Constant Volatility Research Articles

Empirical study based on the model of rough fractional stochastic volatility (RFSV)

To estimate the parameters of the model of option pricing based on the model of rough fractional stochastic volatility (RFSV), we have carried out the empirical analysis during our study on the pricing of SSE 50ETF options in China. First, we have estimated the parameters of option pricing model by adopting the Monte Carlo simulation. Subsequently, we have empirically examined the pricing performance of the RFSV model by adopting the SSE 50ETF option price from January 2019 to December 2020. Our research findings indicate that by leveraging the RFSV model, we are able to attain a more accurate and stable level of option pricing than the conventional Black–Scholes (B-S) model on constant volatility. The errors of option pricing incurred by the B-S model proved to be larger and exhibited higher volatility, revealing the significant impact imposed by stochastic volatility on option pricing.

On the investment strategies in occupational pension plans

Demographic changes increase the necessity to base the pension system more and more on the second and the third pillars, namely the occupational and private pension plans; this paper deals with Target Date Funds (TDFs), which are a typical investment opportunity for occupational pension planners. TDFs are usually identified with a decreasing fraction of wealth invested in equity (a so-called glide path) as retirement comes closer, i.e. wealth is invested more riskily the younger the saver is. We investigate whether this is actually optimal in the presence of non-tradable income risk in a stochastic volatility environment. The retirement planning procedure is formulated as a stochastic optimization problem. We find it is the (random) contributions that induce the optimal path exhibiting a glide path structure, both in the constant and stochastic volatility environment. Moreover, the initial wealth and the initial contribution made to a retirement account strongly influence the fraction of wealth invested in risky assets. The risk aversion of an individual mainly determines the shape of the glide path, while the ratio of the wealth and income influences the steepness of the glide path.

Option Pricing under Randomised GBM Models

By employing a randomisation procedure on the variance parameter of the standard geometric Brownian motion (GBM) model, we construct new families of analytically tractable asset pricing models. In particular, we develop two explicit families of processes that are respectively referred to as the randomised gamma (G) and randomised inverse gamma (IG) models, both characterised by a shape and scale parameter. Both models admit relatively simple closed-form analytical expressions for the transition density and the no-arbitrage prices of standard European-style options whose Black-Scholes implied volatilities exhibit symmetric smiles in the log-forward moneyness. Surprisingly, for integer-valued shape parameter and arbitrary positive real scale parameter, the analytical option pricing formulas involve only elementary functions and are even more straightforward than the standard (constant volatility) Black-Scholes (GBM) pricing formulas. Moreover, we show some interesting characteristics of the risk-neutral transition densities of the randomised G and IG models, both exhibiting fat tails. In fact, the randomised IG density only has finite moments of the order less than or equal to one. In contrast, the randomised G density has a finite first moment with finite higher moments depending on the time-to-maturity and its scale parameter. We show how the randomised G and IG models are efficiently and accurately calibrated to market equity option data, having pronounced implied volatility smiles across several strikes and maturities. We also calibrate the same option data to the wellknown SABR (Stochastic Alpha Beta Rho) model.

Volatility Managed Indexes: The Importance of Intraday Data

Modern volatility managed indexes employ antiquated close-to-close volatility models to scale exposure. This has material consequences on the option pricing dynamics for structured investments, many of which reference a volatility managed index because of option pricing benefits associated with the constant volatility. The author tests two daily close-to-close models and three intraday “realized volatility” models that leverage the rich information dynamics found in intraday data. The results, across three highly liquid US large-cap indexes, suggest that the latter three models produce more consistent volatility profiles, or alternatively, a lower volatility of volatility, without sacrificing performance (i.e., the Sharpe ratio). All else being equal, this lower volatility of volatility should be preferred by product issuers and investors because of the resulting option pricing benefits.

