Option Pricing Formula Research Articles

A Quantum Algorithm for Pricing Asian Options on Valuation Trees

We develop a novel quantum algorithm for approximating the price of a discrete floating-strike Asian option based on an underlying valuation tree. The paths of the tree are encoded in bit-representation into a qubit register, where quantum state preparation is used to load the corresponding distribution onto the states. We implement the expectation value of the option pricing formula as a composition of the price probabilities, the payout and an indicator function, mapping their respective values to amplitudes of additional qubits. Thus, the underlying no longer has to be discretized into the same bit values for different times, resulting in smaller quantum circuits. The algorithm may be used with quantum amplitude estimation, enabling a quadratic speed-up over classical Monte Carlo methods.

A Mellin transform approach to pricing barrier options under stochastic elasticity of variance

AbstractThis article considers a problem of evaluating barrier option prices when the underlying dynamics are driven by stochastic elasticity of variance (SEV). We employ asymptotic expansions and Mellin transform to evaluate the option prices. The approach is able to efficiently handle barrier options in a SEV framework and produce explicitly a semi‐closed form formula for the approximate barrier option prices. The formula is an expansion of the option price in powers of the characteristic amplitude scale and variation time of the elasticity and it can be calculated easily by taking the derivatives of the Black–Scholes price for a barrier option with respect to the underlying price and computing the one‐dimensional integrals of some linear combinations of the Greeks with respect to time. We confirm the accuracy of our formula via Monte‐Carlo simulation and find the SEV effect on the Black–Scholes barrier option prices.

Co-movements, option pricing and risk management: an application to WTI versus Brent spread options

Co-moments of asset returns play a major role in financial contagion during crises. We study the properties of a particular specification of the generalized bivariate normal distribution which allows for co-volatility and co-skewness. With this probability distribution, formulae for single-name and exchange options can be evaluated quickly since they are based on one-dimensional integrals. We provide a very precise approximation formula for spread option prices and derive the corresponding greeks. We perform a day-to-day re-estimation of the probability distribution on a dataset of WTI vs Brent spread options, showing the ability of this specification to capture the salient empirical features observed in the market. Finally, we show the impact of co-movements on portfolio risk management.

CBI-time-changed Lévy processes for multi-currency modeling

We develop a stochastic volatility framework for modeling multiple currencies based on CBI-time-changed Lévy processes. The proposed framework captures the typical risk characteristics of FX markets and is coherent with the symmetries of FX rates. Moreover, due to the self-exciting behavior of CBI processes, the volatilities of FX rates exhibit self-exciting dynamics. By relying on the theory of affine processes, we show that our approach is analytically tractable and that the model structure is invariant under a suitable class of risk-neutral measures. A semi-closed pricing formula for currency options is obtained by Fourier methods. We propose two calibration methods, also by relying on deep-learning techniques, and show that a simple specification of the model can achieve a good fit to market data on a currency triangle.

On Barrier Binary Options in the Telegraph-like Financial Market Model

The article continues the study of the market model based on jump-telegraph processes. It is assumed that the price of a risky asset follows the stochastic exponential of a piecewise linear process, equipped with jumps that occur at the moments of a pattern change. In this case, the standard option pricing formula was derived previously, while exotic options for this model have not yet been explored. Within this framework, we are developing procedures for pricing binary barrier options. This article concerns the “cash-(at hit)-or-nothing” binary barrier option. The main tools of this analysis are methods developed for first-pass probabilities. Some known results related to the ruin probabilities follow directly from these settings.

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

An analytical GARCH valuation model for spread options with default risk

In this paper, an analytical pricing formula for spread options with credit default risk is derived. Assets are set within discrete-time CAPM markets and Heston–Nandi GARCH processes are adopted for capturing the variance dynamics of asset returns. The proposed model decomposes the risk of all assets into idiosyncratic and systematic parts, and considers the impacts on the default intensity of market fluctuations. A corresponding theoretical framework of discrete-time credit risk modeling in reduced-form is also provided. We incorporate the credit risk and systematic factor into an affine-GARCH valuation model and utilize Fourier transform techniques to obtain the analytic spread option prices. Finally, the empirical results and numerical analysis for different model parameters are displayed.

Nonlinear Differential Equations in Preventing Financial Risks

Abstract The nonlinear differential equation option pricing formula is invaluable in financial derivatives investment risk assessment. This article applies the theory of nonlinear differential equations to deal with financial risks in commodity and currency markets. Through this condition, we obtain the fair price process of contingent rights under the classic Black-Scholes model and the price process of the optimal growth investment strategy. The results show that the risk measurement under stable distribution is suitable for investors to manage risk.

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.

An Informational Theory of the Dynamic Value of the Firm

This paper formulates an informational theory of the evolving value of the firm under uncertainties and unknowns in the future payoff structures. In general, the horizon of business firms would last for an indefinitely long period of time, and events in the considerably far future are intrinsically unknown. The existing study of indefinite horizon firms often relies on the assumption of time-invariant structures for the derivation of an optimal solution. In this paper, information about the firm’s future payoffs will be revealed as time goes by. The firm will revise its strategies accordingly, and the process will continue indefinitely. This new approach for the analysis of infinite horizon firms via information updating provides a more realistic and practical alternative to the study of the dynamic value of the firm. Finally, information-based option pricing formulae and non-random walks and cycles in asset price can also be generated with this theory.

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.

