Classical Option Pricing Model Research Articles

Pricing cryptocurrency options with machine learning regression for handling market volatility

Pricing cryptocurrency options, crucial for risk management and market stabilization, presents unique challenges due to specific underlying dynamics like the inversion of the leverage effect. Classical option pricing models like Black–Scholes and Heston struggle to address these dynamics due to their set of assumptions. This study introduces machine learning models for options pricing, specifically regression-tree methods. A data-driven machine learning model can incorporate high-frequency volatility estimators into the input set to enhance pricing accuracy. By integrating these estimators, machine learning models can capture the complex dynamics of cryptocurrency markets more effectively than classical pricing approaches. The comparative analysis reveals that equity options are easier to price, clearly indicating inefficiencies in the cryptocurrency option market, which confirms the challenges in achieving accurate pricing. Our results highlight the effectiveness of machine learning models in adapting to the unique characteristics of emerging asset classes, suggesting a shift towards more data-oriented pricing methodologies

Analysis about the Black-Scholes asset price under the regime-switching framework

Assuming that the macroeconomic environment can be transformed into a two-district system, that is, the path of financial asset prices is uncertain, we track and study the motion of stocks and other asset price process under the conditional Black-Scholes model, and give the economical explanation of the mathematical formula. Further, we derive and analyze an option pricing formula for the Black-Scholes asset model under the condition that the risk-free interest rate is regime-switching too. The method in this article is applied to model the log rate of return of the Tencent stock in a two-district market environment. And the obtained parameter values are used to calculate the option price. In narrowing the gap with actual option prices, our method outperforms the classical option pricing model point by point. Compared with the general and pure mathematical model derived work and the empirical study work, our study does more work on the economic characteristics analysis and interpretation of the mathematical models, and plays a certain role in linking the results of mathematical models with empirical research.

Option pricing generators

We characterize a class of option pricing models by their algebraic structure. Option prices are monoids, that is operators endowed with the commutativity and associativity property and an identity element. If the price of the underlying asset is bounded, the operator corresponds to the concept of t-conorm, while if it is defined on the positive real line the operator is a pseudo-addition. These operators have the same no-arbitrage properties as the classical option pricing models, but are also associative. Each model in this class is characterized by a univariate increasing function that is defined the generator of the model. The generator encodes a synthetic representation of the probability structure of the underlying asset. We provide no arbitrage conditions for the generators and practical guidelines to construct them.

Quantum option pricing and data analysis

The paper proposes to treat financial models using techniques of quantum mechanics. The methodology relies on the Dirac matrix formalism and the Feynman path integral approach. This leads us to reexamine in this framework the classical option pricing models of Cox-Ross-Rubinstein and Black-Scholes. Moreover, financial data are classified with respect to the spectrum of a certain observable and then analyzed to identify price jumps using supervised machine learning tools.

Private equity managers’ fees: estimation and sensitivity analysis using Monte Carlo simulation

Private equity managers’ compensation is a particularly problematic area in terms of its tax treatment in the United States and some European countries. This problem originates from the difficulty of defining the particular forms of incentive and therefore their estimated fair value. Based on the literature, carried interest, which is one of the most common profit-sharing arrangements observed in practice, may be considered as an option characterized by several constraints. The use of classical option-pricing models is inappropriate for taking all these constraints into account. The contribution of this paper is to propose a suitable model to estimate the expected compensation of fund managers and to test the effect of fund characteristics and profit-sharing rules on its fair value. We use the Monte Carlo simulation model and take into account the non-marketability criteria of the carried interest. A sensitivity analysis is performed in order to show the evolution of the carried interest value. The results importantly show the sharp differences between carried interest distributed in the case of venture capital and that in the case of buyout funds and show the importance of the claw-back clause for “deal by deal funds”.

Efficient Option Pricing in Crisis Based on Dynamic Elasticity of Variance Model

Market crashes often appear in daily trading activities and such instantaneous occurring events would affect the stock prices greatly. In an unstable market, the volatility of financial assets changes sharply, which leads to the fact that classical option pricing models with constant volatility coefficient, even stochastic volatility term, are not accurate. To overcome this problem, in this paper we put forward a dynamic elasticity of variance (DEV) model by extending the classical constant elasticity of variance (CEV) model. Further, the partial differential equation (PDE) for the prices of European call option is derived by using risk neutral pricing principle and the numerical solution of the PDE is calculated by the Crank-Nicolson scheme. In addition, Kalman filtering method is employed to estimate the volatility term of our model. Our main finding is that the prices of European call option under our model are more accurate than those calculated by Black-Scholes model and CEV model in financial crashes.

Are classical option pricing models consistent with observed option second-order moments? Evidence from high-frequency data

Abstract As a means of validating an option pricing model, we compare the ex-post intra-day realized variance of options with the realized variance of the associated underlying asset that would be implied using assumptions as in the Black and Scholes (BS) model, the Heston, and the Bates model. Based on data for the S&P 500 index, we find that the BS model is strongly directionally biased due to the presence of stochastic volatility. The Heston model reduces the mismatch in realized variance between the two markets, but deviations are still significant. With the exception of short-dated options, we achieve best approximations after controlling for the presence of jumps in the underlying dynamics. Finally, we provide evidence that, although heavily biased, the realized variance based on the BS model contains relevant predictive information that can be exploited when option high-frequency data is not available.

Are Classical Option Pricing Models Consistent with Observed Option Second-Order Moments? Evidence from High-Frequency Data

As a means of validating an option pricing model, we compare the ex-post intra-day realized variance of options with the realized variance of the associated underlying asset that would be implied using assumptions as in the Black and Scholes (BS) model, the Heston and the Bates model. Based on data for the S&P 500 index, we find that the BS model is strongly directionally biased due to the presence of stochastic volatility. The Heston model reduces the mismatch in realized variance between the two markets, but deviations are still significant. With the exception of short-dated options, we achieve best approximations after controlling for the presence of jumps in the underlying dynamics. Finally, we provide evidence that, although heavily biased, the realized variance based on the BS model contains relevant predictive information that can be exploited when option high-frequency data is not available.

Matching the Sensitivities to Discrete Dividends: A New Approach for Pricing Vanillas

The incorporation of a dividend yield in the classical option pricing model of Black-Scholes results in a minor modification of the Black-Scholes formula, since the lognormal dynamic of the underlying asset is preserved. However, market makers prefer to work with cash dividends with fixed value instead of a dividend yield. Since there is no closed-form solution for the price of a European Call in this case, many methods have been proposed in the literature to approximate it. Here, we present a new approach. We derive an exact analytic formula for the sensitivity to dividends of an European option. We use this result to elaborate a proxy which possesses the same Taylor expansion around 0 with respect to the dividends as the exact price. The obtained approximation is very fast to compute (the same complexity than the usual Black-Scholes formula) and numerical tests show the extreme accuracy of the method for all practical cases.