Effect of spinning solvent on crystallization behavior of confined polymers in electrospun nanofibers

The effect of spinning solvent on the morphology and crystallization behavior of the electrospun nanofibers of polystyrene (PS)/poly(ethylene oxide) (PEO) blends was investigated. The PS/PEO blends, with PEO as the minority component, were electrospun using binary solvent mixtures, involving DMF, dichloromethane and chloroform, with varying solvent dielectric constant and volatility. Even though the bulk morphology of the electrospun nanofibers broadly consisted of PEO domains dispersed in PS matrix, the size and dispersity of the PEO domains depended on the spinning solvent. The high dielectric constant and low volatility of DMF ensured higher jet stretch as well as longer available time for the break-up of PEO domains, for solvent mixtures enriched in the DMF. This resulted in a notable influence on the crystallization behavior of PEO in the blend nanofibers. The fractionated behavior of PEO crystallization was observed to become more noticeable at high DMF content. This was due to the presence of relatively larger population of nanosized PEO domains in nanofibers where the crystallization occurred mostly due to homogenous nucleated crystallization. The present study highlights the role of spinning solvent on crystallization behavior in electrospun nanofibers of polymer mixtures which will ultimately also affect its other properties.

Interest rate structured products: can they improve the risk–return profile?

In this paper, we investigate the contribution of interest rate structured bonds to portfolios of risk-averse retail investors. We conduct our analysis by simulating the term structure according to a multifactor no-arbitrage interest rate model and comparing the performance of a portfolio consisting of basic products (zero-coupon bonds, coupon bonds and floating rate notes) with a portfolio containing more sophisticated exotic products (like constant maturity swaps, collars, spread and volatility notes). Our analysis, performed under different market environments, as well as volatility and correlation levels, takes into account the combined effects of risk premiums required by investors and fees that they have to pay. Our results show that capital protected interest rate structured products allow investors to improve risk–return trade-off if no fees are considered. With fees, our simulations show that structured products add value to the basic portfolio in a very limited number of cases. We believe our paper contributes to understanding the role of structured products in investors portfolios also in light of the current regulatory debate on the use of complex financial products by retail investors.

Implied volatility estimation of bitcoin options and the stylized facts of option pricing

The recently developed Bitcoin futures and options contracts in cryptocurrency derivatives exchanges mark the beginning of a new era in Bitcoin price risk hedging. The need for these tools dates back to the market crash of 1987, when investors needed better ways to protect their portfolios through option insurance. These tools provide greater flexibility to trade and hedge volatile swings in Bitcoin prices effectively. The violation of constant volatility and the log-normality assumption of the Black–Scholes option pricing model led to the discovery of the volatility smile, smirk, or skew in options markets. These stylized facts; that is, the volatility smile and implied volatilities implied by the option prices, are well documented in the option literature for almost all financial markets. These are expected to be true for Bitcoin options as well. The data sets for the study are based on short-dated Bitcoin options (14-day maturity) of two time periods traded on Deribit Bitcoin Futures and Options Exchange, a Netherlands-based cryptocurrency derivative exchange. The estimated results are compared with benchmark Black–Scholes implied volatility values for accuracy and efficiency analysis. This study has two aims: (1) to provide insights into the volatility smile in Bitcoin options and (2) to estimate the implied volatility of Bitcoin options through numerical approximation techniques, specifically the Newton Raphson and Bisection methods. The experimental results show that Bitcoin options belong to the commodity class of assets based on the presence of a volatility forward skew in Bitcoin option data. Moreover, the Newton Raphson and Bisection methods are effective in estimating the implied volatility of Bitcoin options. However, the Newton Raphson forecasting technique converges faster than does the Bisection method.

Robust and accurate construction of the local volatility surface using the Black–Scholes equation

• We present an algorithm to reconstruct a local volatility (LV) function. • We reconstruct the local volatility surface using a nonlinear fitting function. • The results demonstrate the robustness and accuracy of the proposed algorithm. In this study, we develop a numerical method for the robust and accurate construction of a local volatility (LV) surface using the generalized Black–Scholes (BS) equation from the given option price data. The BS equation is a partial differential equation and has been used to model financial option pricing. Constant volatility was used in the classical BS model. However, it is well known that the constant volatility BS model is practically unsuitable because real financial market data demonstrate non-constant volatility behavior. The LV function is dependent on the asset prices and time. One of the difficulties in reconstructing an unknown LV surface is uniqueness. We extend a previous study of reconstructing time-dependent volatility, which is unique, to time- and space-dependent volatility surfaces. We propose an algorithm comprising four steps: the first step is estimating constant implied volatility; the second step is finding the influential region using the probability density function of a log-normal distribution; the third step is calculating the time-dependent volatility function; and the final step is reconstructing the LV surface. We use a finite difference method to numerically solve the BS model and a nonlinear fitting function to compute the LV surface. We perform computational experiments using synthetic and real market data. The numerical results demonstrate the robust and accurate construction of an unknown LV surface using the proposed method.

