Generalized Black-Scholes Model Research Articles

Option Pricing under a Generalized Black–Scholes Model with Stochastic Interest Rates, Stochastic Strings, and Lévy Jumps

We introduce a novel option pricing model that features stochastic interest rates along with an underlying price process driven by stochastic string shocks combined with pure jump Lévy processes. Substituting the Brownian motion in the Black–Scholes model with a stochastic string leads to a class of option pricing models with expiration-dependent volatility. Further extending this Generalized Black–Scholes (GBS) model by adding Lévy jumps to the returns generating processes results in a new framework generalizing all exponential Lévy models. We derive four distinct versions of the model, with each case featuring a different jump process: the finite activity lognormal and double–exponential jump diffusions, as well as the infinite activity CGMY process and generalized hyperbolic Lévy motion. In each case, we obtain closed or semi-closed form expressions for European call option prices which generalize the results obtained for the original models. Empirically, we evaluate the performance of our model against the skews of S&P 500 call options, considering three distinct volatility regimes. Our findings indicate that: (a) model performance is enhanced with the inclusion of jumps; (b) the GBS plus jumps model outperform the alternative models with the same jumps; (c) the GBS-CGMY jump model offers the best fit across volatility regimes.

Predictable forward performance processes in complete markets

We establish existence of Predictable Forward Performance Processes (PFPPs) in conditionally complete markets, which has been previously shown only in the binomial setting. Our market model can be a discrete-time or a continuous-time model, and the investment horizon can be finite or infinite. We show that the main step in construction of PFPPs is solving a one-period problem involving an integral equation, which is the counterpart of the functional equation found in the binomial case. Although this integral equation has been partially studied in the existing literature, we provide a new solution method using the Fourier transform for tempered distributions. We also provide closed-form solutions for PFPPs with inverse marginal functions that are completely monotonic and establish uniqueness of PFPPs within this class. We apply our results to two special cases. The first one is the binomial market and is included to relate our work to the existing literature. The second example considers a generalized Black–Scholes model which, to the best of our knowledge, is a new result.

COMPARISON OF THREE OPTION PRICING MODELS FOR INDIAN OPTIONS MARKET


 
 Black-Scholes option pricing model is used to decide theoretical price of different Options contracts in many stock markets in the world. In can find many generalizations of BS model by modifying some assumptions of classical BS model. In this paper we compared two such modified Black-Scholes models with classical Black-Scholes model only for Indian option contracts. We have selected stock options form 5 different sectors of Indian stock market. Then we have found call and put option prices for 22 stocks listed on National Stock Exchange by all three option pricing models. Finally, we have compared option prices for all three models and decided the best model for Indian Options.
 Motivation/Background:
 In 1973, two economists, Fischer Black, Myron and Robert Merton derived a closed form formula for finding value of financial options. For this discovery, they got a Nobel prize in Economic science in 1997. Afterwards, many researchers have found some limitations of Black-Scholes model. To overcome these limitations, there are many generalizations of Black-Scholes model available in literature. Also, there are very limited study available for comparison of generalized Black-Scholes models in context of Indian stock market. For these reasons we have done this study of comparison of two generalized BS models with classical BS model for Indian Stock market.
 Method:
 First, we have selected top 5 sectors of Indian stock market. Then from these sectors, we have picked total 22 stocks for which we want to compare three option pricing models. Then we have collected essential data like, current stock price, strike price, expiration time, rate of interest, etc. for computing the theoretical price of options by using three different option pricing formulas. After finding price of options by using all three models, finally we compared these theoretical option price with market price of respected stock options and decided that which theoretical price has less RMSE error among all three model prices.
 Result:
 After going through the method described above, we found that the generalized Black-Scholes model with modified distribution has minimum RMSE errors than other two models, one is classical Black-Scholes model and other is Generalized Black-Scholes model with modified interest rate.
 

A compact difference scheme for time-fractional Black-Scholes equation with time-dependent parameters under the CEV model: American options

‎The Black-Scholes equation is one of the most important mathematical models in option pricing theory, but this model is far from market realities and cannot show memory effect in the financial market. This paper investigates an American option based on a time-fractional Black-Scholes equation under the constant elasticity of variance (CEV) model, which parameters of interest rate and dividend yield supposed as deterministic functions of time, and the price change of the underlying asset follows a fractal transmission system. This model does not have a closed-form solution; hence, we numerically price the American option by using a compact difference scheme. Also, we compare the time-fractional Black-Scholes equation under the CEV model with its generalized Black-Scholes model as α = 1 and β = 0. Moreover, we demonstrate that the introduced difference scheme is unconditionally stable and convergent using Fourier analysis. The numerical examples illustrate the efficiency and accuracy of the introduced difference scheme.

