Volatility In Option Pricing Research Articles

Total variation regularization analysis for inverse volatility option pricing problem

We investigate in this paper an inverse problem of recovering the space-dependent volatility in option pricing. To enhance precision across the domain, we transform the original problem into an inverse source problem of a bounded degenerate parabolic equation, utilizing linearization and variable substitution. Unlike classical methods, we apply a total variation regularization combined with a novel generalized finite integration technique. This approach accommodates volatility jumps in an overnight rate scenario. Leveraging an optimal control framework, we demonstrate that the inverse problem can be reformulated as an optimal control problem whose existence, necessary conditions, local uniqueness and stability for the minimizer of control functional are obtained. For numerical verification, we derive the Euler equation and design a discretization algorithm with generalized finite integration technique. Numerical examples showcase the robustness of our approach, highlighting advantages in accuracy and effectiveness over other strategies.

Research on the Development of Implied Volatility in Option Pricing

Option, as an important financial derivative, has received more and more attention from investors in recent years. This paper includes three parts, the first part introduces the concept of options, including the birth and history of options, the trading of options, and different types of options. Then it discusses the history of option pricing and two typical and classic option pricing models and then introduces implied volatility. The second part of this paper introduces the specific development of the option pricing models, the correction process of option pricing models, and volatility models. Different types of volatility, and their comparisons are included. And this paper also discusses the two functions of implied volatility, one is predicting the future and the other is for risk management and portfolio management. The third part concludes the first two-part and discusses the recent progress of option pricing and its volatility.

Numerical Techniques for Determining Implied Volatility in Option Pricing

One of the assumptions of the Black–Scholes model is that volatility is constant in the market. However, in reality, volatility cannot be constant. In this paper we examined an inverse problem of determining the time-dependent non-constant volatility from the observed market price of a European put/call option with a strike price.These unknown non-constant volatilities are from the time the option contract was signed to the time it was exercised (strike price) where the fluctuations of option price is known but the non-constant volatilities that was generated by option price within that time period is unknown.Although many studies have used the portfolio stocks to reconstruct volatility, no study have retrieved volatility from the option market. The aim is to recover unknown non-constant volatilities in vector form from one option contract period using manufacturer observed market data, that is, data from agreed contract period using Black–Scholes Partial Differential Equation (BSPDE). This would be done by using the chain and product rule to take the derivative with respect to volatility in the theoretical(Black-Schoes PDE) model to obtain the non-constant volatilities. By keeping the fluctuations data from observed option prices constant,we differentiate the volatility in the BSPDE to obtain a gradient equation and discretize the gradient equation with the finite difference method to obtain model simulated data. It is expected that non-constant volatility that would be recovered from the market using the gradient descent method would be matched with the non-constant market volatility from the model simulated data and if they are the same or approximately same, the volatility would have deem recovered. All these would be done in the least square sense.

Determining the Volatility in Option Pricing From Degenerate Parabolic Equation

This contribution deals with the inverse volatility problem for a degenerate parabolic equation from numerical perspective. Being different from other inverse volatility problem in classical parabolic equations, the model in this paper is degenerate parabolic equation. Due to solve the deficiencies caused by artificial truncation and control the volatility risk with precision, the linearization method and variable substitutions are applied to transformed the inverse principal term coefficient problem for classical parabolic equation into the inverse source problem for degenerate parabolic equation in bounded region. An iteration algorithm of Landweber type is designed to obtain the numerical solution of the inverse problem. Some numerical experiments are performed to validate that the proposed algorithm is robust and the unknown coefficient is recovered quite well.

Noise trading and stock market bubbles: what the derivatives market is telling us

PurposeThe authors examine the relation between noise trading in equity markets and stochastic volatility by estimating a two-factor jump diffusion model. Their analysis shows that contemporaneous price deviations in the derivatives market are statistically significant in explaining movements in index futures prices and option-market volatility measures.Design/methodology/approachTo understand the impact noise may have in the S&P 500 derivatives market, the authors first measure and evaluate the influence noise exerts on futures prices and then investigate its influence on option volatility.FindingsIn the period from 1996 to 2003, this study finds significant changes in the volatility and mean reversion in the noise level and a significant increase in its relation to implied volatility in option prices. The results are consistent with a bubble in technology stocks that occurred with significant increases in noise trading.Research limitations/implicationsThis study provides estimates for this model during the periods preceding and during the technology bubble. The study analysis shows that the volatility and mean reversion in the noise level are much stronger during the bubble period. Furthermore, the relation between noise trading and implied volatility in the futures market was of a significantly larger magnitude during this period. The study results support the importance of noise trading in market bubbles.Practical implicationsBloomfield, O'Hara and Saar (2009) find that noise traders lower bid–ask spreads and improve liquidity through increases in trading volume and market depth. Such improved market conditions could have positive effects on market quality, and this impact could be evidenced by lower implied volatility when noise traders are more active. Indeed, the results in this study indicate that the level and characteristics of noise trading are fundamentally different during the technology bubble, and this noise trading activity has a larger impact during this period on implied volatility in the options market.Originality/valueThis paper uniquely analyzes derivatives on the S&P 500 Index in order to detect the presence and influence of noise traders. The authors derive and implement a two-factor jump diffusion noise model. In their model, noise rectifies the difference of analysts' opinions, market information and beliefs among traders. By incorporating a reduced-form temporal expression of heterogeneities among traders, the model is rich enough to capture salient time-series characteristics of equity prices (i.e. stochastic volatility and jumps). A singular feature of the authors’ model is that stochastic volatility represents the random movements in asset prices that are attributed to nonmarket fundamentals.

