European Option Pricing Research Articles

A closed-form approximation formula for pricing European options under a three-factor model

AbstractThe double-mean-reverting model, introduced by Gatheral [(2008). Consistent modeling of SPX and VIX options. In The Fifth World Congress of the Bachelier Finance Society London, July 18], is known to be a successful three-factor model that can be calibrated to both CBOE Volatility Index (VIX) and S&P 500 Index (SPX) options. However, the calibration of this model may be slow because there is no closed-form solution formula for European options. In this paper, we use a rescaled version of the model developed by Huh et al. [(2018). A scaled version of the double-mean-reverting model for VIX derivatives. Mathematics and Financial Economics 12: 495–515] and obtain explicitly a closed-form pricing formula for European option prices. Our formulas for the first and second-order approximations do not require any complicated calculation of integral. We demonstrate that a faster calibration result of the double-mean revering model is available and yet the practical implied volatility surface of SPX options can be produced. In particular, not only the usual convex behavior of the implied volatility surface but also the unusual concave down behavior as shown in the COVID-19 market can be captured by our formula.

European Option Pricing Formula in Risk-Aversive Markets

In this study, using the method of discounting the terminal expectation value into its initial value, the pricing formulas for European options are obtained under the assumptions that the financial market is risk-aversive, the risk measure is standard deviation, and the price process of underlying asset follows a geometric Brownian motion. In particular, assuming the option writer does not need the risk compensation in a risk-neutral market, then the obtained results are degenerated into the famous Black–Scholes model (1973); furthermore, the obtained results need much weaker conditions than those of the Black–Scholes model. As a by-product, the obtained results show that the value of European option depends on the drift coefficient μ of its underlying asset, which does not display in the Black–Scholes model only because μ = r in a risk-neutral market according to the no-arbitrage opportunity principle. At last, empirical analyses on Shanghai 50 ETF options and S&P 500 options show that the fitting effect of obtained pricing formulas is superior to that of the Black–Scholes model.

European Option Pricing under Wishart Processes

This study deals with a single risky asset pricing model whose volatility is described by Wishart affine processes. This multifactor model with two dependency matrices describing the correlation between the asset dynamic and Wishart processes makes it more flexible enough to fit the market data for short or long maturities. The aim of the study is to derive and solve the call option pricing problem under the double Wishart stochastic volatility model. The Fourier transform techniques combined with perturbation methods are employed in order to price the European call options. The numerical illustrations on pricing predictions show similar behavior of price movements under the double Wishart model with respect to the market price.

On a strongly continuous semigroup for a Black-Scholes integro-differential operator: European options under jump-diffusion dynamics

We consider a Black-Scholes integro-differential operator associated with a partial integro-differential equation for pricing European options with a jump-diffusion process for the underlying asset. Using the theory of one-parameter semigroups, we prove that the operator is the infinitesimal generator of a strongly continuous semigroup and express the semigroup explicitly as a convolution of a jump function, the Black-Scholes kernel and the payoff function. This is analogous to the Gauss-Weierstrass and Poisson semigroups. Then we investigate the pricing of European options under jump diffusion for two broad classes of payoff functions. A generalised put-call parity relating the functions from both classes is also obtained.

Estimation of the Bid-Ask Prices for the European Discrete Geometric Average and Arithmetic Average Asian Options

Conic finance is a new and exciting development in quantitative finance, which is widely applied to several topics in finance. The theory of conic finance extends the law of one price to the law of two prices, which yields closed forms for bid-ask prices of European options. In this paper, within the framework of conic finance, we derive effective, explicit, approximate formulas to estimate the bid-ask prices for the European discrete geometric average and arithmetic average Asian options. Finally, we give two examples to demonstrate and validate that the approximate closed-form solutions are efficient and accurate.

A Unified Theory of Option Pricing Assuming Stochastic Volatility

Stochastic volatility models have grown in popularity in the past decade or two. However, for many stochastic volatility models, the functional form of volatility along with the description of the diffusion process for volatility have been posed with analytic convenience in mind. Here, we consider that analytic tractability may degenerate as realistic modelling improves and that a more general specification for the stock price and volatility processes may be necessary. This leads to an approximating polynomial for European option prices which is benchmarked to two popular stochastic volatility models, the Stein and Stein, and Heston models, before examining a more general specification which is compared to the corresponding Black Scholes price. Stochastic volatility and European option approximation and Heston and Stein and Stein and Black Scholes

Efficient Estimation of Pricing Kernels and Market-Implied Densities

This paper studies the nonparametric identification and estimation of projected pricing kernels implicit in European option prices and underlying asset returns using conditional moment restrictions. The proposed series estimator avoids computing ratios of estimated risk-neutral and physical densities. Instead, we consider efficient estimation based on the conditional Euclidean empirical likelihood or continuously-updated GMM criterion, which takes into account the informativeness of option prices of varying strike prices beyond observed conditioning variables. In a second step, we convert the implied probabilities into predictive densities by matching the informative part of cross-sections of option prices. Empirically, pricing kernels tend to be U-shaped in the S&P 500 index return given high levels of the VIX, and call and ATM options are more informative about their payoff than put and OTM options.

