European Option Pricing Formula Research Articles

Analytically Pricing European Options under a New Two-Factor Heston Model with Regime Switching

In this paper, we propose a new two-factor stochastic volatility model by introducing a regime switching factor into the Heston model. Despite the complicated model structure, we still manage to derive a closed-form pricing formula for European options, which can save us a lot of time in option pricing and model calibration. The results of our empirical study further indicate that our model is able to provide better performance over existing ones when real market data is employed, demonstrating its possible practical applications.

Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility

<p style='text-indent:20px;'>In this paper, we propose a model to price vulnerable European options where the dynamics of the underlying asset value and the counter-party's asset value follow two jump-diffusion processes with fast mean-reverting stochastic volatility. First, we derive an equivalent risk-neutral measure and transfer the pricing problem into solving a partial differential equation (PDE) by the Feynman-Kac formula. We then approximate the solution of the PDE by pricing formulas with constant volatility via multi-scale asymptotic method. The pricing formula for vulnerable European options is obtained by applying a two-dimensional Laplace transform when the dynamics of the underlying asset value and the counter-party's asset value follow two correlated jump-diffusion processes with constant volatilities. Thus, an analytic approximation formula for the vulnerable European options is derived in our setting. Numerical experiments are given to demonstrate our method by using Laplace inversion.</p>

A New Approach to Minimal Variance Hedging of European Options (Part 2: The Pure-Jump Case)

This paper addresses the following question: How can a financial institution, which has issued a European option, optimally hedge the payoff of this option by investing into the underlying stock and into the option itself? Here, optimality is measured in terms of minimal variance and the associated optimal hedging portfolio is derived by a sufficient stochastic maximum principle. We further obtain a pricing formula for general European options by an application of Fourier transform methods and deduce the time dynamics of the stochastic option price process. We finally apply our theoretical results to several practical examples.

A New Approach to Minimal Variance Hedging of European Options

This paper addresses the following question: How can a financial institution, which has issued a European option, optimally hedge the payoff of this option by investing into the underlying stock and into the option itself? Here, optimality is measured in terms of minimal variance and the associated optimal hedging portfolio is derived by a sufficient stochastic maximum principle. We further obtain a pricing formula for general European options by an application of Fourier transform methods and deduce the time dynamics of the stochastic option price process. We finally apply our theoretical results to several practical examples.

Option Pricing Formulas in a New Uncertain Mean-Reverting Stock Model with Floating Interest Rate

Options play a very important role in the financial market, and option pricing has become one of the focus issues discussed by the scholars. This paper proposes a new uncertain mean-reverting stock model with floating interest rate, where the interest rate is assumed to be the uncertain Cox-Ingersoll-Ross (CIR) model. The European option and American option pricing formulas are derived via the α -path method. In addition, some mathematical properties of the uncertain option pricing formulas are discussed. Subsequently, several numerical examples are given to illustrate the effectiveness of the proposed model.

A closed-form pricing formula for European options under a new stochastic volatility model with a stochastic long-term mean

Based upon the fact that a constant long-term mean could not provide a good description of the term structure of the implied volatility and variance swap curve, as suggested by Byelkina and Levin (in: Sixth world congress of the Bachelier Finance Society, Toronto, 2010) and Forde and Jacquier (Appl Math Finance 17(3):241–259, 2010), this paper presents a new stochastic volatility model, by assuming the long-term mean of the volatility in the Heston model be stochastic. An important feature of our model is that it still preserves the essential advantage of the Heston model, i.e., the analytic tractability, because a closed-form pricing formula for European options can be derived, which could not only facilitate the risk management process but also help save plenty of time in terms of model calibration. The effect of the newly introduced stochastic long-term mean is demonstrated through the numerical comparison with the Heston model. It is also shown that the current model can overall lead to more accurate option prices than the Heston model, through a carefully designed empirical study.

