delta, gamma, of the BS model are also fuzzy numbers. An option portfolio considering these fuzzy numbers will be more accord with actual situations. In this paper, we propose a fuzzy programming model of option portfolio based a ranking criterion of fuzzy numbers, which the fuzzy option portfolio model is converted into a classical linear programming problem. Finally, a numerical example is given to illustrate the validity of the method. Key words: Ranking Criterion; Option portfolio; Fuzzy Linear Programming; Delta-Gamma Neutral

Fast Computation of Vanilla Prices in Time-Changed Models and Implied Volatilities Using Rational Approximations

We present a new numerical method to price vanilla options in time-changed Brownian motion models quickly. The method is based on rational function approximations of the Black-Scholes formula. Detailed numerical results are given for a number of widely used models. In particular, we use the Variance Gamma model, the CGMY model and the Heston model without correlation to illustrate our results. Comparison to the standard fast Fourier transform method (FFT) with respect to accuracy and speed appears to favour the newly developed method in the cases considered. We present error estimates for the option prices.Additionally, we use this method to derive a procedure to compute, for a given set of arbitrage-free European call option prices, the corresponding Black-Scholes implied volatility surface. To achieve this, rational function approximations of the inverse of the Black-Scholes formula are used. We are thus able to work out implied volatilities more efficiently than one can by the use of other common methods. Error estimates are presented for a wide range of parameters.

Asymptotic Expansions of the Lognormal Implied Volatility: A Model Free Approach

We invert the Black-Scholes formula. We consider the cases low strike, large strike, short maturity and large maturity. We give explicitly the first 5 terms of the expansions. A method to compute all the terms by induction is also given. At the money, we have a closed form formula for implied lognormal volatility in terms of a power series in call price.

VALUATION FUNCTIONALS AND STATIC NO ARBITRAGE OPTION PRICING FORMULAS

Often in practice, the implied volatility of an option is calculated to find the option price tomorrow or the prices of ‘nearby’ options. To show that one does not need to adhere to the Black- Scholes formula in this scheme, Figlewski has provided a new pricing formula and has shown that his ‘alternating passive model’ performs as well as the Black-Scholes formula [8]. The Figlewski model was modified by Henderson et al. so that the formula would have no static arbitrage [10]. In this paper, we show how to construct a huge class of such static no arbitrage pricing functions, making use of distortions, coherent risk measures and the pricing theory in incomplete markets by Carr et al. [4]. Through this construction, we provide a more elaborate static no arbitrage pricing formula than Black-Sholes in the above scheme. Moreover, using our pricing formula, we find a volatility curve which fits with striking accuracy the synthetic data used by Henderson et al. [10].

Markov models for commodity futures: theory and practice

The objective of this paper is to develop a generic, yet practical, framework for the construction of Markov models for commodity derivatives. We aim for sufficient richness to permit applications to a broad variety of commodity markets, including those that are characterized by seasonality and by spikes in the spot process. In the first, largely theoretical, part of the paper we derive a series of useful results concerning the low-dimensional Markov representation of the dynamics of an entire term structure of futures prices. Extending previous results in the literature, we cover jump-diffusive models with stochastic volatility as well as several classes of regime-switching models. To demonstrate the process of building models for a specific commodity market, the second part of the paper applies a selection of our theoretical results to the exercise of constructing and calibrating derivatives trading models for USD natural gas. Special attention is paid to the incorporation of empirical seasonality effects in futures prices, in implied volatilities and their ‘smile’, and in correlations between futures contracts of different maturities. European option pricing in our proposed gas model is closed form and of the same complexity as the Black–Scholes formula.

