European Option Pricing Formula Research Articles

Option pricing under jump diffusion model

We provide an European option pricing formula written in the form of an infinite series of Black–Scholes-type terms under double Lévy jumps model, where both the interest rate and underlying price are driven by Lévy processes. The series solution converges with a radius of convergence, and it is complemented by some numerical experiments to demonstrate its speed of convergence.

Variance and volatility swaps and options under the exponential fractional Ornstein–Uhlenbeck model

Considering the fair strike values of variance and volatility swaps, we use a stochastic volatility model in which the log volatility is given by a fractional Ornstein–Uhlenbeck process with two versions; a stationary version and a version with a deterministic initial value. Under these versions, the fair strike formulas are obtained in exact form for variance swaps and approximated fair strike formulas are derived for volatility swaps based on the fact that an aggregation of log-normal variables is well-approximated by shifted log-normal or log-normal distribution. In addition, we obtain two approximate pricing formulas for European options on the realized variance and volatility. The accuracy and robustness of the approximated fair strike formulas are examined via Monte-Carlo computations. We conduct calibration experiments to show that the Hurst exponent and the mean-reversion property of the fractional Ornstein–Uhlenbeck process are able to produce various shapes resembling the market term-structures of variance swaps when they are put together.

European Option Pricing Based on FSDE Driven by Fractional Brownian Motion

In the actual financial market, the classical Black-Scholes (B-S) model cant perfectly describe the process of stock price. Besides, memory effect is an important phenomenon in financial systems. Thus, in this paper, we establish a fractional order stochastic differential equations (FSDE) which is driven by fractional Brownian motion (fBm) to describe the effect of noise memory and trend memory in financial pricing. Finally, we derive a European option pricing formula based on the established model. After conducting an empirical analysis based on the SSE 50ETF, we find that the established model performs better than the traditional one.

A closed-form pricing formula for European options under a multi-factor nonlinear stochastic volatility model with regime-switching

A closed-form pricing formula for European options under a multi-factor nonlinear stochastic volatility model with regime-switching

Pricing European Options under a Fuzzy Mixed Weighted Fractional Brownian Motion Model with Jumps

This study investigates the pricing formula for European options when the underlying asset follows a fuzzy mixed weighted fractional Brownian motion within a jump environment. We construct a pricing model for European options driven by fuzzy mixed weighted fractional Brownian motion with jumps. By converting the partial differential equation (PDE) into a Cauchy problem, we derive explicit solutions for both European call options and European put options. The figures and tables demonstrating the effectiveness of the results highlight the suitability of the fuzzy mixed weighted fractional Brownian motion with jump model for option pricing.

Nonparametric estimates of option prices via Hermite basis functions

We consider approximate pricing formulas for European options based on approximating the logarithmic return’s density of the underlying by a linear combination of rescaled Hermite polynomials. The resulting models, that can be seen as perturbations of the classical Black-Scholes one, are nonpararametric in the sense that the distribution of logarithmic returns at fixed times to maturity is only assumed to have a square-integrable density. We extensively investigate the empirical performance, defined in terms of out-of-sample relative pricing error, of this class of approximating models, depending on their order (that is, roughly speaking, the degree of the polynomial expansion) as well as on several ways to calibrate them to observed data. Empirical results suggest that such approximate pricing formulas, when compared with simple nonparametric estimates based on interpolation and extrapolation on the implied volatility curve, perform reasonably well only for options with strike price not too far apart from the strike prices of the observed sample.

Analytically pricing European options under a hybrid stochastic volatility and interest rate model with a general correlation structure

AbstractIn this paper, an additional factor is introduced into the Heston–Hull–White (HHW) hybrid model, which originally combines the Heston stochastic volatility model and the Hull–White stochastic interest rate model, to capture the correlation between the underlying price and the interest rate, while at the same time preserve the analytical tractability. With the analytical solution to the characteristic function of the underlying price being successfully derived, a closed‐form pricing formula for European options under the two‐factor HHW hybrid model is presented. Numerical experiments are also carried out to show the effect of the newly introduced factor. To further demonstrate the importance of introducing the correlation, we have also conducted an empirical study, comparing the performance of our model and that of the HHW model, based on European options written on S&P 500 index.