A semi-analytical pricing formula for European options under the rough Heston-CIR model

We combine the rough Heston model and the CIR (Cox–Ingersoll–Ross) interest rate together to form a rough Heston-CIR model, so that both the rough behaviour of the volatility and the stochastic nature of the interest rate can be captured. Despite the convoluted structure and non-Markovian property of this model, it still admits a semi-analytical pricing formula for European options, the implementation of which involves solving a fractional Riccati equation. The rough Heston-CIR model is more general, taking both the rough Heston model and the Heston-CIR model as special cases. The influence of rough volatility and stochastic interest rate is shown to be significant through numerical experiments.
 
 doi:10.1017/S1446181120000024

A revised option pricing formula with the underlying being banned from short selling

An important issue in derivative pricing that hasn't been explored much until very recently is the impact of short selling to the price of an option. This paper extends a recent publication in this area to the case in which a ban of short selling of the underlying alone is somewhat less ‘effective’ than the extreme case discussed by Guo and Zhu [Equal risk pricing under convex trading constraints. J. Econ. Dyn. Control, 2017, 76, 136–151]. The case presented here is closer to reality, in which the effect of a ban on the underlying of an option alone may quite often be ‘diluted’ due to market interactions of the underlying asset with other correlated assets. Under a new assumption that there exists at least a correlated asset in the market, which is allowed to be short sold and thus can be used by traders for hedging purposes even though short selling of the underlying itself is banned, a new closed-form equal-risk pricing formula for European options is successfully derived. The new formula contains two distinguishable advantages; (a) it does not induce any significantly extra burden in terms of numerically computing option values, compared with the effort involved in using the Black–Scholes formula, which is still popularly used in finance industry today; (b) it remains simple and elegant as only one additional parameter beyond the Black–Scholes formula is introduced, to reflect the dilution effect to the ban as a result of market interactions.

A semianalytical formula for European options under a hybrid Heston–Cox–Ingersoll–Ross model with regime switching

AbstractIn this paper, we consider the pricing of European options under a regime‐switching Heston–Cox–Ingersoll–Ross (CIR) hybrid model, where the mean‐reversion levels of both the stochastic volatility and interest rate are assumed to change among different states. Albeit difficult, we have still managed to derive an semianalytical pricing formula for European options after the generalized moment generating function of this particular model is worked out. Numerical experiments are also carried out to demonstrate the accuracy of the newly derived formula as well as the influence of the introduction of the regime‐switching mechanics on option prices. Finally, through a preliminary empirical study, our model is shown to be superior to the Heston‐CIR model, which demonstrates the importance of introducing the regime‐switching mechanics.

A SEMI-ANALYTICAL PRICING FORMULA FOR EUROPEAN OPTIONS UNDER THE ROUGH HESTON-CIR MODEL

We combine the rough Heston model and the CIR (Cox–Ingersoll–Ross) interest rate together to form a rough Heston-CIR model, so that both the rough behaviour of the volatility and the stochastic nature of the interest rate can be captured. Despite the convoluted structure and non-Markovian property of this model, it still admits a semi-analytical pricing formula for European options, the implementation of which involves solving a fractional Riccati equation. The rough Heston-CIR model is more general, taking both the rough Heston model and the Heston-CIR model as special cases. The influence of rough volatility and stochastic interest rate is shown to be significant through numerical experiments.

A closed-form pricing formula for vulnerable European options under stochastic yield spreads and interest rates

Abstract This paper develops a three-factor valuation model of vulnerable European options incorporating stochastic yield spreads and interest rates, which extends a constant yield spread and deterministic interest rate proposed in Hull and White [12]. The dynamics of the short-term interest rate are represented implicitly by a stochastic bond price process. Therefore, the stochastic factors in the model are the spot price of the underlying asset that follows a geometrical Brownian motion process, the yield spreads and the default-free unit discount bond that are modeled by a mean-reverting Ornstein-Uhlenbeck stochastic process. Furthermore, we exploit Mellin transform techniques to derive a closed-form solution for vulnerable European options under the Black-Scholes model, which is simply computed using the standard normal cumulative distribution function so that the pricing and hedging of vulnerable European options can be computed very accurately and rapidly. Numerical experiments demonstrate how credit risk and interest rate risk affect the prices of European options.

A New Binomial Tree Method for European Options under the Jump Diffusion Model

In this paper, the binomial tree method is introduced to price the European option under a class of jump-diffusion model. The purpose of the addressed problem is to find the parameters of the binomial tree and design the pricing formula for European option. Compared with the continuous situation, the proposed value equation of option under the new binomial tree model converges to Merton’s accurate analytical solution, and the established binomial tree method can be proved to work better than the traditional binomial tree. Finally, a numerical example is presented to illustrate the effectiveness of the proposed pricing methods.

European option pricing model based on uncertain fractional differential equation

In this paper, we investigate a new version of stock model under uncertain circumstances for uncertain stock markets. Firstly, solutions to some uncertain fractional differential equations are presented by employing the Mittag-Leffler function. Then, a new uncertain stock model with mean-reverting process is formulated on the basis of uncertain fractional differential equations. Finally, European option pricing formulas based on the proposed model are investigated as well as some numerical examples.

