Option Pricing Formula Research Articles

THE EDGEWORTH AND GRAM–CHARLIER DENSITIES

This paper is the first to define the Edgeworth density and comprehensively compare it to the Gram–Charlier density in the context of option pricing. The two densities allow additional cumulants to the normal distribution; although similar, they are not the same when truncated. Many academics have misidentified the two. This paper clearly distinguishes the two, presents the derivation of both, and develops a general option pricing model which can be used for both densities with an arbitrary number of additional cumulants. The option pricing formula for each density is also calibrated and compared to more typical models with the most advanced being the affine jump-diffusion model (stochastic volatility with double jumps).

Pricing exchange options under hybrid stochastic volatility and interest rate models

This paper investigates the pricing of exchange options under hybrid models integrating stochastic volatility and stochastic interest rates. It aims to achieve two primary objectives. First, we derive a closed-form pricing formula for exchange options under a two-factor Heston–Hull–White hybrid model, which accounts for long-term volatility and exhibits relatively broad correlations among the dynamics of asset prices, volatilities, and interest rates. Second, we explore the Heston model’s integration with a generalized single-factor stochastic interest rate model, illustrating that the price is not dependent on the specific form of the interest rate process. A closed-form pricing formula for exchange options under this framework is also derived. Our numerical experiments support the proposed formulas and elucidate the effects of various parameters on option prices.

Extreme ATM skew in a local volatility model with discontinuity: joint density approach

This paper concerns a local volatility model in which the volatility takes two possible values, and the specific value depends on whether the underlying price is above or below a given threshold. The model is known, and a number of results have been obtained for it. In particular, option pricing formulas and a power-law behaviour of the implied volatility skew have been established in the case when the threshold is taken at the money. In this paper, we derive an alternative representation of option pricing formulas. In addition, we obtain an approximation of option prices by the corresponding Black–Scholes prices. Using this approximation streamlines obtaining the aforementioned behaviour of the skew. Our approach is based on the natural relationship of the model with skew Brownian motion and consists of the systematic use of the joint distribution of this stochastic process and some of its functionals.

Dynamic Asset Pricing in a Unified Bachelier–Black–Scholes–Merton Model

We present a unified, market-complete model that integrates both Bachelier and Black–Scholes–Merton frameworks for asset pricing. The model allows for the study, within a unified framework, of asset pricing in a natural world that experiences the possibility of negative security prices or riskless rates. Unlike the classical Black–Scholes–Merton, we show that option pricing in the unified model differs depending on whether the replicating, self-financing portfolio uses riskless bonds or a single riskless bank account. We derive option price formulas and extend our analysis to the term structure of interest rates by deriving the pricing of zero-coupon bonds, forward contracts, and futures contracts. We identify a necessary condition for the unified model to support a perpetual derivative. Discrete binomial pricing under the unified model is also developed. In every scenario analyzed, we show that the unified model simplifies to the standard Black–Scholes–Merton pricing under specific limits and provides pricing in the Bachelier model limit. We note that the Bachelier limit within the unified model allows for positive riskless rates. The unified model prompts us to speculate on the possibility of a mixed multiplicative and additive deflator model for risk-neutral option pricing.

A Study of Black-Scholes-Merton Model

This study introduces the Option Pricing Theory improved by Black, Scholes, and Merton, which is also known as the Black-Scholes-Merton model, in detail and investigates its derivation process and historical development process. Despite the content above, it also includes its characteristics and mechanism. The newest Option Pricing formula proposed by Black, Scholes, and Merton was derived from mathematical methods such as the stochastic integral equation and Ito theorem. This essay concludes that the Black Scholes Merton model paved the way for the development and innovation of the subject of Financial Mathematics, which was found by the person who first came up with the Option Pricing Theory, which also contributes to the investment field to reduce the risk.

Valuing American options using multi-step rebate options

The determination of optimal exercise boundaries is a critical aspect of pricing American options, which often requires costly numerical methods. This paper proposes a new approach that employs multi-step rebate options to approximate American option prices. Since the rebate options offer payoffs when the multi-step boundaries are touched, the prices of American options are estimated by maximizing the multi-step rebate option prices, and the optimal multi-step barriers replace the true optimal exercise boundaries. To this end, the closed-form pricing formulas for multi-step rebate options are derived and utilized to approximate several American option prices. Through extensive numerical experiments, we demonstrate the validity and performance of our approach.

