Black-Scholes Option Pricing Formula Research Articles

Exotic option pricing model of the Black–Scholes formula: a proactive investment strategy

The option is an important derivative tool in financial market, and after decades of development, the option has emerged in various forms. This paper studies an exotic option with a proactive investment strategy. Compared with the classic option theory, we assume that the option holders continuously trade the underlying assets according to a predetermined investment strategy. Taking the European call option as an example, we first give some assumptions and define the loss function in terms of a logarithmic investment strategy. Then, the specific pricing expression of the exotic option is derived from the Black-Scholes option pricing formula. Next, numerical simulations are presented to visualize the mechanism of the exotic option in 2D and 3D forms by selecting appropriate parameters. The results indicate that the exotic option has a significant price advantage (up to 43.9% under specific parameter settings) over the classic option. Moreover, the empirical results illustrate a perfect fit between 50 ETF (issued by the Shanghai Stock Exchange) and our exotic option. The new proposed exotic option extends the Black-Scholes option theory from a no-trading condition (do not buy or sell underlying assets during the validity period) to a dynamic investment condition and has important practical significance in real life.

Proactive Hedging European Option Pricing with a General Logarithmic Position Strategy

This study proposes an exotic option that extends the classical European option by requiring option holders to continuously trade in underlying assets according to a predesignated trading strategy with a general logarithmic position. The pricing formula for the exotic option with a general logarithmic strategy is derived from the Black–Scholes option pricing formula, and its price advantage is compared (based on simulations) to the classical European option and to the exotic option with a linear position. By varying key parameters, we found that the exotic option with a general logarithmic position has a significant price advantage (up to 34% under certain parameter settings) over the classical European option. Moreover, the exotic option with a general logarithmic strategy can save 5.5% more of the option premium than applying a linear position strategy. Our simulation results indicate that the price advantage of this proactive hedging option with a general logarithmic strategy depends heavily on the initial amount of capital; in particular, this exotic option is more suitable for traders with limited initial amounts of capital.

Merton’s equation and the quantum oscillator II: Option pricing

Merton has proposed a model for contingent claims on a firm as an option on the firms value, and is based on a generalization of the Black–Scholes stochastic equation (Merton, 1974). A special case of Merton’s model is proposed – based on the quantum oscillator – for pricing options. Two cases of the option price are obtained: both these cases yield possible candidates for the generalization of the Black–Scholes option pricing formula. However, one of the proposed option prices does not obey the martingale condition and the other does not yield the correct discounting of future cash flows. For these reasons, the option prices do not obey put–call parity. The options can, however, be used to approximately price market traded options. The oscillator model for the option price has an extra parameter that is absent for the Black–Scholes case. Similar to the model studied by Baaquie et al. (2014), which that does not obey put–call parity, the option’s price can be studied empirically and the extra parameter in the model could, in principal, generate implied volatility.

Application of Credit Risk Management Model in Chinese Banks

The main objective of this paper is to perform empirical analysis and research on the KMV and Zeta models, discussing whether banks in China could adopt both models in their credit risk management practices. In order to measure credit risk, the KMV model focuses on “Expected Default Probability” (EDP) that is calculated using Black-Scholes Option Pricing Formula. On the other hand, the Zeta Model focuses on determining the probability of a company going bankrupt two years prior to the event. Previous research on risk management has shown that the primary risk the banks generally face is credit risk as an increasingly greater number of banks suffer losses because of credit issues. This paper therefore aims to add to the existing literature a strong case for the relevance of both the KMV and Zeta models to be considered in the topic of banks’ credit risk management.

Pricing European Options with Triangular Fuzzy Parameters: Assessing Alternative Triangular Approximations in the Spanish Stock Option Market

The aim of this paper is contributing from a practical and empirical perspective to option pricing under fuzziness. When we evaluate Black–Scholes option pricing formula with triangular fuzzy numbers, we obtain a non-triangular price that may be slightly difficult to use in practical applications. We improve the applicability of the fuzzy version of that formula by proposing and testing three triangular approximations when the subjacent asset price, its volatility and free interest rate are triangular fuzzy numbers. To check the goodness of these approximations, we firstly evaluate their closeness to the actual values of fuzzy Black and Scholes model. We find that the quality of those approximations depends on options maturity and moneyness grade and if we are pricing call or put options. We also assess the capability of those approximating methods to reflect satisfactorily real market prices and obtain good results. To develop all empirical applications, we use a sample of options on IBEX35 traded in the Spanish derivatives market on 3/1/2017.