European option pricing problem based on a class of Caputo-Hadamard uncertain fractional differential equation

<abstract><p>Uncertain fractional differential equation (UFDE) is very suitable for describing the dynamic change in uncertain environments. In this paper, we consider the European option pricing problem by applying the Caputo-Hadamard UFDEs to simulate the dynamic change of stock price. First, an uncertain stock model with the mean-reverting process is studied, and the European option pricing formulas are given. Then, the effect of uncertain interference on the bond is considered, and the corresponding European option pricing formulas are presented. Finally, some numerical examples are given to illustrate the effectiveness of pricing formulas.</p></abstract>

A simple European option pricing formula with a skew Brownian motion

Zhu and He [(2018). A new closed-form formula for pricing European options under a skew Brownian motion. The European Journal of Finance 24(12): 1063–1074] provided an innovative closed-form solution by replacing the standard Brownian motion in the Black–Scholes framework using a particular skew Brownian motion. Their formula involves numerically integrating the product of the Guassian density and corresponding distribution function. Being different from their pricing formula, we derive a much simpler formula that only involves the Gaussian distribution function and Owen's $T$ function.

A closed-form pricing formula for European options under a new three-factor stochastic volatility model with regime switching

A closed-form pricing formula for European options under a new three-factor stochastic volatility model with regime switching

Robust and Nearly Exact Option Pricing with Bilateral Gamma Processes

Bilateral gamma processes generalize the variance gamma process and allow one to capture, more precisely, the differences between upward and downward moves of financial returns, notably in terms of jump speed, frequency, and size. Like in most other pure jump models, option pricing under bilateral gamma processes relies heavily on numerical evaluation of Fourier integrals. In this article, we combine the Mellin transform and residue calculus to establish closed-form pricing formulas for several vanilla and exotic European options. These formulas take the form of series whose terms are straightforward to evaluate in practice and achieve an arbitrary degree of precision, without requiring sophisticated numerical tools; moreover, the convergence of the series is particularly accelerated for short maturity options, which are the most challenging to price for competing Fourier methods. Accuracy of the formulas is assessed thanks to several comparisons with state-of-the-art Fourier methods, with reference prices provided for future research.

Option pricing with state‐dependent pricing kernel

AbstractWe introduce a new volatility model for option pricing that combines Markov switching with the realized generalized autoregressive conditional heteroskedasticity (GARCH) framework. This leads to a novel pricing kernel with a state‐dependent variance risk premium and a pricing formula for European options, which is derived with an analytical approximation method. We apply the Markov‐switching Realized GARCH model to Standard and Poor's 500 index options from 1990 to 2019 and find that investors' aversion to volatility‐specific risk is time‐varying. The proposed framework outperforms competing models and reduces (in‐sample and out‐of‐sample) option‐pricing errors by 15% or more.

A closed-form pricing formula for European options in an illiquid asset market

This article addresses the problem of pricing European options when the underlying asset is not perfectly liquid. A liquidity discounting factor as a function of market-wide liquidity governed by a mean-reverting stochastic process and the sensitivity of the underlying price to market-wide liquidity is firstly introduced, so that the impact of liquidity on the underlying asset can be captured by the option pricing model. The characteristic function is analytically worked out using the Feynman–Kac theorem and a closed-form pricing formula for European options is successfully derived thereafter. Through numerical experiments, the accuracy of the newly derived formula is verified, and the significance of incorporating liquidity risk into option pricing is demonstrated.

The Variance Gamma++ process and applications to energy markets

AbstractThe purpose of this article is to introduce a new Lévy process, termed the Variance Gamma++ process, to model the dynamics of assets in illiquid markets. Such a process has the mathematical tractability of the Variance Gamma process and is obtained by applying the self‐decomposability of the gamma law. Compared to the Variance Gamma model, it has an additional parameter representing the measure of the trading activity. We give a full characterization of the Variance Gamma++ process in terms of its characteristic triplet, characteristic function, and transition density. In addition, we provide efficient path simulation algorithms, both forward and backward in time. We also obtain an efficient “integral‐free” explicit pricing formula for European options. These results are instrumental to apply Fourier‐based option pricing and maximum likelihood techniques for the parameter estimation. Finally, we apply our model to illiquid markets, namely to the calibration of European power futures market data. We accordingly evaluate exotic derivatives using the Monte Carlo method and compare these values to those obtained using the Variance Gamma process and give an economic interpretation. Finally, we illustrate an extension to the multivariate framework.