Fair Terms and Fair Pricing for Multiple Warrant Issues

Some firms utilize one or more tranches of warrant issues to supplement their capital base. Unlike exchange-traded options, the exercise of warrants requires the issuance of stock by the company, resulting in a form of dilution. Some previous studies of warrant valuation relied on “the value of the firm,” which is nonobservable, making it difficult to apply the corresponding valuation formula. This paper derives closed-form formulas to value single and multiple tranches of warrants based on the underlying stock price, its volatility, and other known parameter values. The paper first establishes the equivalence of the Black-Scholes formula for both call options and warrants in the case of a single tranche. Thereafter, it considers the impact on the value of previously issued warrants that results when a new tranche of warrants is subsequently issued, showing in each case that fair treatment of the first-issued warrant holders requires an adjustment (due to dilution) in the terms of those warrants and a corresponding modification in the warrants’ value once a second tranche of warrants is issued. To promote such fair treatment, terms of a warrant indenture would specify the nature of the adjustment required when future warrants are issued or exercised, analogous to the antidilution terms related, for example, to stock dividends. Unlike multiple issues of traded options, which are valued independently of one another, multiple warrant issues will be shown to have prices dependent on other warrants outstanding. Also examined is the sensitivity of the fair-value adjustment to changes in the underlying variables, and the theoretical fair-value prices are compared with Black-Scholes prices and with market prices of warrants in the case of two publicly traded companies, each with two warrant issues outstanding. As warrant issues modify the equity structure of a firm, the methodology of valuing warrants presented here will be useful to investment actuaries in situations in which a comprehensive market value for all of a firm’s securities is called for. In addition, risk management practices may sometimes include the use of warrant transactions to hedge stock positions similar to the way that call options are used for that purpose. This may include hedging the risk in equity-linked insurance contracts when the equity position includes stock in companies that have one or more warrant issues that are traded. The methods developed here are also applicable to multiple issues of executive stock options (ESOs) or to combinations of warrant issues and ESOs.

An Intuitive Understanding of the Black-Scholes Formulas

Black-Scholes formulas for European call and put options are the core of the option theory. However, Black-Scholes formulas are often considered difficult by many students. In this short note, we present an intuitive understanding of the Black-Scholes formulas. This intuitive understanding makes the option theory more accessible to many people.

Pricing European options with a log Student’s t-distribution: A Gosset formula

The distributions of returns for stocks are not well described by a normal probability density function (pdf). Student’s t-distributions, which have fat tails, are known to fit the distributions of the returns. We present pricing of European call or put options using a log Student’s t-distribution, which we call a Gosset approach in honour of W.S. Gosset, the author behind the nom de plume Student. The approach that we present can be used to price European options using other distributions and yields the Black–Scholes formula for returns described by a normal pdf.

Matching the Sensitivities to Discrete Dividends: A New Approach for Pricing Vanillas

The incorporation of a dividend yield in the classical option pricing model of Black-Scholes results in a minor modification of the Black-Scholes formula, since the lognormal dynamic of the underlying asset is preserved. However, market makers prefer to work with cash dividends with fixed value instead of a dividend yield. Since there is no closed-form solution for the price of a European Call in this case, many methods have been proposed in the literature to approximate it. Here, we present a new approach. We derive an exact analytic formula for the sensitivity to dividends of an European option. We use this result to elaborate a proxy which possesses the same Taylor expansion around 0 with respect to the dividends as the exact price. The obtained approximation is very fast to compute (the same complexity than the usual Black-Scholes formula) and numerical tests show the extreme accuracy of the method for all practical cases.

Approximate Hedging of Contingent Claims under Transaction Costs for General Pay-offs

In 1985 Leland suggested an approach to price contingent claims under proportional transaction costs. Its main idea is to use the classical Black–Scholes formula with a suitably enlarged volatility for a periodically revised portfolio whose terminal value approximates the pay-off h(S T ) = (S T − K)+ of the call option. In subsequent studies, Lott, Kabanov and Safarian, and Gamys and Kabanov provided a rigorous mathematical analysis and established that the hedging portfolio approximates this pay-off in the case where the transaction costs decrease to zero as the number of revisions tends to infinity. The arguments used heavily the explicit expressions given by the Black–Scholes formula leaving open the problem whether the Leland approach holds for more general options and other types of price processes. In this paper we show that for a large class of the pay-off functions Leland's method can be successfully applied. On the other hand, if the pay-off function h(x) is not convex, then this method does not work.

Stochastic financial models

Portfolio Choice Introduction Utility Mean-variance analysis The Binomial Model One-period model Multi-period model A General Discrete-Time Model One-period model Multi-period model Brownian Motion Introduction Hitting-time distributions Girsanov's theorem Brownian motion as a limit Stochastic calculus The Black-Scholes Model Introduction The Black-Scholes formula Hedging and the Black-Scholes equation Path-dependent claims Dividend-paying assets Interest-Rate Models Introduction Survey of interest-rate models Gaussian random-field model Appendix A: Mathematical Preliminaries Appendix B: Solutions to the Exercises Further Reading References Index Exercises appear at the end of each chapter.