Bernstein’s inequalities and their extensions for getting the Black–Scholes option pricing formula

In this paper we show how the results of Bernstein (1943) and recent results of Zubkov and Serov (2012) on the normal approximation to the binomial distribution lead to an alternative derivation of the Black–Scholes formula from a binomial option pricing model.

上证50ETF期权的定价方法与风险控制研究

基于直接法、广义自回归条件异方差模型(GARCH模型)和马尔科夫转换模型对上证50ETF年对数收益率的波动率进行估计，并通过Black-Scholes期权定价模型计算期权价格。实证分析的结果显示GARCH模型适用于标的资产比较稳定的情况而马尔科夫转换模型在标的资产表现出明显上升或下降趋势时表现良好，从而给出选取定价方式的标准。然后采用VaR方法衡量在期权投资中产生的金融风险。其中具体的计算方法包括历史模拟法以及采用波动率序列的参数解析法，我们也讨论两种方法的优缺点。 Based on Direct Method, GARCH Model and Markov Switching Model, the annual logarithm yields of Shanghai 50ETF were estimated and therefore the option price can be computed by Black-Scholes option pricing formula. Empirical results show that GARCH Method is more suitable when the un-derlying assets are relatively stable while Markov model performs better when the underlying as-sets show obvious trend of rising or falling and this could be the criterion of choosing pricing me-thods. After pricing options, VaR method is used to measure the financial risk of different options. The concrete computation methods used here are historical simulation VaR and the parametrically method with variance sequence which have their own advantages respectively.

A Pathologically Eclectic, Lazy Derivation of the Black Scholes Formula

The classical Black Scholes option-pricing formula which has been an indispensable tool in the hands of the practitioners of quantitative finance has been derived numerous ways. Some of these derivations have been very theoretical with the full machinery of Girsanov's theorem, some informal and based on practical hedging arguments. We suggest yet another derivation (Actually, we just do certain things while computing the drift in the changed measure very differently), which is based on sound theoretical framework, but does not use much of advanced theory. Instead the derivation suggested here makes repeated use of the technique of measure change that is rather a universal and standard tool in derivative pricing. The approach is lazy, because it defers doing certain things, unless absolutely necessary, while at the same time elegant enough to be pedagogically interesting in its own right.

Option pricing: Stock price, stock velocity and the acceleration Lagrangian

The industry standard Black–Scholes option pricing formula is based on the current value of the underlying security and other fixed parameters of the model. The Black–Scholes formula, with a fixed volatility, cannot match the market’s option price; instead, it has come to be used as a formula for generating the option price, once the so called implied volatility of the option is provided as additional input. The implied volatility not only is an entire surface, depending on the strike price and maturity of the option, but also depends on calendar time, changing from day to day. The point of view adopted in this paper is that the instantaneous rate of return of the security carries part of the information that is provided by implied volatility, and with a few (time-independent) parameters required for a complete pricing formula.An option pricing formula is developed that is based on knowing the value of both the current price and rate of return of the underlying security which in physics is called velocity. Using an acceleration Lagrangian model based on the formalism of quantum mathematics, we derive the pricing formula for European call options. The implied volatility of the market can be generated by our pricing formula. Our option price is applied to foreign exchange rates and equities and the accuracy is compared with Black–Scholes pricing formula and with the market price.

Some Misconceptions in Derivative Pricing

This paper has used the Arbitrage Theorem (Gordan Theorem) to clarify some misconceptions in the literature of derivative pricing. First, unlike the claim of the irrelevancy of the underlying asset’s (stock’s) expected return, it is found that the value of an option depends on the probability of the underlying asset (stock) rising or falling. Using the relationship between relative price ratio between the two states: and the probability of the up move, the paper also derives discrete-time versions of the Greeks. Second, since with no arbitrage, is a function of and, the Black-Scholes option pricing formula contains the underlying asset’s expected rate of return. Third, with a two-step contract, it has been shown that within a company, there is no first claim or seniority between bond and stock, but there is first claim among fixed-income assets (e.g., labor and bond), and labor is senior to bond.

