Option Pricing Under Multifractional Process and Long-Range Dependence

We obtain a Black-Scholes formula for the arbitrage-free pricing of European Call options with constant coefficients when the underlying stock generates dividends. To hedge the Call option, we will always borrow money from bank. We see the influence of the dividend term on the option pricing via the comparison theorem of BSDE(backward stochastic differential equation [5], [7]). We also consider the option pricing problem in terms of the borrowing rate R which is not equal to the interest rater. The corresponding Black-Scholes formula is given. We notice that it is in fact the borrowing rate that plays the role in the pricing formula.

Risk-neutral pricing of European call options is investigated from a mathematical point-of-view and is found to be a specious concept1. Risk-neutral pricing of European call options is an approximation in which all terms of order are ignored, where is the risk premium and σ is the volatility.

In this paper, some problems about Black-Scholes model with fuzzy number coefficients is studied. Firstly, by using the Ito formula, the asset prices at $T$ or $t$ times are obtained from the Black-Scholes model with fuzzy number coefficients. Then, by operation rules of fuzzy numbers, the expression of the price of European call option in form of level set of fuzzy number is obtained. At last, an example is given to show the feasibility of the method.

The Indonesian stock market, especially PT Bank Central Asia (BCA) stock experiences high volatility. Therefore, this research is very important to provide a deeper understanding of the pricing of put and call options as a risk management instrument. This study aims to analyze the pricing of put and call options on PT Bank Central Asia (BCA) shares using the Black-Scholes model and identify factors that affect stock price fluctuations on option prices. This study aims to understand the changes in put and call option prices on PT Bank Central Asia (BCA) shares caused by fluctuations in stock prices. This study uses historical data of PT Bank Central Asia (BCA) shares, volatility, and current interest rates as the basis for analysis. The Black-Scholes method is used as a framework to analyze the pricing of put and call options by considering volatility and interest rates. The results of this study analyze the price changes of put and call options in the face of stock price fluctuations and the factors that affect option prices so as to provide insight into risk management and investment decision making. This research is expected to provide practical guidance for investors and risk managers in making decisions related to put and call options on PT Bank Central Asia (BCA) shares.

We present a new approach to pricing American-style derivatives that is applicable to any Markovian setting (i.e., not limited to geometric Brownian motion) for which European call-option prices are readily available. By approximating the value function with an appropriately chosen interpolation function, the pricing of an American-style derivative with arbitrary payoff function is converted to the pricing of a portfolio of European call options, leading to analytical expressions for those cases where analytical European call prices are available (e.g., the Merton jump-diffusion process). Furthermore, in many settings, the approach yields upper and lower analytical bounds that provably converge to the true option price. We provide computational results to illustrate the convergence and accuracy of the resulting estimators.

Investment is a saving activity with the aim of overcoming price increases or often called inflation. Investments can be in the form of gold, property, silver or stock investments. Stock investment is considered more profitable than just saving at a bank. Currency values are declining due to inflation. This results in a tendency to invest in shares. Stock investment carries a great risk. Therefore, in 2004 stock options began to trade. Stock options are contracts that give the holder the right to buy / sell shares at the agreed time, at a certain price. Stock option prices tend to be cheaper than stock prices. Therefore, determining the right price of stock options is needed. In this study, we will focus on the European type of buying options, the right to buy shares at an agreed price at maturity. The purpose of this study is the completion of the Black-Scholes model of European type option prices at a constant market, assuming stock movements meet the stochastic differential equation, fixed risk-free interest rates, companies distributing dividends, no taxes, no transaction costs, and free market arbitration. The results of this research are in the form of differential equations and the settlement of the Black-Scholes model of European type call option prices, and a case study used by stock option contracts with a maturity of January 4, 2010, PT Aqua Golden Mississippi Tbk.

In this paper, the problem of European call option pricing is studied based on binary tree model in which the rising and falling factors (coefficients) are fuzzy numbers. By converting the fuzzy number coefficients binary tree model into four classical binary tree models, a trapezoidal fuzzy number option pricing (TFN-option pricing) is defined, and two calculation formulas to solve the TFN-option pricing are obtained. One calculation formula is that the more general conditions hold. But the other calculation formula is established under some special conditions. And we give an example to illustrate the feasibility of our method.KeywordsTrapezoid fuzzy numberBinary tree modelStock pricesPricing of European call options