On a Free Boundary Problem for American Options Under the Generalized Black–Scholes Model

We consider the problem of pricing American options using the generalized Black–Scholes model. The generalized Black–Scholes model is a modified form of the standard Black–Scholes model with the effect of interest and consumption rates. In general, because the American option problem does not have an exact closed-form solution, some type of approximation is required. A simple numerical method for pricing American put options under the generalized Black–Scholes model is presented. The proposed method corresponds to a free boundary (also called an optimal exercise boundary) problem for a partial differential equation. We use a transformed function that has Lipschitz character near the optimal exercise boundary to determine the optimal exercise boundary. Numerical results indicating the performance of the proposed method are examined. Several numerical results are also presented that illustrate a comparison between our proposed method and others.

On the integral relationship between the early exercise boundary and the value function of the American put option

We prove in this paper a new integral relationship between the American put option early exercise boundary and its value function in the generalized Black–Scholes model. Based on this relationship we show that it is possible to construct the L2-approximation to the unknown early exercise boundary provided that we have at hand any uniform approximation of the American put option value function.

T-Stability of the Euler-Maruyama Algorithm for the Generalized Black-Scholes Model with Fractional Brownian Motion

T-Stability of the Euler-Maruyama Algorithm for the Generalized Black-Scholes Model with Fractional Brownian Motion

Numerical solution of generalized Black–Scholes model

This paper presents a numerical scheme that approximates the option prices for different option styles, governed by the generalized Black–Scholes equation in its degenerate form. The proposed method uses the HODIE scheme in the spacial direction and the two-step backward differentiation formula in the temporal direction. It is proved that the method has second order convergence in space as well as in time. Numerical experiments validate the theoretical results.

Optimal mean–variance efficiency of a family with life insurance under inflation risk

We study an optimization problem of a family under mean–variance efficiency. The market consists of cash, a zero-coupon bond, an inflation-indexed zero-coupon bond, a stock, life insurance and income-replacement insurance. The instantaneous interest rate is modeled as the Cox–Ingersoll–Ross (CIR) model, and we use a generalized Black–Scholes model to characterize the stock and labor income. We also take into account the inflation risk and consider our problem in the real market. The goal of the family is to maximize the mean of the surplus wealth at the retirement or death of the breadwinner and minimize its variance by finding a portfolio selection. The efficient frontier and optimal strategies are derived through the dynamic programming method and the technique of solving associated nonlinear HJB equations. We also present a numerical illustration to explore the impact of economical parameters on the efficient frontier.

Optimal investment, consumption and life insurance under stochastic framework

In this paper, we consider an optimal control problem of investment, consumption and life insurance for a wage earner who has constant relative risk aversion (CRRA) preferences. The wage earner can invest in zero-coupon bond, stock and life insurance. The interest rate is Vasicek model, the stock follows an extension of Heston stochastic volatility model. We use a generalized Black-Scholes model to characterize labor income. Risks of the incomes increasing rate and interest rate are cointegrated. The optimal strategies of the problem are derived by dynamic programming method and technique of solving associated HJB equations. We also present a sensitivity analysis to explore the impact of economical parameters on the optimal strategies.

High-order Numerical Method for Generalized Black-Scholes Model

This work presents a high order numerical method for the solution of generalized Black-Scholes model for European call option. The numerical method is derived using a two-step backward differentiation formula in the temporal discretization and a High-Order Difference approximation with Identity Expansion (HODIE) scheme in the spatial discretization. The present scheme gives second order accuracy in time and third order accuracy in space. Numerical experiments are conducted in support of the theoretical results.

On Some Auto-Induced Regime Switching Double-Threshold Glued Diffusions

Regime switching processes are usually defined with an external random source driving the regime changes. In this article, we define and study a regime switching diffusion considering two thresholds, and regime switching occurring, by a change in the diffusion drift and volatility, whenever the trajectory touches the upper threshold after having crossed, or touched, the lower threshold or touches the lower threshold after having crossed, or touched, the upper threshold. We develop an estimation procedure for the thresholds and for the regime parameters of the diffusions. We show that a generalized Black–Scholes model with the regime switching diffusion as the law of the risky asset is arbitrage free and complete under an additional hypothesis on the diffusion coefficients of the two regime diffusions.