Markov Chain Monte Carlo Method for Estimating Implied Volatility in Option Pricing

Using market covered European call option prices, the Independence Metropolis-Hastings Sampler algorithm for estimating Implied volatility in option pricing was proposed. This algorithm has an acceptance criteria which facilitate accurate approximation of this volatility from an independent path in the Black Scholes Model, from a set of finite data observation from the stock market. Assuming the underlying asset indeed follow the geometric brownian motion, inverted version of the Black Scholes model was used to approximate this Implied Volatility which was not directly seen in the real market: for which the BS model assumes the volatility to be a constant. Moreover, it is demonstrated that, the Implied Volatility from the options market tends to overstate or understate the actual expectation of the market. In addition, a 3-month market Covered European call option data, from 30 different stock companies was acquired from Optionistic.Com, which was used to estimate the Implied volatility. This accurately approximate the actual expectation of the market with low standard errors ranging between 0.0035 to 0.0275.

Comparing Implied Volatility of B-S Model and Dupire Style Local Volatility in Option Pricing

본 논문에서는 BS모형에 의해 유도된 내재변동성과 Dupire 스타일 국지 변동성모형간의 관계를 시뮬레이션을 통하여 비교 분석한다. 이러한 차이점을 분석함으로써 BS모형에 의해 내재된 변동성과 유사한 변동성 모형을 탐색하고 보다 나은 현실적인 변동성모형을 발견하며 고정변동성 변수에 종속되지 않은 현실적인 가격과정을 응용하는데 초점을 맞추고 있다. 본 논문의 동기는 옵션시장의 스마일 현상과 Dupire(1994)가 제안한 국지변동성모형을 사용하여 스마일 현상을 보다 더 설명할수 있는 해법을 제시하는데 있다. 일반적으로 BS모형의 내재변동성은 체계적인 옵션행사가격과 만기 까기 기간에 의존하는 경향이 있다. Derman과 Kan (1994), Dupire (1994) 및 Rubinstein (1994)등의 연구에 따르면 이러한 과정은 BS모형에서 제시한 고정변동성 가정이 현실적으로 맞지 않다는 데 기인하고 있다. 본 논문에서는 콜 가격에 대한 Dupire 스타일 국지변동성 모형을 유도하고 내재변동성과 관련된 Dupire 스타일 국지변동성을 제시한다. 이러한 비교를 통하여 본 논문의 결과는 Dupire스타일 국지변동성모형이 옵선 가격 결정 시 고정 변동성 또는 내재변동성에 대한 대안적 변동성 모형이 될 수 있다는 것을 발견하였다.

Numerical Methods for Solving Implied Volatility in Option Pricing

Implied volatility is an important variable affecting the option price. In the B-S model, other factors can be obtained directly, but implied volatility is not directly. According to B-S model, build on the volatility equation, using Newton iteration to obtain its approximate solution and compare with the classical formula of demand volatility. The example shows that good results can be obtained.

How Long Does the Market Think You Will Live? Implying Longevity from Annuity Prices

We use life annuity prices to extract information about human longevity by formulating a model that links the term structure of mortality and interest rates. We then invert the model and perform nonlinear least squares to obtain implied longevity forecasts. Methodologically, we assume a Cox-Ingersoll-Ross (CIR) model for the underlying yield curve and a cohort-varying Gompertz-Makeham (GM) law for mortality. Our main result is that over the last decade markets have been implying an improvement in longevity of an average of 6 weeks per year for males and 3 weeks for females. In the year 2004 market prices implied a 40.1% probability of survival to the age 90 for a 75-year old male (51.2% for a female) annuitant. By the year 2013 the implied survival probability had increased to 46.1% (and 53.1%). The corresponding implied life expectancy has increased (at the age of 75) from 13.09 years for males (15.08 years for females) to 14.28 years (and 15.61 years). Although these values are implied directly from markets, they are consistent with demographic projections. Similar to implied volatility in option pricing, we believe that our implied survival probabilities (ISP) and implied life expectancy (ILE) are relevant for the financial management of assets post-retirement and very important for the optimal timing and allocation to annuities; procrastinators are swimming against an uncertain but rather strong longevity trend.