Option Pricing under the Subordinated Market Models

This paper aims to study option pricing problem under the subordinated Brownian motion. Firstly, we prove that the subordinated Brownian motion controlled by the fractional diffusion equation has many financial properties, such as self‐similarity, leptokurtic, and long memory, which indicate that the fractional calculus can describe the financial data well. Then, we investigate the option pricing under the assumption that the stock price is driven by the subordinated Brownian motion. The closed‐form pricing formula for European options is derived. In the comparison with the classic Black–Sholes model, we find the option prices become higher, and the “volatility smiles” phenomenon happens in the proposed model. Finally, an empirical analysis is performed to show the validity of these results.

An analytical approximation formula for European option prices under a liquidity-adjusted non-affine stochastic volatility model

<abstract><p>In this paper, we investigate the pricing of European options under a liquidity-adjusted non-affine stochastic volatility model. An analytical European option pricing formula is successfully derived with the COS method, based on an approximation for the characteristic function of the underlying log-asset price. Numerical analysis reveals that our results are very efficient and of reasonable accuracy, and we also present some sensitivity analysis to demonstrate the effects of different parameters on option prices.</p></abstract>

A closed-form pricing formula for European options with market liquidity risk

In this paper, the impact of liquidity on the underlying asset is taken into account when pricing European options through a discounting factor which depends on two factors, i.e., market liquidity risk modeled as a mean-reverting stochastic process and the sensitivity of the underlying to market liquidity. A closed-form pricing formula for liquidity-adjusted European options is derived in the form of an infinite series using Karhunen–Loève expansion for the Ornstein–Uhlenbeck process. The convergence of the series solution is theoretically proved to guarantee closedness so that market practitioners can adopt the new formula when they need to account in market liquidity risk. The speed of convergence is demonstrated through numerical experiments. Finally, the accuracy of the newly derived formula is shown by comparing option prices calculated with our formula and those obtained from Monte-Carlo simulation, and various properties of our formula are also investigated.

Multiscale stochastic elasticity of variance for options and equity linked annuity; A Mellin transform approach

It is important to impose a persistent stochastic factor on the underlying asset model to obtain the fair value of financial derivatives with long time-to-maturities. Our empirical study, including the Covid-19 pandemic crisis period, indicates the presence of both fast and slow-scale in the elasticity of variance of S&P 500. This paper extends the elasticity in terms of multiscale stochastic process and obtains a closed form analytic pricing formula for European options and then derive the fair value of Equity-Linked-Annuity (ELA). The Mellin transform method for solving the relevant partial differential equations provides a computationally-efficient pricing formula for the options and the ELA. The prices can be easily calculated simply by taking derivatives of the Black–Scholes option price. Our results reveal the sensitivity of the ELA term structure to the fast-scale or slow-scale related group parameters.

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.

Valuation of European options with stochastic interest rates and transaction costs

The celebrated Black–Scholes model is well known for its elegant pricing formula for European options. However, like many other models, the Black–Scholes model is not perfect, which is largely due to the fact that assumptions in the model are idealized and not all empirically valid. One of the assumptions is that the market does not have transaction costs, which is not satisfied in a real market. Leland [H. Leland, Option pricing and replication with transactions costs, J. Financ. 40 (1985), pp. 1283–1301] pioneers a modified replicating strategy for European options by incorporating transaction costs. In this paper, we further consider the problem of pricing European options under a stochastic interest rate and stochastic volatility model with transaction costs, and derive a nonlinear partial differential equation (PDE) from this model. Then, we apply the finite-difference scheme to solve this PDE and conduct numerical experiments.