Application of Moore-Penrose inverse in deciding the minimal martingale measure

The Moore-Penrose inverse is an important tool in algebra. This paper shows that the Moore-Penrose inverse is also an efficient technique in determining the minimal martingale measure if a security price follows a semi-martingale which satisfies some structure condition. We extend a result of Dzhaparidze and Spreij concerning the Moore-Penrose inverse to the case that the Moore-Penrose inverse of any matrix-valued predictable process is still predictable. Furthermore, we obtain an explicit formula of the minimal martingale measure by employing the Moore-Penrose inverse. Specifically, the minimal martingale measure in a generalized Black-Scholes model is found.

The Instantaneous Volatility and the Implied Volatility Surface for a Generalized Black–Scholes Model

Takaoka (Asia–Pacific Financial Markets 11:431–444, 2004) proposed a generalization of the Black–Scholes stock price model by taking a weighted average of geometric Brownian motions of different variance parameters. The model can be classified as a local volatility model, though its local volatility function is not explicitly given. In the present paper, we prove some properties concerning the instantaneous volatility process, the implied volatility curve, and the local volatility function of the generalized model. Some numerical computations are also carried out to confirm our results.

A Comparative Analysis of the Value of Information in a Continuous Time Market Model with Partial Information: The Cases of Log-Utility and CRRA.

In this paper we study the question what value an agent in a generalized Black-Scholes model with partial information attributes to the complementary information. To do this, we study the utility maximization problems from terminal wealth for the two cases partial information and full information. We assume that the drift term of the risky asset is a dynamic process of general linear type and that the two levels of observation correspond to whether this drift term is observable or not. Applying methods from stochastic filtering theory we derive an analytical tractable formula for the value of information in the case of logarithmic utility. For the case of constant relative risk aversion (CRRA) we derive a semi-analytical formula, which uses as an input the numerical solution of a system of ODE's. For both cases we present a comparative analysis.

Delta-hedging vega risk?

In this article we compare the profit and loss arising from the delta-neutral dynamic hedging of options, using two possible values for the delta of the option. The first is the Black–Scholes implied delta, while the second is the local delta, namely the delta of the option in a generalized Black–Scholes model with a local volatility, recalibrated to the market smile every day. We explain why, in negatively skewed markets, the local delta should provide a better hedge than the implied delta during slow rallies or fast sell-offs, and a worse hedge, although to a smaller extent, during fast rallies or slow sell-offs. Since slow rallies and fast sell-offs are more likely to occur than fast rallies or slow sell-offs in negatively skewed markets (provided we have physical as well as implied negative skewness), we conclude that, on average, the local delta provides a better hedge than the implied delta in negatively skewed markets. We obtain the same conclusion in the case of positively skewed markets. We illustrate these results using both simulated and real time-series of equity-index data, which have had a large negative implied skew since the stock market crash of October 1987. Moreover, we check numerically that the conclusions we draw are true when transaction costs are taken into account. In the last section we discuss the case of barrier options.

Estimation of local volatilities in a generalized Black–Scholes model

This paper studies a parameter estimation problem for a generalized Black–Scholes equation, which is used for option pricing. In estimating the volatility function from a set of market observations, we use an implicit finite difference scheme. The function space parameter estimation convergence (FSPEC) is proved and numerical simulations were performed.

Calibration of the local volatility in a trinomial tree using Tikhonov regularization

Following an approach introduced by Lagnado and Osher (Lagnado R and OsherS 1997 J. Comput. Finance 1 13–25), we study the application of Tikhonovregularization to the financial inverse problem of calibrating a local volatilityfunction from observed vanilla option prices. Moreover, we provide a unifiedtreatment for this problem in two different settings: first, the generalizedBlack–Scholes model, and second, a trinomial tree discretization. We presentserial and parallel implementations of the method in the discrete setting,using a probabilistic interpretation to compute, at significantly reducedcost, the gradient of the cost criterion. We illustrate the stability of thisregularized calibration procedure by numerical examples. Finally we extend thismethodology to the problem of calibration with American option prices.