Closed‐form approximated pricing of multivariate derivatives under switching regime models

AbstractMarkov switching regime models have played an increasingly important role in finance and economics, especially for business cycles and long swings in currencies. Regime‐switching models provide a simple way to capture stochastic volatility and thus overcomes the drawback of the classical lognormality assumption characterized by constant volatility. This paper considers multivariate Black and Scholes type models with a Markov regime‐switching mechanism. We show that the pricing of some multivariate derivatives under models where the Markov chain has two or three states, can be approximated accurately in closed‐form, based on linear and quadratic Taylor polynomials. Closed form approximation methods are computationally advantageous as they perform in constant time, compared with alternative methods such as Monte‐Carlo, where the accuracy of the estimation is directly linked to the number of executed simulations.

Infinite Black-Scholes Mixture

We consider a slight extension of the constant volatility Black-Scholes model where the integrated variance is assumed random, uncorrelated to the spot process and uniformly distributed over a finite range. We analytically calculate the expectation over the random variance and obtain a closed form formula for the call price under this infinite mixture model. Based on the call formula every quantity of interest (CDF, PDF, etc.) can be calculated in closed form.

TREND-CYCLE MODELING OF TURKISH INFLATION: DOES STOCHASTIC VOLATILITY PLAY A ROLE?

Inflation dynamics in Turkey have been notoriously volatile for several decades. Starting in the 1970 decade, inflation rates frequently reached three digit levels, remained high until 2000 and decreased to relatively low levels after 2001 as a consequence of a series of reforms. Considering the volatile and alternating behavior of inflation dynamics in Turkey, the purpose of this work is to utilize different specifications, including time-varying trend and stochastic volatility setups, in order to test, to which extent stochastic volatility is relevant for modeling inflation in Turkey for the period 1955-2020. Our results suggest that inflation volatility is indeed time-varying and that the period 1980-2000 was characterized by increasing trend inflation and inflation volatility. Also, we show that specifications with stochastic volatility are preferred over those with constant volatility.

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 collective investment problem in a stochastic volatility environment: The impact of sharing rules

It is typical in collectively administered pension funds that employees delegate fund managers to invest their contributions. In addition, many pension funds still need to sustain guarantees (prescribed by law) in spite of the current low interest environment. In this paper, we consider an optimal collective investment problem for a pool of investors who (implicitly) demand minimum guarantees by deriving utility from the wealth exceeding their guarantees in two financial market settings, one with a stochastic and one with a constant volatility. We find that individual investors’ well-being will not be worsened through the collective investment in both financial markets, as individual optimal solutions are attainable if a financially fair state-dependent sharing rule is applied. When more prevailing sharing rules like linear rules are applied, this holds no longer. Furthermore, the degree of sub-optimality imposed by linear sharing rules is more pronounced in the stochastic volatility market than in the constant volatility market.

Time-varying autoregressive distributed lag model with changing volatility for stress test

We present a novel time-varying autoregressive distributed lag (TV-ADL) model that allows for changes in both transmission mechanisms and innovation volatilities. The forecasting performance of the TV-ADL model has been substantially improved by removing the unrealistic traditional assumptions of constant volatility and constant inter-variable relationship. Our model is further adapted to stress tests mandated by the US Federal Reserve to generate conditional forecasts of the pre-provision net revenue of financial holding companies with large assets. The improvement of forecasting performance is demonstrated by the significant reduction of out-of-sample forecast errors at different horizons.