Bond and Option Prices under Skew Vasicek Model with Transaction Cost

This paper studies the European option pricing on the zero-coupon bond in which the Skew Vasicek model uses to predict the interest rate amount. To do this, we apply the skew Brownian motion as the random part of the model and show that results of the model predictions are better than other types of the model. Besides, we obtain an analytical formula for pricing the zero-coupon bond and find the European option price by constructing a portfolio that contains the option and a share of the bond. Since the skew Brownian motion is not a martingale, thus we add transaction costs to the portfolio, where the time between trades follows the exponential distribution. Finally, some numerical results are presented to show the efficiency of the proposed model.

The closed-form option pricing formulas under the sub-fractional Poisson volatility models

A new fractional process called the sub-fractional Poisson process NH(t) is proposed, which has continuous sample paths, long- memory, leptokurtosis and heavy tail distribution, is of convenience to price options and simulate the variance process of risk asset return. Based on the sub-fractional Poisson process NH(t) the new fractional variance processes have been proposed, which may capture the skewness and the long-memory as well as mean-reverting in the stock price volatilities. In particular, the characteristic function method for option pricing is given, and the analytical formulas for European option price C(t,St) have been obtained under the risk-neutral probability measure.

Binomial tree method for option pricing: Discrete Carr and Madan formula approach

This paper suggests a new Fourier analysis approach to evaluate the option prices and its sensitivities (Greeks) using the binomial tree model. In the last half of this paper, we show that option prices are efficiently and effectively evaluated using a semi-closed form formula for European option prices. We can compute option prices in a broad class of jump-diffusion models because we calculate the characteristic function for an underlying asset price numerically. Furthermore, we also compute the price of European options in the exp-Levy model. This numerical experiment gives new insights into option pricing in the nonparametric Levy model. The option prices and sensitivities can be computed very accurately and efficiently, even in binomial tree models with jumps.

Utility-indifference pricing of European options with proportional transaction costs

Pricing options in a market with transaction costs is an important research topic in quantitative finance. There are basically two categories of approaches to deal with transaction costs. One is discrete hedging strategies under the Black–Scholes framework, another is utility indifference approaches. Hedging approaches are easy to implement, but preference-independent, whereas utility indifference approaches incorporate risk preferences, in general involving lengthy calculations. In a market with transaction costs, trading stocks involves risk so investors’ risk preferences must be taken into consideration. In this paper, we price European options with proportional transaction costs using a utility indifference approach which updates the fraction of one’s total wealth in the risky asset at regular intervals. Our HJB equation for the portfolio without option is two-dimensional instead of three-dimensional as in standard utility indifference approaches. Since the HJB equations could only be solved by a numerical method, our approach provides saving of computing time. Results from our numerical experiments via the finite difference method are presented to illustrate the effects of key option parameters on option prices.

Pricing of Adverse Development Covers Using Option Pricing Methods

The market for Adverse Development Covers and Loss Portfolio Transfers has been growing in the past few years. Despite this growth, reinsurers are still struggling to define a standard method for pricing such covers. In this context, this article aims at providing an innovative method for pricing such contracts. The proposed method is based on the famous Mack model and fits a Constant Elasticity of Variance (CEV) model to the Mack results (expected value and standard deviation) for each future development year of each accident/underwriting year. Having fitted the CEV model, it is possible to estimate the value of the Adverse Development Covers for each accident/underwriting year using standard European option pricing techniques and to compare this valuation with usual non-life insurance valuation techniques.

REPLICATION SCHEME FOR THE PRICING OF EUROPEAN OPTIONS

This paper proposes an efficient method for calculating European option prices under local, stochastic, and fractional volatility models. Instead of directly calculating the density function of a target underlying asset, we replicate it from a simpler diffusion process with a known analytical solution for the European option. For this purpose, we derive six functions that characterize the density function of a diffusion process, for both the original and simpler processes and match these functions so that the latter mimics the former. Using the analytical formula, we then approximate the option price of the target asset. By comparison with previous works and numerical experiments, we show that the accuracy of our approximation is high, and the calculation is fast enough for practical purposes; hence, it is suitable for calibration purposes.