A rational finance explanation of the stock predictability puzzle

AbstractWe address the stock predictability puzzle, a challenge in the stock market often discussed in behavioral finance. Our approach formulates a statistical model within rational finance, avoiding reliance on behavioral finance assumptions, and integrates stock return predictability into the Black–Scholes option pricing framework. Empirical analysis focuses on the predictability of stock prices by option and spot traders, introducing a forward‐looking measure we term “implied excess predictability.” Results show that option traders' predictability of stock returns positively correlates with moneyness, whereas for spot traders, this relationship is inverse. These findings suggest a potential asymmetry in stock price predictability between spot and option traders. Additionally, we demonstrate the importance of incorporating stock return predictability into option pricing formulas, particularly for options with strike prices significantly different from the stock price. Conversely, when moneyness is close to unity, predictability is not integrated into option pricing, indicating equal information among spot and option traders. Comparison of volatility measures reveals the difference between implied and realized variances or variance risk premia as potential predictors of stock returns.

Counting jumps: does the counting process count?

In this paper, we develop a ‘jump diffusion type’ financial model based on renewal processes for the discontinuous part of the risk driver, and study its ability to price options and accurately reproduce the corresponding implied volatility surfaces. In this model, the log-returns process displays finite mean and variance, and non-vanishing skewness and excess kurtosis over a long period. The proposed construction is parsimonious, and it allows for a simple and intuitive pricing formula for European vanilla options; furthermore, it offers an efficient Monte Carlo sample path generator for the pricing of exotic options. We illustrate the performance of the proposed framework using observed market data, and we study the features of the best fitting model specifications.

Explicit formulae for the valuation of European options with price impacts

In this work, we examine the consequences of trading a large position in vanilla European options within a multi-period binomial model framework for the underlying asset price, S. Given the significant size of the transaction, we expect both the derivative's price and the underlying asset's price to be affected by market impacts. Consequently, derivative valuation should incorporate these effects. To address this, we not only utilize a multi-period binomial model to represent the price process S but also incorporate trading impacts in a multiplicative manner.Moreover, we conduct our analysis in discrete time to better capture the influence of price impacts. Our findings suggest, for instance, that the strike price should be determined by both the trade's magnitude and parameterized market impacts. We present explicit formulas for European option prices under market impacts and offer numerical examples to elucidate our findings. Upon request, we can provide code implemented in the statistical package R.

Pricing VIX options based on mean-reverting models driven by information

Financial time series are dynamic and influenced by different types of information from the market. In this study, we propose new models for SPX and VIX options using the Hawkes process, jump process with stochastic intensity, and tempered stable process to capture these changes in financial time series based on three distinct characteristics of market information. We calculate the VIX option pricing formula using these models and find that the simplified VIX model based on VIX characteristics has significantly less pricing error than the consistent VIX model derived from the SPX model. Additionally, our findings suggest that the tempered stable process effectively models the volatility of VIX, sparse large jumps, and infinitesimal jumps. It also shows potential as an alternative to Brownian motion for representing volatility. Conversely, jump processes with stochastic jump intensities adeptly describe asymmetric jumps, and their integration with Brownian motion provides a more accurate depiction of the VIX’s volatility and jump dynamics. Finally, the introduction of jump processes into mean-reverting models for the VIX indicates a relatively low correlation between volatility magnitudes and the current VIX levels. This research contributes to the theory of SPX and VIX options and offers guidance for the development of other information-driven economic models.

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.

Calibration of European option pricing model in uncertain environment: Valuation of uncertainty implied volatility

Uncertain differential equations have been widely used in modeling financial markets, and option pricing formulae have been obtained by employing these equations. However, according to the existing literature, the parameter estimation of the option pricing model driven by the uncertain differential equation has not been evaluated so far, and the parameters have been assumed to be arbitrary numbers for such problems. Based on the mentioned gap, this study presents an approach for parameter estimation of European option pricing model in the uncertain environment for the first time. For this purpose, some of the most common uncertain models are considered, and the corresponding empirical examples are presented. Since implied volatility as an instrument provides traders a range of prices that securities change among them, this study evaluates the implied volatility of the European option under the geometric Liu process for the first time. Empirical comparison for both in sample and out of sample dataset based on the uncertainty framework and Black–Scholes model demonstrates that the uncertain models are more accurate for evaluating the European option prices and the implied volatilities.