Modeling investment guarantees in Japan: A risk-neutral GARCH approach

The variable annuity market in Japan is still young, but growing rapidly. Most variable annuities in Japan are sold with one or more investment guarantees, such as a Guaranteed Minimum Maturity Benefit (GMMB), which guarantees that the ultimate annuity principal will not fall below a pre-set level regardless of the underlying investment performance. Of interest to financial institutions selling variable annuities is the cost associated with such a guarantee. Although the Black-Scholes option pricing formula can be applied readily, the resulting price might be inaccurate because returns on the Japanese stock price index do not seem to behave as assumed in the formula. In this study, we propose a method for valuing investment guarantees on the basis of a GARCH-type model, which better captures the characteristics of the stock price index. A difficulty in option-pricing with GARCH is the identification of a risk-neutral probability measure. We solve this problem by considering the conditional Esscher transform. Computational details of the proposed method are illustrated by valuing costs of two popular investment guarantees in Japan.

Measuring and Managing Foreign Exchange Risk at a Large Pharmaceutical Firm: Application of Value at Risk

The present study has been carried out on the express request of a major pharmaceutical firm. The firm has substantial foreign exchange exposure and it needs a tool which can be used for managing foreign exchange risk. In this paper as the calculation of Value at Risk (VaR) is to deal with the risk involved with the options, their non-linear payoff nature, it is more challenging to incorporate all the option sensitivities in the model that we adopt to measure VaR. As the portfolio consists of currency pairs, it was more important to preserve correlation of the currency pairs while measuring VaR using any approach. We calculated delta of each option of the portfolio by using Black-Scholes option pricing formula and thereby calculated the unhedged amount for each asset in the portfolio. We have used variance-covariance approach, historical simulation and Monte Carlo simulation to calculate VaR. After comparison of results by weighing their advantages and disadvantages, historical simulation was ruled out to implement as it will not consider correlation between the assets. We have chosen to implement Variance-Covariance approach against the Monte Carlo simulation method mainly because it takes correlation into consideration and the latter is less transparent. Variance-Covariance approach is also faster and relatively easy to implement when compared to the Monte Carlo simulation. We designed a template using Excel VBA and MATLAB to calculate weekly VaR with the input variables (Current price of the underlying, Strike rate, Time to maturity, Volatility, Risk free interest rates of the currency pairs) and from the result we can interpret whether we are over hedged or under hedged using options as our hedging strategy. This study provides the insight of what can be the value still at risk when the firm is hundred percent hedged using options because of the option sensitivities. It also provides a lucid interpretation to the risk management executive to understand his position on the option hedging strategies.

Option Pricing Model for Autocorrelated Stock Returns

This paper develops an analytic option pricing model for the case of serially correlated asset returns. This model provides a valuable insight into dependence of option price on the return autocorrelation. A surprising relationship is found between asset price volatility, asset return volatility and asset return autocorrelation coefficient. The analytical solution obtained here reduces to the well known Black Scholes option pricing formula for the special case of no autocorrelation in asset returns. However, in the case of serially correlated asset returns, our model provides superior results by controlling for the difference between asset price volatility and asset return volatilities. Empirical tests obtain statistically and economically significant results demonstrating the superiority of our model when compared to the traditional Black Scholes formula. Additionally, a Monte Carlo method is developed to price option for assets with autocorrelated returns. Numerical comparison between these two methods demonstrate strikingly similarity, being equal to within the computational error. This serves to affirm the validity of our method.

Empirical Study of the Effect of Including Skewness and Kurtosis in Black Scholes Option Pricing Formula on S&P CNX Nifty Index Options

The most popular model for pricing options, both in financial literature as well as in practice has been the Black-Scholes model. In spite of its wide spread use the model appears to be deficient in pricing deep in the money and deep out of the money options using statistical estimates of volatility. This limitation has been taken into account by practitioners using the concept of implied volatility. The value of implied volatility for different strike prices should theoretically be identical, but is usually seen in the market to vary. In most markets across the world it has been observed that the implied volatilities of different strike prices form a pattern of either a ‘smile’ or ‘skew’. Theoretically, since volatility is a property of the underlying asset it should be predicted by the pricing formula to be identical for all derivatives based on that same asset. Hull [1993] and Nattenburg [1994] have attributed the volatility smile to the non normal Skewness and Kurtosis of stock returns. Many improvements to the Black-Scholes formula have been suggested in academic literature for addressing the issue of volatility smile. This paper studies the effect of using a variation of the BS model (suggested by Corrado & Sue [1996] incorporating non-normal skewness and kurtosis) to price call options on S&P CNX Nifty. The results strongly suggest that the incorporation of skewness and kurtosis into the option pricing formula yields values much closer to market prices. Based on this result and the fact that this approach does not add any further complexities to the option pricing formula, we suggest that this modified approach should be considered as a better alternative.