A note on “A closed-form pricing formula for European options under the Heston model with stochastic interest rate”

He and Zhu (2018) provide an innovative closed-form pricing formula in an infinite series for European options under the Heston-CIR hybrid model, based on the technique of numeraire change. In contrast to their formula, we give an alternative closed-form pricing formula, which does not need to use the technique of numeraire change.

BSM model for ML-payoff function through PDTM

This paper contributes to the valuation of Black–Scholes–Merton (BSM) European option pricing formula for Modified Log-payoff (ML-payoff) function, [Formula: see text] through Projected Differential Transform Method (PDTM). The ML-payoff function is closely related with the entropy function [Formula: see text] in Information Theory. It turns out that the present BSM formula is quite close to the celebrated plain vanilla option.

A Stock Model with Varying Stock Diffusion for Uncertain Market

The option-pricing problem is an important topic in modern finance. In this paper, we propose a stock model with varying stock diffusion based on uncertainty theory. The European option pricing formulas are derived from the proposed uncertain stock model, and some mathematical properties of these formulas are investigated. Moreover, extended uncertain stock models are introduced and discussed. Finally, numerical examples are given to illustrate the proposed model.

A general closed form option pricing formula

A new method to retrieve the risk-neutral probability measure from observed option prices is developed and a closed form pricing formula for European options is obtained by employing a modified Gram-Charlier series expansion, known as the Gauss-Hermite expansion. This expansion converges for fat-tailed distributions commonly encountered in the study of financial returns. The expansion coefficients can be calibrated from observed option prices and can also be computed, for example, in models with the probability density function or the characteristic function known in closed form. We investigate the properties of the new option pricing model by calibrating it to both real-world and simulated option prices and find that the resulting implied volatility curves provide an accurate approximation for a wide range of strike prices. Based on an extensive empirical study, we conclude that the new approximation method outperforms other methods both in-sample and out-of-sample.

Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales

In this paper, we consider the valuation of vulnerable options under a Markov-modulated jump-diffusion model, where the option writer's asset value is subject to price pressure from other financial institutions due to distressed selling. A change of numeraire technique, proposed by Geman et al. [ 14 ], is employed to obtain a semi-analytical pricing formula for an vulnerable European option in the presence of regime switching effect. The method is numerically implemented using the multinomial approach in Costabile et al. [ 6 ]. We study the impacts of distressed selling and regime switching on the European option prices via numerical experiments.

CDS Option Valuation with Double-Exponential Jumps

We demonstrate how the Double-Exponential Jump-Diffusion (DEJD) process can be used to value iTraxx CDS options based on historical returns of the underlying CDS index. In the first step we find Maximum Likelihood estimates for the volatility of the normal component of returns and the Poisson frequencies and mean sizes of upward and downward exponential jumps. In results similar to Ramezani and Zeng’s (2006) application of DEJD to equities, we find that the DEJD provides a better fit than either a normal or single-jump specification. We take the additional step of using parameter estimates as inputs to the semi-closed European option pricing formula proposed by Kou (2002) under DEJD. We compare the model and market option prices across strikes and find that both the level and shape of the implied volatility smile match closely. Our findings suggest that the DEJD provides a realistic description of the joint role of positive and negative economic surprises in credit markets. In addition, the use of Maximum Likelihood with the option pricing model is a practical way to analyse divergence between realised and market-implied distributions of credit returns, and can be used to check pricing.

Estimation of market prices of risks in the G.A.R.C.H. diffusion model

In this paper we propose an estimation procedure which uses joint data on the underlying asset and option prices to extract market prices of return and volatility risks in the context of the G.A.R.C.H. diffusion model. The procedure is flexible and simple to implement. Firstly, a quasi-closed form pricing formula for European options in the G.A.R.C.H. diffusion model is derived. This result greatly eases the computational burden for computing option prices, and well suited for our model estimation. Then, based upon the joint data, we develop an efficient importance sampling-based maximum likelihood (E.I.S.-M.L.) estimation method for the objective and risk-neutral parameters of the G.A.R.C.H. diffusion model and a particle filter algorithm for latent state variable. Hence, this allows us to infer the market prices of risks that link the objective measure and the risk-neutral measure. Finally, we illustrate our approach using actual data on the Hang Seng Index (H.S.I.) and index warrant prices. The results show that both the return and volatility risks are priced by the market. Moreover, an option pricing study demonstrates that the market price of the volatility risk plays an important role in fitting option prices.