Option Pricing with a Dividend General Equilibrium Model

In this paper we summarize the main idea and results of Yuen and Yang (2009, 2010a, 2010b) and provide some results on pricing of Parisian options under the Markov regime-switching model (MRSM). The MRSM allows the parameters of the market model depending on a Markovian process, and the model can reflect the information of the market environment which cannot be modeled solely by linear Gaussian process. However, when the parameters of the stock price model are not constant but governed by a Markovian process, the pricing of the options becomes complex. We present a fast and simple trinomial tree model to price options in MRSM. In recent years, the pricing of modern insurance products, such as Equity-Indexed annuity (EIA) and variable annuities (VAs), has become a popular topic. We show here that our trinomial tree model can been used to price EIA with strong path dependent exotic options in the regime switching model.

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.

We introduced a new method to compute the European Call (and Put) Option price under the assumption of multifractional Brownian motion (mBm). The reason why we need a procedure for estimating the Option price is due to the absence of a closed formula for this process. To compute the Option price, we first simulated the logarithmic price under mBm and, by using a discount factor, we computed the option’s pay-off. Then, we fitted the best probability distribution associated to the discounted pay-off, computing the European Call Option price as its average.

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.

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.

The detecting regime structure of securities that is chosen by investors have started to gain importance day by day. In this research, regime structure of BIST-100 Index, which is one of the most intensively traded index of Borsa Istanbul, has been analyzed by Markov Regime Switching (MRS) Model. It will be determined the number of regimes of BIST-100 Index by using a daily closing value data set from 03.01.1997 to 31.07.2017 (4951 daily data). Firstly, return series of BIST-100 Index is calculated as percentage for analyzes then MRS model applied to this series in order to detect the number of regimes. Based on the findings, it has been determined three different regimes for BIST-100 Index. All of these regimes, which were defined as, low, normal and high regimes statistically significant at %5 significance level. Upon investigating the transition probabilities of MRS model, BIST-100 Index is generally observed to indicate a tendency towards a shift to normal regime. According to transition probabilities results, while the index is in the high regime, it prefers to stay in the high regime with a probability of 19.9%, wants to shift from high regime to normal regime with the probability of 64.96% and wants to shift from high regime to low regime with the probability of 15%. Providing that the index is within the normal regime, next period index will remain in the normal regime with %97.5 probability while ?t will shift from normal regime to low regime with %1.5 probability. In addition, it was detected that the possibility of index shift from normal regime to high regime is %1. When the regime is low, on the other hand, next period it will be the same as %24.31 probability while the regime will shift from low to high with %17.8 probability. The probability of shift from low regime to normal regime is %57.86. After determined the transition probabilities of indices, it was also calculated the expected durations of each regime separately. Findings have indicated that, duration of low regime is 1,32 days and high regime is 1.24 days whereas expected duration of normal regime is 41.22 days. Finally, BIST-100 Index is proposed as one of the indices that can be invested due to the normal regime structure for investors who do not prefer to take the risk and seek stability.

Purpose ―Current paper assesses the impact of macroeconomic variables on Pakistan's economic growth. Method ― This study analyzed the data using the Markov Regime switching (MS) model using monthly data for 1981-2020. Firstly, BDS and CUSUM square tests were applied to detect the non-linearity of the model. Results ―The model is non-linear, so the Markov regime-switching model is used for analysis. Each regime's mean and variance are highly significant and show a high growth regime with high volatility and a low growth regime with low volatility. Furthermore, the results show that inflation, interest rate, and trade openness negatively impact while real effective exchange rates positively affect development in both regimes. The negative effect of interest rate, exchange rate, inflation, and trade openness become more pronounced in low growth regimes. Implication ― This study suggests that policymakers should consider the non-linear behaviour of macroeconomics. This will help to formulate better policies for the economy's economic growth. Originality ―The current research adds to the existing literature by identifying the non-linear effect of growth indicators on economic growth, which was previously neglected in the case of Pakistan.

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.

A non-parametric valuation framework (ANN-MRS) using artificial neural networks for pricing financial derivatives has been developed whilst the volatility of underlying asset return dynamics are modelled by Markov regime switching model. Its immediate application is on pricing of the Chinese warrants. To access the potential of neural network pricing with volatility in regime switching, weekly data of Jiangtong Stock returns are used to calculate the volatilities by using the maximum likelihood estimation. The ability of neural network for predicting the warrant prices is compared to the Black-Scholes model. Comparisons reveal that the mean squared error for the neural network is less than that of the Black-Scholes model in both in sample and out of sample estimations. The result indicates the neural network model coupled with Markov regime switching (for volatility estimation) has a superior performance comparing the warrant pricing by using the Black-Scholes model with historical volatility.

This article focuses on the pricing problem of foreign equity options under a Markovian regime-switching model with liquidity risk and default risk. Some basic parameters of foreign equity price, foreign exchange rate, and counterparty’s asset value are controlled by a continuous-times Markov chain. We adopt the liquidity discount factor to describe effects of the market liquidity on foreign equity price and counterparty’s asset value. The regime-switching Esscher transform is used to derive a risk-neutral measure. Based on the proposed pricing framework, we provide an explicit pricing formula for foreign equity options with default risk via the Fourier transform technique. Finally, numerical analysis is presented to illustrate the effects of regime-switching, liquidity risk, and default risk on the option price.