An adaptive Monte Carlo algorithm for European and American options

Abstract. In this paper, a new adaptive Monte Carlo algorithm is proposed to solve systems of linear algebraic equations (SLAEs). The corresponding properties of the algorithm and its advantages over the conventional and previous adaptive Monte Carlo algorithms are discussed and theoretical results are established to justify the convergence of the algorithm. Furthermore, the algorithm is used to solve the SLAEs obtained from finite difference method for the problem of European and American options pricing. Numerical tests are performed on examples with matrices of different size and on SLAEs coming from option pricing problems. Comparisons with standard numerical and stochastic algorithms are also done which demonstrate the computational efficiency of the proposed algorithm.

A regression-based Monte Carlo method to solve two-dimensional forward backward stochastic differential equations

The purpose of this paper is to investigate the numerical solutions to two-dimensional forward backward stochastic differential equations(FBSDEs). Based on the Fourier cos-cos transform, the approximations of conditional expectations and their errors are studied with conditional characteristic functions. A new numerical scheme is proposed by using the least-squares regression-based Monte Carlo method to solve the initial value of FBSDEs. Finally, a numerical experiment in European option pricing is implemented to test the efficiency and stability of this scheme.

A fractional Black-Scholes model with stochastic volatility and European option pricing

In this paper, we introduce the stochastic volatility into the FMLS (finite moment log-stable) model to capture the effect of both jumps and stochastic volatility. However, this additional stochastic source adds another degree of complexity in seeking for analytical formula when pricing European options, as the involved FPDE (fractional partial differential equation) system governing option prices is now of three dimensions. Albeit difficult, we have still managed to present an analytical solution expressed in terms of Fourier cosine series, after a two-step solution procedure is developed for the target FPDE system. This solution is different from the most literature as it is truly explicit, involving no Fourier inversion. It is also shown through the numerical experiments that it converges very rapidly and has potential to be applied in practice.

A Few Uncomfortable Truths About the Black and Scholes (B-S) Equation

We examine the performance of the Black-Scholes (B-S) formulas around (i.e., before, during and after) two periods of market stress: the subprime crisis (October, 2018) and the onset of the COVID-19 pandemic (March, 2020). We find, in agreement with previous studies under different circumstances, that the accuracy of these formulas is very poor. We also demonstrate that the usual derivation of the B-S formulas is unnecessarily complicated as one can arrive at the same expressions invoking less restrictive assumptions, dispensing with stochastic calculus altogether, and using only undergraduate-level statistics. Moreover, we challenge: (i) the relative merits of assuming that stock prices are better described by means of log-normal distributions, as opposed to normal distributions (both assumptions seem equally inadequate); and (ii) the idea of estimating the price of European options by focusing on modeling the asset price-process between t = 0 and expiration (T). We show that what is relevant is estimating the asset price at T; not the price trajectory between 0 and T. In short, in reference to the asset price, what really matters is the destination, not the journey. This represents an important shift in terms of how we should think about European-options pricing schemes.

AN ANALYTICAL OPTION PRICING FORMULA FOR MEAN-REVERTING ASSET WITH TIME-DEPENDENT PARAMETER

AbstractWe present an analytical option pricing formula for the European options, in which the price dynamics of a risky asset follows a mean-reverting process with a time-dependent parameter. The process can be adapted to describe a seasonal variation in price such as in agricultural commodity markets. An analytical solution is derived based on the solution of a partial differential equation, which shows that a European option price can be decomposed into two terms: the payoff of the option at the initial time and the time-integral over the lifetime of the option driven by a time-dependent parameter. Finally, results obtained from the formula have been compared with Monte Carlo simulations and a Black–Scholes-type formula under various kinds of long-run mean functions, and some examples of option price behaviours have been provided.

SPECTRALLY ACCURATE OPTION PRICING UNDER THE TIME-FRACTIONAL BLACK–SCHOLES MODEL

AbstractWe propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular$L1$finite difference approximation, which converges with order$\mathcal {O}((\Delta \tau )^{2-\alpha })$for functions which are twice continuously differentiable. However, when using the$L1$scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach.

OPTION PRICING UNDER THE FRACTIONAL STOCHASTIC VOLATILITY MODEL

AbstractWe 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.