Mean square error for the Leland–Lott hedging strategy: convex pay-offs

Leland’s approach to the hedging of derivatives under proportional transaction costs is based on an approximate replication of the European-type contingent claim VT using the classical Black–Scholes formula with a suitably enlarged volatility. The formal mathematical framework is a scheme of series, i.e., a sequence of models with transaction cost coefficients kn=k0n−α, where α∈[0,1/2] and n is the number of portfolio revision dates. The enlarged volatility \(\widehat{\sigma}_{n}\) in general depends on n except for the case which was investigated in detail by Lott, to whom belongs the first rigorous result on convergence of the approximating portfolio value \(V^{n}_{T}\) to the pay-off VT. In this paper, we consider only the Lott case α=1/2. We prove first, for an arbitrary pay-off VT=G(ST) where G is a convex piecewise smooth function, that the mean square approximation error converges to zero with rate n−1/2 in L2 and find the first order term of the asymptotics. We are working in a setting with non-uniform revision intervals and establish the asymptotic expansion when the revision dates are \(t_{i}^{n}=g(i/n)\), where the strictly increasing scale function g:[0,1]→[0,1] and its inverse f are continuous with their first and second derivatives on the whole interval, or g(t)=1−(1−t)β, β≥1. We show that the sequence \(n^{1/2}(V_{T}^{n}-V_{T})\) converges in law to a random variable which is the terminal value of a component of a two-dimensional Markov diffusion process and calculate the limit. Our central result is a functional limit theorem for the discrepancy process.

A new analytical approximation for European puts with stochastic volatility

In this paper, we apply singular perturbation techniques to price European puts with a stochastic volatility model, and derive a simple and elegant analytical formula as an approximation for the value of European put options. In contrast to the existing Heston’s semi-analytical formula, this approximation has the following unique feature: the latter only involves the standard normal distribution function, which is as fast and easy to implement as the Black–Scholes formula; whereas the former requires the evaluation of a logarithm with a complex argument during the involved Fourier inverse transform, which may sometimes result in numerical instability. Various numerical experiments suggest that our new formula can achieve a high order of accuracy for a large class of option derivatives with relatively short tenor.

Option hedging for small investors under liquidity costs

Following the framework of Cetin et al. (Finance Stoch. 8:311–341, 2004), we study the problem of super-replication in the presence of liquidity costs under additional restrictions on the gamma of the hedging strategies in a generalized Black–Scholes economy. We find that the minimal super-replication price is different from the one suggested by the Black–Scholes formula and is the unique viscosity solution of the associated dynamic programming equation. This is in contrast with the results of Cetin et al. (Finance Stoch. 8:311–341, 2004), who find that the arbitrage-free price of a contingent claim coincides with the Black–Scholes price. However, in Cetin et al. (Finance Stoch. 8:311–341, 2004) a larger class of admissible portfolio processes is used, and the replication is achieved in the L 2 approximating sense.

Option Valuation Using Asymptotic Expansion

In this paper, the asymptotic expansion technique is proposed in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. We study the pricing of defaultable derivatives, such as bonds, bond options, and credit default swaps in the reduced form framework of intensity-based models. We use asymptotic expansions from which we derive approximations for the pricing functions of these derivatives. In particular, we assume an Ornstein-Uhlenbeck process for the interest rate, and a two-factor diffusion model for the intensity of default. The classical Black-Scholes formula gives the price of call options when the underlying is a geometric Brownian motion with a constant volatility. The underlying might be the price of a stock or an index and a constant volatility corresponds to a fixed standard deviation for the random fluctuations in the returns of the underlying. Modern market phenomena make it important to analyze the situation when this volatility is not fixed but rather is heterogeneous and varies with time. The volatility is modeled as stochastic process that varies on two characteristic time scales: fast mean reverting as evident by Fouque and el.(2000). Analysis of market data, however, shows the need for introducing also a slowly varying factor in the model for the stochastic volatility. In particular, the introduction of the slow factor gives a much better fit for options with longer maturities. The resulting approximation is still independent of the particular details of the volatility model and gives more flexibility in the parameterization of the implied volatility surface. The resulting approximation is still independent of the particular details of the volatility model and gives more flexibility in the parameterization of the implied volatility surface.