Option pricing model is a wildly interested topic in an area of financial Mathematics. The pioneer model was introduced by Fischer Black and Myron Scholes which is known as the Black-Scholes model. This model was derived under various assumptions such as liquidity and no transaction costs for which a underlying asset price in stock market might not be satisfied. With this fact, the underlying asset price models were remodeled, in order to determine an option value. This research aims to extend the Black-Scholes model by relaxing the assumption of no transaction costs in illiquid markets. Also, jumps of asset price are considered in this work. To do this, a differential form of asset price with transaction costs and jumps in illiquid markets is introduced and then used to construct the extended option pricing model. Furthermore, a numerical result of a call option price under a new situation is provided.

Forecasting crude oil price volatility via a HM-EGARCH model

Market traders often buy and sell stocks to maximize their total return. For each purchase and sale, there is often a return commission. Having trading technology plays an important role in quantitative trading. In this paper, we first use the XGBoost model to learn the historical price fluctuation data of gold and bitcoin, and the prediction accuracy R2 is between 0.998 and 0.999, which is a good fit. Then LT algorithm and PS algorithm are used to identify the rules, the Markov regime switching model is used to determine the bear and bull market, and finally, the DQN model is used to plan the trading strategy and add a machine learning algorithm to it to optimize the strategy to get the initial $1000 in the future investment strategy.

In this study, we investigate the determinants of credit spread using a Markov regime-switching model. We consider corporate governance variables and credit risk to analyze the determinants of credit spread. The corporate governance mechanism is an indicator of company sustainability, and credit spread is the main factor in profits obtained by banks. However, the relationship between credit spread and corporate governance is seldom discussed. We focus on loans from banks in Taiwan between 2000 and 2019 and apply a Markov regime-switching model, which is superior to other models in capturing different effects in various regimes. We specify two regime types: corporate governance and credit risk regimes. Furthermore, we consider four aspects of corporate governance: firm ownership structure, board structure, deviation, and information environment. In this study, the determinants of credit spread are investigated more thoroughly than in previous studies. Moreover, in this study, we examine the effects of monetary policy and economic status on credit spread using a Markov regime-switching model; such models are not employed to their full extent in related studies of credit spread. Empirical results indicate that credit spread has different effects in various regimes. Thus, understanding the determinants of credit spread in different regimes is crucial for financial analysts, investors, economic policymakers, and banks. Consequently, we expect that this study can improve the management and measurement of credit risk and be of value to financial institutions.

Contents: Acknowledgements General Introduction PART I HARRY M. MARKOWITZ Introduction to Part I Howard R. Vane and Chris Mulhearn 1. Harry Markowitz (1952a), 'Portfolio Selection' 2. Harry Markowitz (1952b), 'The Utility of Wealth' 3. H. Levy and H.M. Markowitz (1979), 'Approximating Expected Utility by a Function of Mean and Variance' 4. Harry M. Markowitz and Eric L. van Dijk (2003), 'Single-Period Mean-Variance Analysis in a Changing World' PART II MERTON H. MILLER Introduction to Part II Howard R. Vane and Chris Mulhearn 5. Franco Modigliani and Merton H. Miller (1958), 'The Cost of Capital, Corporation Finance and the Theory of Investment' 6. Franco Modigliani and Merton H. Miller (1959), 'The Cost of Capital, Corporation Finance and the Theory of Investment: Reply' 7. Merton H. Miller and Franco Modigliani (1961), 'Dividend Policy, Growth, and the Valuation of Shares' 8. Franco Modigliani and Merton H. Miller (1963), 'Corporate Income Taxes and the Cost of Capital: A Correction' 9. Merton H. Miller and Franco Modigliani (1966), 'Some Estimates of the Cost of Capital to the Electric Utility Industry, 1954-57' 10. Merton H. Miller (1977), 'Debt and Taxes' PART III WILLIAM F. SHARPE Introduction to Part III Howard R. Vane and Chris Mulhearn 11. William F. Sharpe (1963), 'A Simplified Model For Portfolio Analysis' 12. William F. Sharpe (1964), 'Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk' 13. William F. Sharpe (1966), 'Mutual Fund Performance' 14. William F. Sharpe (1978), 'Bank Capital Adequacy, Deposit Insurance, and Security Values' PART IV ROBERT C. MERTON Introduction to Part IV Howard R. Vane and Chris Mulhearn 15. Robert C. Merton (1969), 'Lifetime Portfolio Selection Under Uncertainty: The Continuous-Time Case' 16. Robert C. Merton (1971), 'Optimum Consumption and Portfolio Rules in a Continuous-Time Model' 17. Robert C. Merton (1973a), 'Theory of Rational Option Pricing' 18. Robert C. Merton (1973b), 'An Intertemporal Capital Asset Pricing Model' 19. Robert C. Merton (1974), 'On the Pricing of Corporate Debt: The Risk Structure of Interest Rates' 20. Robert C. Merton (1977), 'On the Pricing of Contingent Claims and the Modigliani-Miller Theorem' PART V MYRON S. SCHOLES Introduction to Part V Howard R. Vane and Chris Mulhearn 21. Fischer Black and Myron Scholes (1972), 'The Valuation of Option Contracts and a Test of Market Efficiency' 22. Fischer Black and Myron Scholes (1973), 'The Pricing of Options and Corporate Liabilities' 23. Fischer Black and Myron Scholes (1974), 'The Effects of Dividend Yield and Dividend Policy on Common Stock Prices and Returns' 24. Myron Scholes and Joseph Williams (1977), 'Estimating Betas from Nonsynchronous Data' 25. Merton H. Miller and Myron S. Scholes (1978), 'Dividends and Taxes' Name Index