Bias Reduction in European Option Pricing

This paper is mainly about the research object of early exercise of American options by two-period binomial models. In this work, we use the two-period binomial model to analyze the price of American options and European options, decide whether to exercise early or not, and compare the differences between American options and European options. What is more, we use Python to help us with our calculations. Options are a kind of financial derivatives that help one to arbitrage, speculate, and hedge. To make more profits, people can use the binomial model to calculate the price of options. Through this analysis, we got to know each price of options in different situations, and we know that the price of American options is always equal to or larger than the price of European options. This work can help us to calculate the price of the option and decide if it is exercised early, and make sure to get more profits and apply it to daily life.

Option pricing is used in the greater part of commonly traded instruments in the financial market that predicts the future stock price. The option’s fair price can be found either analytically or by applying suitable numerical techniques. Black-Scholes (B-S) seminal approach is one of them, but it has less accuracy in pricing all types of options. Therefore, a lattice model, namely a trinomial model, was developed to overcome these valuation problems for American or European option pricing. However, Trinomial model is not capable of providing maximum accuracy for option pricing as the system of equations used in the model involves a non-linear growth of the stock price. It is, therefore, important to develop a new system of equations to enhance the performance of trinomial framework in contemplation of American or European options. This study aims to improve the performance of the trinomial model for American option by developing a new system of equations with a linear growth rate of the stock price in different states of the regime so that all the data can be stored in the amalgamation of a trinomial tree. The performance was assessed by comparing the trinomial model to the binomial and Black-Scholes (B-S) pricing for the American option. The results demonstrated that the trinomial option pricing gives higher accuracy than Black-Scholes (B-S) pricing and is faster than the binomial pricing.

<abstract><p>In this paper, the pricing of European options under a new two-factor non-affine stochastic volatility model is studied. In order to reduce the computational complexity, we use the Taylor expansion and Fourier-cosine method to derive an analytical approximation formula for European option prices. Numerical experiments prove that the proposed formula is fast and efficient for pricing European options compared with Monte Carlo simulations. The sensitivity of the parameters is analyzed to explain the rationality of the model. Finally, we present some preliminary empirical analysis revealing that the pricing performance of our proposed model is superior to that of the single-factor model.</p></abstract>

The impact of market frictional costs on pricing insurance and financial products in a regime-switching environment has not been well-explored. This paper introduces a general pricing model for European options which incorporates market frictional costs, regime switches and model uncertainty. Regime switches are due to changes in an economic environment. Model uncertainty is attributed to misspecification of transition intensities for economic regimes. The selling and buying prices of a European option are determined through stochastic optimal control and nonlinear partial differential equations. A fair value is determined by a closed-form solution to a minimization problem based on a relative entropy. The fair value is consistent with the one obtained using the Esscher transform, which is an important tool in actuarial science. Numerical methods and results for implementing the pricing model are presented. The results indicate that after controlling for the model uncertainty, market frictional costs are more significant than regime switches in accounting for the fair, selling and buying prices.

An analytical approximation formula for European option pricing under a new stochastic volatility model with regime-switching

Approach to the Delta Greek of nonlinear Black–Scholes equation governing European options

The main purpose of this paper is to study the stock price jump process is more common than Poisson process -a kind of special renewal process-the incident time interval is independent random variable sequence and identical subordinate to Gamma distribution. The object of study is European bi-direction option, It also known as the double two-way options. Also, European bi-direction option is an exotic option. Then, build up stochastic differential equation under the circumstance of the market no arbitrage based on stochastic analysis and martingale theory. Based on the above discussion, we obtain the European bi-direction option pricing formulas under jump diffusion model by simple mathematical induce by means of martingale method.

Derivatives have become widely accepted as tools for hedging and risk-management, as well as speculation to some extent. A more recent trend has been gaining ground, namely, arbitrage in derivatives. The critical parameter in derivatives pricing is the volatility of the underlying asset. Exchanges often overestimate volatility in order to cover any sudden changes in market behavior, leading to systematic overpricing of derivatives. Accurate forecasting of volatility would expose systematic overpricing. Unfortunately, volatility is not an easy phenomenon to predict or forecast. One class of models that have proved successful in forecasting volatility in many situations is the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) family of models. The objective of the present study is to analyze systematic bias in the pricing of options derivatives. In order to perform the analysis, data were collected for a sample of stock options traded on the National Stock Exchange (NSE) of India and their underlying stocks. In the study, GARCH models are used to forecast underlying stock volatility, and the forecasted volatility is used in the Black-Scholes model in order to determine whether the corresponding options were fairly priced. Any systematic bias in options pricing would provide evidence for arbitrage opportunities.