In the financial world, the thing that is interesting for investors today is to predict a reasonable Asian Call Option price that does not cause the seller or buyer of the option to experience losses. The constant interest assumption is more often used by previous research as an assumption in predicting the price of Asian Call Options. This is not in line with the fact that interest rate movements are not constant. The use of incorrect assumptions will affect the accuracy of prediction results. In overcoming this problem, this study aims to predict the price of Asian Call Options with non-constant interest rates using Monte Carlo simulation. The non-constant interest rate model used is Cox-Ingersoll-Ross (CIR) because this model produces positive interest rates. This study begins with a screening of Normally distributed interest rate data. The data is used to determine the parameters of the CIR interest rate model. Next, an Asian Call Option price simulation with non-constant interest rates was carried out and ended by analyzing a graph of the Asian Call Option price simulation results with non-constant interest rates. The simulated price result of the Asian Call Option with non-constant interest rates converged at the value of $170.82. Based on the price chart of Asian Call Options with different strike prices, the larger the strike price causes, the larger the Asian Call Option price, which corresponds to the price characteristics of the Asian Call Option. In addition, by extending the maturity period, the price of Asian Call Options with non-constant interest rates is greater according to the price characteristics of Asian Call Options.

Option Pricing with a Dividend General Equilibrium Model

We present option pricing under the double stochastic volatility model with stochastic interest rates and double exponential jumps with stochastic intensity in this article. We make two contributions based on the existing literature. First, we add double stochastic volatility to the option pricing model combining stochastic interest rates and jumps with stochastic intensity, and we are the first to fill this gap. Second, the stochastic interest rate process is presented in the Hull–White model. Some authors have concentrated on hybrid models based on various asset classes in recent years. Therefore, we build a multifactor model with the term structure of stochastic interest rates. We also approximated the pricing formula for European call options by applying the COS method and fast Fourier transform (FFT). Numerical results display that FFT and the COS method are much faster than the numerical integration approach used for obtaining the semi-closed form prices. The COS method shows higher accuracy, efficiency, and stability than FFT. Therefore, we use the COS method to investigate the impact of the parameters in the stochastic jump intensity process and the existence of the process on the call option prices. We also use it to examine the impact of the parameters in the interest rate process on the call option prices.

Volumetric Risk Hedging Strategies and Basis Risk Premium for Solar Power

We propose an analytically tractable local volatility model for asset price dynamics leading to volatility smile/skew and fatter-tailed probability distribution. The proposed local volatility model is based on stochastic process of hyperbolic-sine type. We derive the transition probability density function for hyperbolic-sine model and justify that this function has delta function terminal condition at initial time. We compare the probability density functions in Black-Scholes and hyperbolic-sine models to demonstrate that the probability distribution in hyperbolic sine model has some features of fat-tailed distributions. Risk neutral valuation technique is applied to find explicit valuation formula for European call option price in hyperbolic-sine model. In hyperbolic-sine model European call option is more valuable than an identical option in Black-Scholes model for ATM options. We verify that in hyperbolic-sine model Breeden-Litzenberger formula (relation between European call option price and probability density function) holds true. We also examine that Dupire formula correctly recovers volatility function from European call option price in hyperbolic-sine model.

The Information Content of The Implied Volatility Surface: Can Option Prices Predict Jumps?

We develop a new high-performance spectral collocation method for the computation of American put and call option prices. The proposed algorithm involves a carefully posed Jacobi-Newton iteration for the optimal exercise boundary, aided by Gauss-Legendre quadrature and Chebyshev polynomial interpolation on a certain transformation of the boundary. The resulting scheme is straightforward to implement and converges at a speed several orders of magnitude faster than existing approaches. Computational effort depends on required accuracy; at precision levels similar to, say, those computed by a finite-difference grid with several hundred steps, the computational throughput of the algorithm in the Black-Scholes model is typically close to 100 000 option prices per second per CPU. For benchmarking purposes, Black-Scholes American option prices can generally be computed to ten or eleven significant digits in less than one-tenth of a second.

Call option prices in the Black–Scholes model, viewed as functions of strike and maturity, are totally positive of order two (), meaning that the price ratio of a higher‐strike call to a lower‐strike call increases with maturity, with adjustments for dividends and interest. We develop conditions for this property in other models and contrast it with full total positivity, which holds only for out‐of‐the‐money strikes in the Black–Scholes model. Related properties apply to puts. We give a simple sufficient condition for based on the unimodality of ratios of densities of the underlying asset at different dates. We show that the property entails a strengthening of monotonicity of the underlying asset in the convex order and thus a strengthening of the absence of static arbitrage. We construct examples illustrating the gaps between these properties. We develop connections between and the shape of the implied volatility surface—in particular, connections with supermodularity of implied variance, a condition implying that lines of implied variance for different maturities fan out at high strikes. An examination of S&P 500 options market data indicates that violations are infrequent and typically reverse quickly.