Sailing in rough waters: Examining volatility of fMRI noise

BackgroundThe assumption that functional magnetic resonance imaging (fMRI) noise has constant volatility has recently been challenged by studies examining heteroscedasticity arising from head motion and physiological noise. The present study builds on this work using latest methods from the field of financial mathematics to model fMRI noise volatility. MethodsMulti-echo phantom and human fMRI scans were used and realised volatility was estimated. The Hurst parameter H ∈ (0,1), which governs the roughness/irregularity of realised volatility time series, was estimated. Calibration of H was performed pathwise, using well-established neural network calibration tools. ResultsIn all experiments the volatility calibrated to values within the rough case, H < 0.5, and on average fMRI noise was very rough with 0.03 < H < 0.05. Some edge effects were also observed, whereby H was larger near the edges of the phantoms. DiscussionThe findings suggest that fMRI volatility is not only non-constant, but also substantially more irregular than a standard Brownian motion. Thus, further research is needed to examine the impact such pronounced oscillations in the volatility of fMRI noise have on data analyses.

Vietnam Land Market: Definitions, Characteristics, and Effects

The land is a vital resource, and the world’s powers that want to develop must rely on land as their core. Vietnam is in economic renewal, so whether the country can produce as expected or not depends on how land is used. The land conversion is continuously changing, leading to the constant volatility of the land market. Besides, the land is a unique commodity, and in a socialist country like Vietnam, only land use rights can be bought and sold. The article uses descriptive and explanatory methods to highlight the results that need research. This article also shows Vietnam’s land market characteristics through the primary market and the secondary market. This market decentralization creates a two-price mechanism, meaning that a piece of land is priced differently depending on the organization that uses it. This mechanism has created difficulties in producing an available land bank, leading to a reduction in investment attraction in this potential country.

What is the volatility of an Asian option?

We propose a simple approximation for pricing Asian options on an underlying asset with an implied volatility smile by substituting an appropriately adjusted volatility into a Black-Scholes pricer which takes a constant volatility. For Asian options with strikes close to the at-the-money point, the adjustment is independent of the shape of the volatility smile at leading order in an expansion in log-strike, which makes it robust against calibration errors. The smile adjusted valuation becomes exact for Asian options with continuous time averaging in the short maturity limit in the local volatility model.

On The Investment Strategies in Occupational Pension Plans

Demographic changes increase the necessity to base the pension system more and more on the second and the third pillar, namely the occupational and private pension plans; this paper deals with Target Date Funds (TDFs), which are a typical investment opportunity for occupational pension planners. TDFs are usually identified with a decreasing fraction of wealth invested in equity (a so-called glide path) as retirement comes closer, i.e., wealth is invested more risky the younger the saver is. We investigate whether this is actually optimal in the presence of non-tradable income risk in a stochastic volatility environment. The retirement planning procedure is formulated as a stochastic optimization problem. We find it is the (random) contributions that induce the optimal path exhibiting a glide path structure, both in the constant and stochastic volatility environment. Moreover, the initial wealth and the initial contribution made to a retirement account strongly influence the fractional amount of wealth to be invested in risky assets. The risk aversion of an individual mainly determines the steepness of the glide path.

Pricing Various Types of Power Options under Stochastic Volatility

The exotic options with curved nonlinear payoffs have been traded in financial markets, which offer great flexibility to participants in the market. Among them, power options with the payoff depending on a certain power of the underlying asset price are widely used in markets in order to provide high leverage strategy. In pricing power options, the classical Black–Scholes model which assumes a constant volatility is simple and easy to handle, but it has a limit in reflecting movements of real financial markets. As the alternatives of constant volatility, we focus on the stochastic volatility, finding more exact prices for power options. In this paper, we use the stochastic volatility model introduced by Schöbel and Zhu to drive the closed-form expressions for the prices of various power options including soft strike options. We also show the sensitivity of power option prices under changes in the values of each parameter by calculating the resulting values obtained from the formulas.

ELS pricing and hedging in a fractional Brownian motion environment

An equity-linked security (ELS) is a debt instrument with several payments and maturities linked to equity markets. This paper is a study of the pricing and hedging of the ELS when the underlying asset price moves in a geometric fractional Brownian motion environment. We develop two different methods for calibrating fractional implied volatility, obtain an empirical result on the Hurst exponent, and introduce a new Greek called Eta to find the sensitivity of the ELS price to the Hurst parameter. We propose three Delta hedging strategies and compare them with each other and the classical Black-Scholes Delta hedging strategy. Their performance is shown to depend on market circumstance (bull or bear). Our results with constant volatility and Hurst exponent provide a building basis for more stable hedging strategies in the non-Markov environment.