VARIANCE GAMMA PROCESS WITH MONTE CARLO SIMULATION AND CLOSED FORM APPROACH FOR EUROPEAN CALL OPTION PRICE DETERMINATION

The Option is widely applied in the financial sector. The Black-Scholes-Merton model is often used in calculating option prices on a stock price movement. The model uses geometric Brownian motion which assumes that the data is normally distributed. However, in reality, stock price movements can cause sharp spikes in data, resulting in nonnormal data distribution. So we need a stock price model that is not normally distributed. One of the fastest growing stock price models today is the process exponential model. The process has the ability to model data that has excess kurtosis and a longer tail (heavy tail) compared to the normal distribution. One of the members of the process is the Variance Gamma (VG) process. The VG process has three parameters which each of them, to control volatility, kurtosis and skewness. In this research, the secondary data samples of options and stocks of two companies were used, namely zoom video communications, Inc. (ZM) and Nokia Corporation (NOK). The price of call options is determined by using closed form equations and Monte Carlo simulation. The Simulation was carried out for various values until convergent result was obtained.

Empirical study based on the model of rough fractional stochastic volatility (RFSV)

To estimate the parameters of the model of option pricing based on the model of rough fractional stochastic volatility (RFSV), we have carried out the empirical analysis during our study on the pricing of SSE 50ETF options in China. First, we have estimated the parameters of option pricing model by adopting the Monte Carlo simulation. Subsequently, we have empirically examined the pricing performance of the RFSV model by adopting the SSE 50ETF option price from January 2019 to December 2020. Our research findings indicate that by leveraging the RFSV model, we are able to attain a more accurate and stable level of option pricing than the conventional Black–Scholes (B-S) model on constant volatility. The errors of option pricing incurred by the B-S model proved to be larger and exhibited higher volatility, revealing the significant impact imposed by stochastic volatility on option pricing.

Limit theorems for prices of options written on semi-Markov processes

We consider plain vanilla European options written on an underlying asset that follows a continuous time semi-Markov multiplicative process. We derive a formula and a renewal type equation for the martingale option price. In the case in which intertrade times follow the Mittag-Leffler distribution, under appropriate scaling, we prove that these option prices converge to the price of an option written on geometric Brownian motion time-changed with the inverse stable subordinator. For geometric Brownian motion time changed with an inverse subordinator, in the more general case when the subordinator’s Laplace exponent is a special Bernstein function, we derive a time-fractional generalization of the equation of Black and Scholes.

Projection of the two-dimensional Black-Scholes equation for options with underlying stock and strike prices in two different currencies

The two-variable Black-Scholes equation is used to study the option exercise price of two different currencies. Due to the complexity of dealing with several variables, reduction methods have been implemented to deal with these problems. This paper proposes an alternative reduction by using the so-called Zwanzig projection method to one-dimension, successfully developed to study the diffusion in confined systems. In this case, the option price depends on the stock price and the exchange rate between currencies. We assume that the exchange rate between currencies will depend on the stock price through some model that bounds such dependence, which somehow influences the final option price.As a result, we find a projected one-dimensional Black-Scholes equation similar to the so-called Fick-Jacobs equation for diffusion on channels. This equation is an effective Black-Scholes equation with two different interest rates, whose solution gives rise to a modified Black-Scholes formula. The properties of this solution are shown and were graphically compared with previously found solutions, showing that the corresponding difference is bounded.

On Market Completions Approach to Option Pricing

Option pricing is one of the most important problems of contemporary quantitative finance. It can be solved in complete markets with non-arbitrage option price being uniquely determined via averaging with respect to a unique risk-neutral measure. In incomplete markets, an adequate option pricing is achieved by determining an interval of non-arbitrage option prices as a region of negotiation between seller and buyer of the option. End points of this interval characterise the minimum and maximum average of discounted pay-off function over the set of equivalent risk-neutral measures. By estimating these end points, one constructs super hedging strategies providing a risk-management in such contracts. The current paper analyses an interesting approach to this pricing problem, which consists of introducing the necessary amount of auxiliary assets such that the market becomes complete with option price uniquely determined. One can estimate the interval of non-arbitrage prices by taking minimal and maximal price values from various numbers calculated with the help of different completions. It is a dual characterisation of option prices in incomplete markets, and it is described here in detail for the multivariate diffusion market model. Besides that, the paper discusses how this method can be exploited in optimal investment and partial hedging problems.

Proximity of Bachelier and Samuelson Models for Different Metrics

This paper proposes a method of comparing the prices of European options, based on the use of probabilistic metrics, with respect to two models of price dynamics: Bachelier and Samuelson. In contrast to other studies on the subject, we consider two classes of options: European options with a Lipschitz continuous payout function and European options with a bounded payout function. For these classes, the following suitable probability metrics are chosen: the Fortet-Maurier metric, the total variation metric, and the Kolmogorov metric. It is proved that their computation can be reduced to computation of the Lambert in case of the Fortet-Mourier metric, and to the solution of a nonlinear equation in other cases. A statistical estimation of the model parameters in the modern oil market gives the order of magnitude of the error, including the magnitude of sensitivity of the option price, to the change in the volatility.

SPX Calibration of Option Approximations under Rough Heston Model

The volatility of stock return does not follow the classical Brownian motion, but instead it follows a form that is closely related to fractional Brownian motion. Taking advantage of this information, the rough version of classical Heston model also known as rough Heston model has been derived as the macroscopic level of microscopic Hawkes process where it acts as a high-frequency price process. Unlike the pricing of options under the classical Heston model, it is significantly harder to price options under rough Heston model due to the large computational cost needed. Previously, some studies have proposed a few approximation methods to speed up the option computation. In this study, we calibrate five different approximation methods for pricing options under rough Heston model to SPX options, namely a third-order Padé approximant, three variants of fourth-order Padé approximant, and an approximation formula made from decomposing the option price. The main purpose of this study is to fill in the gap on lack of numerical study on real market options. The numerical experiment includes calibration of the mentioned methods to SPX options before and after the Lehman Brothers collapse.

Option pricing under the fractional stochastic volatility model

We investigate the European call option pricing problem under the fractional stochastic volatility model. The stochastic volatility model is driven by both fractional Brownian motion and standard Brownian motion. We obtain an analytical solution of the European option price via the Itô’s formula for fractional Brownian motion, Malliavin calculus, derivative replication and the fundamental solution method. Some numerical simulations are given to illustrate the impact of parameters on option prices, and the results of comparison with other models are presented. doi:10.1017/S1446181121000225

Pricing timer options: second-order multiscale stochastic volatility asymptotics

We study the pricing of timer options in a class of stochastic volatility models, where the volatility is driven by two diffusions—one fast mean-reverting and the other slowly varying. Employing singular and regular perturbation techniques, full second-order asymptotics of the option price are established. In addition, we investigate an implied volatility in terms of effective maturity for the timer options, and derive its second-order expansion based on our pricing asymptotics. A numerical experiment shows that the price approximation formula has a high level of accuracy, and the implied volatility in terms of its effective maturity is illustrated. doi:10.1017/S1446181121000249