Distributional divergence, statistical experiments and consequences in 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.

Forward Measures in a Ho and Lee Jump Diffusion Model

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.

Background/Objectives: The main objective of this paper is to present an n-step binomial model which can be used to price an option under any exotic conditions. Methods/Statistical Analysis: Mathematical models have been presented using which an n-step binomial model can be developed. The model can be used for estimating price of options under n of number exotic conditions that influence the option price. Findings: Pricing of exotic options like Asian, American etc., undertaken through Binomial Trees using only one-step considering maximum and minimum values that can be taken by the underling at the maturity leads to a rough approximation of the option price. The approximation is possible by assuming stock price movements to be in one or two binomial steps during the life of the option. A binomial tree extended to an N-step Model can be used to price various exotic options. A study of the convergence in European option price with respect to Number of steps (N) and variation in price of Asian and American options with respect to confidence factor (k) (proxy for implied volatility) using the maximum and minimum boundaries on the value of k gives the investors the ability to change the value of k so that they can have their own opinions concerning the risk-neutral probability distribution.

Bias Reduction in European Option Pricing

We consider the pricing of options when the dynamics of the risky underlying asset are driven by a Markov-modulated jump-diffusion model. We suppose that the market interest rate, the drift and the volatility of the underlying risky asset switch over time according to the state of an economy, which is modelled by a continuous-time Markov chain. The measure process is defined to be a generalized mixture of Poisson random measure and encompasses a general class of processes, for example, a generalized gamma process, which includes the weighted gamma process and the inverse Gaussian process. Another interesting feature of the measure process is that jump times and jump sizes can be correlated in general. The model considered here can provide market practitioners with flexibility in modelling the dynamics of the underlying risky asset. We employ the generalized regime-switching Esscher transform to determine an equivalent martingale measure in the incomplete market setting. A system of coupled partial-differential-integral equations satisfied by the European option prices is derived. We also derive a decomposition result for an American put option into its European counterpart and early exercise premium. Simulation results of the model have been presented and discussed.

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

<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>

Efficient pricing of options in jump–diffusion models: Novel implicit–explicit methods for numerical valuation

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.

European Option Pricing under Jump Diffusion with Proportional Transaction Costs

In this paper, using the utility based option pricing approach, pioneered by Hodges and Neuberger, we propose a new model in a continuous-time market with proportional transaction costs and several trading restrictions. According to this model, we study the problem of option pricing and prove that the value function of our model is a unique constrained viscosity solution of a HJB equation.

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

Equity basket correlation can be estimated both using the physical measure from stock prices, and also using the risk neutral measure from option prices. The difference between the two estimates motivates a so-called “dispersion strategy”. We study the performance of this strategy on the German market and propose several profitability improvement schemes based on implied correlation (IC) forecasts. Modelling IC conceals several challenges. Firstly the number of correlation coefficients would grow with the size of the basket. Secondly, IC is not constant over maturities and strikes. Finally, IC changes over time. We reduce the dimensionality of the problem by assuming equicorrelation. The IC surface (ICS) is then approximated from the implied volatilities of stocks and the implied volatility of the basket. To analyze the dynamics of the ICS we employ a dynamic semiparametric factor model.