Option valuation model with adaptive fuzzy numbers

In this paper, we consider moment properties for a class of quadratic adaptive fuzzy numbers defined in Dubois and Prade [D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980]. The corresponding moments of Trapezoidal Fuzzy Numbers (Tr.F.N’s) and Triangular Fuzzy Numbers (T.F.N’s) turn out to be special cases of the adaptive fuzzy number [S. Bodjanova, Median value and median interval of a fuzzy number, Information Sciences 172 (2005) 73–89]. A numerical example is presented based on the Black–Scholes option pricing formula with quadratic adaptive fuzzy numbers for the characteristics such as volatility parameter, interest rate and stock price. Our approach hinges on a characterization of imprecision by means of fuzzy set theory.

A streamlined derivation of the Black-Scholes option pricing formula

This article presents a simplified derivation of the Black-Scholes formula for option valuation via the interplay of three essential tools. The pedagogical discussion utilises only elementary integration techniques, basic facts from normal distribution theory and concept of indicator function and is therefore accessible to any junior undergraduate pursuing a degree in financial mathematics, actuarial science, economics and business. Illustrating the use of these mathematical techniques adds rigour to student’s critical thinking and brings a perspective that finance studies is an interdisciplinary area.

Review Paper. A survey of mathematical finance

Finance is one of the fastest growing areas in mathematics. In some senses it is not a discipline in its own right, but rather an application area in which mathematicians with backgrounds in probability theory, statistics, optimal control, convex and functional analysis and partial differential equations can bring to bear experiences and results from their own fields to problems of real world interest. In this survey we begin with the simplest possible financial model, and then give an account of the Black–Scholes option pricing formula, in which the key ideas are the replication of option pay–offs and pricing under the risk–neutral measure. Then we move on to discuss other important problems in finance, including the general theory for semi–martingale price processes, pricing in incomplete markets, interest–rate models and credit risk. The emphasis is on techniques and methodologies from stochastic processes.

APLIKASI FORMULA PENILAIAN OPSI BLACK-SCHOLES UNTUK ESTIMASI NILAI CALL OPSI INDEKS SAHAM LQ-45 DI BURSA EFEK JAKARTA

Universitas Indonesia APPLICATION OF THE BLACK-SCHOLES OPTION PRICING FORMULA ON VALUATION OF LQ-45 STOCK INDEX OPTION. The objective of this research is to investigate the applicability of the Black- Scholes Option Pricing Model (BSOPM) on options on market index at the Jakarta Stock Exchange (JSX). A simulation is conducted using actual JSX LQ-45 index data between January 1997 and April 1999. Each month, a simulated premium of a one- month calf option is calculated based on BSOPM and men compared with its payoff at its maturity date. The results show that the average profit of the simulated long stock index call option is negative but not statistically significant. It means that the BSOPM, although not rejectable statistically, cannot be applied blindly on the valuation of JSX stock index options. Kata kunci: Opsi, formula Black-Scholes, indeks LQ-45.

Does the behaviour of the asset tell us anything about the option price formula? A cautionary tale

If Y = (Y 1,…,Y N) are the log-returns of an asset on succeeding days, then under the assumptions of the Black-Scholes option pricing formula, these are independent normal random variables with common mean and variance in the risk-neutral measure. If we can show empirically that Y does not have these properties in the realworld measure, does this mean that the Black-Scholes option pricing formula fails? It does not; as we show in this note, so long as the joint distribution of Y in the realworld measure has a strictly positive density, then the Black-Scholes option price formula may still be correct. We conclude that attempts to argue that the Black-Scholes formula must fail because observed log returns appear to be fat-tailed, or appear to have nonconstant volatility, or appear to have serial correlation are fallacious.

Pricing Options under Stochastic Interest Rates: A New Approach

We will generalize the Black-Scholes option pricing formula by incorporating stochastic interest rates. Although the existing literature has obtained some formulae for stock options under stochastic interest rates, the closed-form solutions have been known only under the Gaussian (Merton type) interest rate processes. We will show that an explicit solution, which is an extended Black-Scholes formula under stochastic interest rates in certain asymptotic sense, can be obtained by extending the asymptotic expansion approach when the interest rate volatility is small. This method, called the small-disturbance asymptotics for Ito processes, has recently been developed by Kunitomo and Takahashi (1995, 1998) and Takahashi (1997). We found that the extended Black-Scholes formula is decomposed into the original Black-Scholes formula under the deterministic interest rates and the adjustment term driven by the volatility of interest rates. We will illustrate the numerical accuracy of our new formula by using the Cox–Ingersoll–Ross model for the interest rates.