Jump-diffusion option pricing with non-IID jumps

This paper investigates how violations of the assumption that jumps are identically and independently distributed (IID) affect option prices. We characterize several types of IID jumps violations including jumps with time-varying means, time-varying variances, and time-varying autocorrelations, and analyze the combined effect of these violations on option prices. These IID violations affect option prices in different ways. Different effects of IID violations can either offset each other to eliminate the effect of isolated IID violations or can act as leverages increasing the magnitude of mispricing created by a single violation of the IID assumption. Our investigation is performed inside a new framework for the pricing of options that relaxes the assumption that jumps are IID. We derive three families of jump-diffusion option pricing formulae with non-IID jumps. Numerically and empirically, we show that the non-IID models can improve the BSM on the S&P 500 options for both in-samples and out-of-the-samples data in terms of SSE, AE and ARE.

Quantile Hedging in the complete financial market under the mixed fractional Brownian motion model and the liquidity constraint

This article proposes a pricing framework for European option that utilizes a Quantile hedging strategy in a complete financial market. The methodology involves applying the long memory Geometric Brownian motion model, utilizing the generalized mixed fractional Girsanov theorem, and incorporating relevant findings related to quasi-conditional expectation. The first step in this framework is to derive the option price formula using the equivalent martingale measure. This approach enables investors to use the entire option price without any restrictions to hedge their contingent claim. Then, we add the initial wealth condition to hedge and determine the option price with the Quantile hedging strategy. To do this, we find the maximum probability of success in complete risk hedging for different amounts of model parameters and then use the probability values to determine the option price. Moreover, we apply the minimum variance algorithms and the error correction method to find the hedge ratio. Finally, we illustrate the hedging strategy and provide numerical results that assist investors in determining the required capital for covering the option risk under market data. This calculation takes into account both the level of risk tolerance and the anticipated return on investment.

Option Pricing with the Logistic Return Distribution

The Black–Scholes model and many of its extensions imply a log-normal distribution of stock total returns over any finite holding period. However, for a holding period of up to one year, empirical stock return distributions (both conditional and unconditional) are not log-normal, but rather much closer to the logistic distribution. This paper derives analytic option pricing formulas for an underlying asset with a logistic return distribution. These formulas are simple and elegant and employ exactly the same parameters as B&S. The logistic option pricing formula fits empirical option prices much better than B&S, providing explanatory power comparable to much more complex models with a larger number of parameters.

Pricing vanilla, barrier, and lookback options under two-scale stochastic volatility driven by two approximate fractional Brownian motions

<p>In this paper, we proposed a stochastic volatility model in which the volatility was given by stochastic processes representing two characteristic time scales of variation driven by approximate fractional Brownian motions with two Hurst exponents. We obtained an approximate closed-form formula for a European vanilla option price and the corresponding implied volatility formula based on singular and regular perturbations and a Mellin transform. The explicit formula for the implied volatility allowed us to find the slope of the implied volatility skew with respect to the Hurst exponent and time-to-maturity. The proposed model allows the market volatility behavior to be captured uniformly in time-to-maturity. We conducted an empirical analysis to find the validity of the proposed model by comparing it with other models and Monte Carlo simulation. Further, we extended the pricing result for the vanilla option to two path-dependent exotic (barrier and lookback) options and obtained the corresponding price formulas explicitly.</p>

Pricing geometric average Asian options in the mixed sub-fractional Brownian motion environment with Vasicek interest rate model

<p>Considering the characteristics of long-range correlations in financial markets, the issue of valuing geometric average Asian options is examined, assuming that the variations of the underlying asset follow the mixed sub-fractional Brownian motion, and the dynamics of short-term interest rate satisfies the mixed sub-fractional Vasicek model. Based on the principle of no arbitrage, the definite solution of PDE of a zero-coupon bond for geometric average Asian options under the circumstance of the mixed sub-fractional is given by the delta hedging technique. The derivation of the explicit pricing formula for geometric average Asian options with fixed strike price is achieved through the utilization of multiple variable substitutions. Furthermore, we perform numerical calculations to analyze the performance of the model.</p>

Fractal barrier option pricing under sub-mixed fractional Brownian motion with jump processes

<p>In this work, we mainly focused on the pricing formula for fractal barrier options where the underlying asset followed the sub-mixed fractional Brownian motion with jump, including the down-and-out call option, the down-and-out put option, the down-and-in call option, the down-and-in put option, and so on. To start, the fractal Black-Scholes type partial differential equation was established by using the fractal Itô's formula and a self-financing strategy. Then, by transforming the partial differential equation to the Cauchy problem, we obtained the explicit pricing formulae for fractal barrier options. Finally, the effects of barrier price, fractal dimension, Hurst index, jump intensity, and volatility on the value of fractal barrier options were exhibited through numerical experiments.</p>