Barrier Option Pricing in the Sub-Mixed Fractional Brownian Motion with Jump Environment

This paper investigates the pricing formula for barrier options where the underlying asset is driven by the sub-mixed fractional Brownian motion with jump. By applying the corresponding Ito^’s formula, the B-S type PDE is derived by a self-financing strategy. Furthermore, the explicit pricing formula for barrier options is obtained through converting the PDE to the Cauchy problem. Numerical experiments are conducted to test the impact of the barrier price, the Hurst index, the jump intensity and the volatility on the value of barrier option, respectively.

Skew-Brownian motion and pricing European exchange options

This article derives a closed-form pricing formula for European exchange options under a non-Gaussian framework for the underlying assets, intending to resolve mispricing associated with a geometric Brownian motion. The dynamics of each of the two correlated underlying assets are assumed to be governed by the exponential of a skew-Brownian motion, which is specified as a sum of a standard Brownian motion and an independent reflected Brownian motion. The proposed pricing formula does not incur additional computational costs than the standard Black–Scholes framework, which one can quickly recover as a particular case of the proposed framework. Finally, we present some numerical experiments followed by a valuable discussion on the results.

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.

AN EMPIRICAL ANALYSIS OF OPTION PRICING WITH SHORT SELL BANS

Short sell bans are often imposed during a financial crisis as a desperate measure to stabilize financial markets. Yet, the impact of short sell bans on option pricing and hedging is not well studied, at least quantitatively, until very recently when Guo & Zhu [(2017) Equal risk pricing under convex trading constraints, Journal of Economic Dynamics and Control 76, 136–151] and He & Zhu [(2020) A revised option pricing formula with the underlying being banned from short selling, Quantitative Finance 20 (6), 935–948] formulated a new pricing framework with the underlying being either completely or partially banned from short selling. However, no empirical results were provided to substantiate the usefulness of the formulae, as well as to deepen our understanding on the effects of short sell bans. This paper provides a comprehensive empirical study on the effects of short sell bans to the standard option pricing theory by carrying out both cross-sectional and options time series model calibration of the model devised by He & Zhu (2020) [A revised option pricing formula with the underlying being banned from short selling, Quantitative Finance 20 (6), 935–948]. Overall, our empirical results indicate that the alternative option pricing formula considering short sell restrictions has the ability to capture highly-quoted implied volatility, with an evident improvement of 39% out-of-sample performance compared to the benchmark Black–Scholes model during the period of short sell ban.

APPROXIMATE OPTION PRICING FORMULA FOR BARNDORFF-NIELSEN AND SHEPHARD MODEL

For the Barndorf-Nielsen and Shephard model, we present approximate expressions of call option prices based on the decomposition formula developed by [T. Arai (2021) Alos type decomposition formula for Barndor-Nielsen and Shephard model, Journal of Stochastic Analysis 2 (2), 3]. Besides, some numerical experiments are also implemented to make sure how effective our approximations are.

Pricing VXX options by modeling VIX directly

AbstractIn this paper, we first develop a theoretical and model‐free VXX formula in terms of Volatility Index (VIX) futures in both discrete and continuous forms. The discrete form of VXX can quantify the roll yield of VXX, which can be used to explain VXX's underperformance. Using the log‐normal Ornstein–Uhlenbeck (LOU) diffusion model, we show how the number of rolls of VIX futures affects the VXX option pricing formula and its implied volatility (IV). To further verify the nonflat IV of VXX, the VXX option pricing formula under the LOU with stochastic volatility model is also derived. Finally, we analyze their pricing performance, and the ability to forecast implied volatilities.

Homotopy perturbation method to solve Black Scholes differential equation for ML-payoff function

The Homotopy Perturbation Method (HPM) is a semi analytical method for solving linear and non linear ordinary as well as partial differential equations. This paper contributes to the solution of Black Scholes equation for modified log payoff (ML-payoff) functions using HPM. It should be emphasized that these formulas are quite close to the closed form solutions of Black Scholes equation for ML-payoff functions which are very close to the celebrated option pricing formulas for plain vanilla payoff functions.

Vulnerable European option pricing in a Markov regime-switching Heston model with stochastic interest rate

This paper considers pricing of European-style vulnerable options under the Heston stochastic volatility and stochastic interest rate model in which the mean-reversion levels of both variance and interest rate processes are modulated by a continuous-time Markov process with a finite state space. An analytical pricing formula is derived by using the Esscher transform, joint characteristic function and multivariate Fourier transform technique, where the closed-form solution of the characteristic function is obtained by the law of iterated expectation. Then we provide the efficient approximation to calculate the analytical pricing formula of option using the FFT approach and examine the accuracy of the approximation by Monte Carlo simulation. Finally, the sensitivity analysis of different parameters in the proposed model on the vulnerable call option price and its Delta value are provided, and the difference between the proposed model and the Heston and stochastic interest rate model with non-Markov regime-switching are presented by some numerical experiments, which shows the influence of introducing regime-switching into Heston model with stochastic interest rate.

Valuing vulnerable Asian options with liquidity risk under Lévy processes

AbstractIn this paper, we study the pricing of vulnerable Asian options with liquidity risk. We employ general Lévy processes to capture the changes in the liquidity discount factors and the information processes of all assets. In the proposed pricing model, we obtain the closed-form pricing formula of vulnerable Asian options using the Fourier transform methods. Finally, the derived pricing formula is used to illustrate the effects of asymmetric jump risk, and the effects are relatively stable on (vulnerable) Asian options with different moneynesses.