Comparison of Black–Scholes Formula with Fractional Black–Scholes Formula in the Foreign Exchange Option Market with Changing Volatility

In this paper, the authors discuss the fractional option pricing with Black–Scholes formula, deduce the Fractional Black–Scholes formula, show the empirical results by using China merchants bank foreign exchange call option price, and find when the volatility is smaller, the asymptotic mean squared error of Fractional Black–Scholes is bigger than the Traditional Black–Scholes’, while the volatility is bigger—the market mechanism has a full play, the result is reverse. Namely when the market mechanism is given a full scope, the estimating effect of Fractional Black–Scholes is better than Traditional Black–Scholes’.

Scaling and long-range dependence in option pricing I: Pricing European option with transaction costs under the fractional Black–Scholes model

This paper deals with the problem of discrete time option pricing by the fractional Black–Scholes model with transaction costs. By a mean self-financing delta-hedging argument in a discrete time setting, a European call option pricing formula is obtained. The minimal price C min ( t , S t ) of an option under transaction costs is obtained as timestep δ t = ( 2 π ) 1 2 H ( k σ ) 1 H , which can be used as the actual price of an option. In fact, C min ( t , S t ) is an adjustment to the volatility in the Black–Scholes formula by using the modified volatility σ 2 ( 2 π ) 1 2 − 1 4 H ( k σ ) 1 − 1 2 H to replace the volatility σ , where k σ < ( π 2 ) 1 2 , H > 1 2 is the Hurst exponent, and k is a proportional transaction cost parameter. In addition, we also show that timestep and long-range dependence have a significant impact on option pricing.

Black-Scholes Formula in Subdiffusive Regime

In the classical approach the price of an asset is described by the celebrated Black-Scholes model. In this paper we consider a generalization of this model, which captures the subdiffusive characteristics of financial markets. We introduce a subdiffusive geometric Brownian motion as a model of asset prices exhibiting subdiffusive dynamics. We find the corresponding fractional Fokker-Planck equation governing the dynamics of the probability density function of the introduced process. We prove that the considered model is arbitrage-free and incomplete. We find the corresponding subdiffusive Black-Scholes formula for the fair prices of European options and show how these prices can be evaluated using Monte-Carlo methods. We compare the obtained results with the classical ones.

The Black–Scholes model as a determinant of the implied volatility smile: A simulation study

The paper represents an initial effort to shed light on the determinants of the implied volatility smile in financial (derivative) markets. It fully details the implications of the institutionalization of the Black–Scholes model in an uncertain world populated by individuals who are bounded by the amount of calculation or accounting which is technically possible. Combining model simulations, empirical analysis, and mathematical derivations, the paper proposes that the determinants of the volatility smile might be related to the behavior of traders. In pricing options, they use the widely accepted Black–Scholes formula with a measure of stock volatility that they derive from their subjective beliefs. Moreover, heterogeneity of traders’ beliefs and the way traders update their expectations have nontrivial effects, both on equilibrium prices and on the emergence of the implied volatility smile.

Empirical Performance of the Constant Elasticity Variance Option Pricing Model

In this essay, we empirically test the Constant–Elasticity-of-Variance (CEV) option pricing model by Cox (1975, 1996 ) and Cox and Ross (1976), and compare the performances of the CEV and alternative option pricing models, mainly the stochastic volatility model, in terms of European option pricing and cost-accuracy based analysis of their numerical procedures. In European-style option pricing, we have tested the empirical pricing performance of the CEV model and compared the results with those by Bakshi et al. (1997). The CEV model, introducing only one more parameter compared with Black-Scholes formula, improves the performance notably in all of the tests of in-sample, out-of-sample and the stability of implied volatility. Furthermore, with a much simpler model, the CEV model can still perform better than the stochastic volatility model in short term and out-of-the-money categories. When applied to American option pricing, high-dimensional lattice models are prohibitively expensive. Our numerical experiments clearly show that the CEV model performs much better in terms of the speed of convergence to its closed form solution, while the implementation cost of the stochastic volatility model is too high and practically infeasible for empirical work. In summary, with a much less implementation cost and faster computational speed, the CEV option pricing model could be a better candidate than more complex option pricing models, especially when one wants to apply the CEV process for pricing more complicated path-dependent options or credit risk models.