Profitability Decomposition and Operating Risk

The DuPont decomposition of profitability into profit margin and asset turnover has been applied extensively to forecast future profits. This paper provides a complementary analysis under which the DuPont decomposition also informs about the risk that profits may differ from expectation. In the standard decomposition, the profit margin describes how sales translate into operating profit. Correspondingly, the analysis here shows how the profit margin informs about how variance in sales growth rates translates into variance in the growth rate of operating profit. We identify a measure that captures the effect. That measure also points to a role for the asset turnover. In the standard decomposition, the asset turnover describes how operating assets yield sales, while our analysis shows how the asset turnover combines with the profit margin to indicate the variability of operating profit resulting from those sales. The empirical results validate that the DuPont decomposition is useful for forecasting the variance in future growth rates in operating profits and also for forecasting the variance in future stock returns. Moreover, the decomposition also explains the implied volatility in option prices.

Calibration of the Volatility in Option Pricing Using the Total Variation Regularization

In market transactions, volatility, which is a very important risk measurement in financial economics, has significantly intimate connection with the future risk of the underlying assets. Identifying the implied volatility is a typical PDE inverse problem. In this paper, based on the total variation regularization strategy, a bivariate total variation regularization model is proposed to estimate the implied volatility. We not only prove the existence of the solution, but also provide the necessary condition of the optimal control problem—Euler-Lagrange equation. The stability and convergence analyses for the proposed approach are also given. Finally, numerical experiments have been carried out to show the effectiveness of the method.

Option valuation based on the neural regression model

In this article, a neural regression (NR) model, which produces nonlinear coefficients of multiple regression model based on neural networks, is introduced to capture the option valuation's nonlinear characteristics effectively. The traditional linear regression uses the least-squares estimator to estimate the coefficient of a linear regression and thus may only produce suboptimal solutions. However, Applying neural networks to forecast volatility in option pricing has increased in popularity in recent years since many studies have indicated that the conventional option pricing models are not accurate enough. Our proposed neural regression model devotes to evaluate option values to improve on the tracking error in the measurement of hedging capability. The NR model uses the variables introduced by the Black-Scholes Model and applies the multiple regressions (MR) model to re-price option values. It is worth noting that each corresponding weight coefficient in MR is constructed by a complete neural network rather than by a scalar value. By capturing the nonlinear behaviors of option pricing, our proposed NR model has lower tracking error and better hedging capability than the BS model and other studies.

An inverse problem of determining the implied volatility in option pricing

In the Black–Scholes world there is the important quantity of volatility which cannot be observed directly but has a major impact on the option value. In practice, traders usually work with what is known as implied volatility which is implied by option prices observed in the market. In this paper, we use an optimal control framework to discuss an inverse problem of determining the implied volatility when the average option premium, namely the average value of option premium corresponding with a fixed strike price and all possible maturities from the current time to a chosen future time, is known. The issue is converted into a terminal control problem by Green function method. The existence and uniqueness of the minimum of the control functional are addressed by the optimal control method, and the necessary condition which must be satisfied by the minimum is also given. The results obtained in the paper may be useful for those who engage in risk management or volatility trading.

Long-memory volatility in derivative hedging

The aim of this work is to take into account the effects of long memory in volatility on derivative hedging. This idea is an extension of the work by Fedotov and Tan [Stochastic long memory process in option pricing, Int. J. Theor. Appl. Finance 8 (2005) 381–392] where they incorporate long-memory stochastic volatility in option pricing and derive pricing bands for option values. The starting point is the stochastic Black–Scholes hedging strategy which involves volatility with a long-range dependence. The stochastic hedging strategy is the sum of its deterministic term that is classical Black–Scholes hedging strategy with a constant volatility and a random deviation term which describes the risk arising from the random volatility. Using the fact that stock price and volatility fluctuate on different time scales, we derive an asymptotic equation for this deviation in terms of the Green's function and the fractional Brownian motion. The solution to this equation allows us to find hedging confidence intervals.

Temporal Price Relation Between Stock and Option Markets and a Bias of Implied Volatility in Option Prices

We show that if a particular temporal relation exists between the option and spot markets, the implied volatility in option prices can be biased depending on the level of the true volatility. The higher the true volatility, the more upward (downward) biased the implied volatility will be, if the option market leads (lags) the spot market. Using intraday data of the S&P 500 index options, we show that the option market leads the spot market at least in the sample. More importantly, the implied volatility is biased due to the lead-lag relationship, and the bias is more profound when the market is more volatile.

A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm

In this work, we develop a reduced-basis approach for the efficient computation of parametrized expected values, for a large number of parameter values, using the control variate method to reduce the variance. Two algorithms are proposed to compute online, through a cheap reduced-basis approximation, the control variates for the computation of a large number of expectations of a functional of a parametrized Ito stochastic process (solution to a parametrized stochastic differential equation). For each algorithm, a reduced basis of control variates is pre-computed offline, following a so-called greedy procedure, which minimizes the variance among a trial sample of the output parametrized expectations. Numerical results in situations relevant to practical applications (calibration of volatility in option pricing, and parameter-driven evolution of a vector field following a Langevin equation from kinetic theory) illustrate the efficiency of the method.