In this paper, we consider the problem of option pricing when return rate and volatility are ambiguous. Firstly we illustrate how to describe this ambiguous option pricing model by using set-valued differential inclusion and how to change the discussion of pricing bound problems of options into that of maximal and minimal conditional expectations. Secondly we discuss the properties of maximal and minimal conditional expectations, especially the representation theorem of maximal and minimal expectations. Finally we give the bounds of the European option pricing by using above theorems.KeywordsSet-valued stochastic differential inclusionMartingale measuresMaximal and minimal conditional expectationsBounds of option prices

ABSTRACTDistributional Divergence and Statistical Experiments are used herein for a positive stochastic process. This framework provides, under mild assumptions, Risk Neutral Probability (-ies) for a stock price process which does not have necessarily either to satisfy a Stochastic Differential Equation or to follow a model, both non-realistic assumptions. The results contribute in understanding the relation between statistical contiguity and market's informational efficiency. -price of European option is obtained, confirming the universal quote of the Black–Scholes–Merton price for the class of calm stock prices that includes log-normal price. Other consequences are presented.

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.

These days, option pricing is crucial in trading to make a decision that would lead to the best benefits. There are some conditions that must be considered in choosing volatility functions in simulating European option price model under the Ornstein-Uhlenbeck stochastic volatility assumption. To approximate the option price based on that assumption, we use the Euler-Maruyama scheme to approximate a call option prices based on the European option model. The volatility functions for the model must meet a Holder condition requirement to achieve error convergence when the option price is approximated. Regarding that purpose, we consider some volatility functions and investigate how the chosen functions affect the performance of the model.

In this paper, we analyze the pricing of European option when the riskfree interest rate follows a jump process. An expression for European call price is first obtained in the case of constant volatility. Then we present a general formulation which takes care of the variation of volatility. In this general formulation we utilize a point process filtering technique to estimate the state process. Finally we carry out some numerical simulation of our results

The aim of this thesis is to develop efficient valuation methods for financial contracts under models with jumps and stochastic volatility, and to present their rigorous mathematical underpinning. For efficient risk management, large books of exotic options need to be priced and hedged under models that are flexible enough to describe the observed option prices at speeds close to real time. To do so, hundreds of vanilla options, which are quoted in terms of implied volatility, need to be calibrated to market prices quickly and accurately on a regular basis. With this in mind we develop efficient methods for the evaluation of (i) vanilla options, (ii) implied volatility and (iii) common path-dependent options. Firstly, we derive a new numerical method for the classical problem of pricing vanilla options quickly in time-changed Brownian motion models. 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 option pricing method with respect to speed appears to favour our newly developed method in the cases considered. Secondly, 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. In order 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 is possible using other common methods. Error estimates are presented for a wide range of parameters. Thirdly, we develop a new Monte Carlo variance reduction method to estimate the expectations of path-dependent functionals, such as first-passage times and occupation times, under a class of stochastic volatility models with jumps. The method is based on a recursive approximation of the first-passage time probabilities and expected occupation times of Levy bridge processes that relies in part on a randomisation of the timeparameter. We derive the explicit form of the recursive approximation in the case of bridge processes corresponding to the class of Levy processes with mixed-exponential jumps, and

This paper reviews the past research on option pricing, including the Bachelier option pricing formula, Black-Scholes model, and the Cox-Ross-Rubinstein (CRR) binomial option pricing model. Then some practical calculations are performed on the Black-Scholes model and CRR binomial model and the differences between the results of these two models are analyzed for investors to choose the appropriate model for option prices. From the practical calculation results, it is verified that the Black-Scholes model can give out the analytical solutions of option prices and decrease the calculation of solution, while it can only be used for European options with stock prices regarded as a continuous process. For the CRR binomial model, although it can only give numerical solutions and requires lots of calculation because of too many branches, but it can solve the American option prices with early exercises and the European option prices with stock prices regarded as a discrete process.