Constant Volatility Research Articles

Valuing supply‐chain responsiveness under demand jumps

As the time between the decision about what to produce and the moment when demand is observed (the decision lead time) increases, the demand forecast becomes more uncertain. Uncertainty can increase gradually in decision lead time, or can increase as a dramatic change in median demand. Whether the forecast evolves gradually or in jumps has important implications for the value of responsiveness, which we model as the cost premium worth paying to reduce the decision lead time (the justi fied cost premium). Demand uncertainty arising from jumps rather than from constant volatility increases the justifi ed cost premium when an average jump increases median demand, but decreases the justifi ed cost premium when an average jump decreases median demand. We fi t our model to two data sets, fi rst publicly available demand data from Reebok, then point-of-sale data from a supermarket chain. Finally, we present two special cases of the model, one covering a sudden loss of demand, and the other a one-time adjustment to median demand.

Valuing investment decisions of renewable energy projects considering changing volatility

Volatility is an important parameter when evaluating investments using the real options method. For renewable energy investments, the volatility of cash flow continuously changes, because as new information and knowledge are gathered, there is less foreseen variation. This paper proposes an extended recombining trinomial tree model, where the changing volatility is used to generate transition probabilities. The changing volatility is generated using a consolidation process where multiple random variables, including the market price of electricity, carbon price, and lending rate, are integrated into a low-dimension stochastic process. A two-factor learning curve is used to model the changes of investment cost. We apply the proposed model to analyze solar photovoltaic (PV) power generation investment in China. The results show volatility with changing feature. Compared with constant volatility, changing volatility may advance investment decisions and change the project value. Complete grid parity policy in the solar PV industry is infeasible because the opportunity cost brought by the option of delaying cannot be offset. The changing volatility may produce a lower and equally effective subsidy level compared with constant volatility. A carbon emission trading scheme is helpful in advancing investments in renewable energy, which is reflected in improvements in project value, advancements in investment decisions, and reductions in the required subsidy level. • Propose an extended recombining trinomial tree model considering changing volatility. • Integrate multiple random variables into a low-dimension process to generate the changing volatility. • Analyze the effects of changing volatility on project value, investment decision and the needed subsidy.

Over-Fitting in The TVP Model: A Comparison of Shrinkage Priors in Inflation Forecasting

The study is empirically motivated to analyze the performance of new class of Bayesian shrinkage priors that are powerful in reducing time-varying parameters to static ones to avoid over-fitting problem in time-varying parameter models. We utilized newly improved shrinkage priors in a non-centered parameterization form following Bitto and Fruhwirth-Schnatter (2019) who introduced variance selection using normal-gamma prior which nests the Bayesian LASSO prior of Belmonte et al. (2014) and spike and slab prior as in Schnatter and Wagner (2010). Therefore, the priors are able to discriminate time-varying coefficients from the static ones and the coefficients that can be shrunk to zero. In the empirical exercise, we estimated the generalized Phillip curve for three inflation-targeting countries and produced evidence of significant time-variation in most of the predictors. We compare the performance of the priors in density forecast of inflation allowing for constant and stochastic volatility in the model estimation. Evidence from the estimates of the log predictive density score shows that the hierarchical Normal Gamma shrinkage prior produces the best results for Canada and South Africa whilst the Normal Bayesian LASSO produces the best results for New Zealand, and that adding stochastic volatility improves the performance of models in density forecast.

Impact of FOMC Cycle on Market Uncertainty: Evidence From Interest Rate Derivatives

This paper investigates how Federal Reserve (Fed) actions influence market uncertainty. We consider two kinds of Fed events: the day of the Federal Open Market Committee (FOMC) meeting -- which includes a policy statement, press conference and release of a Summary of Economic Projections -- and the day the minutes of the FOMC minutes are released -- which is typically set 3 weeks after the meeting. Unlike related papers focused on the issue, we measure market uncertainty by the implied volatility extracted from interest rate options, specifically swaptions. We use 1-month constant maturity volatility for swaptions over tenors ranging from one up to 30 years as they are reflective of how these volatilities are marked by dealers/market-makers and cover only one FOMC meeting/minutes release at a time. We use an event study approach along with extensive graphical analysis to evaluate the impact of Fed actions. The results show that 1-month constant maturity implied volatility increases marginally going into these events and falls much more significantly afterwards. Remarkably, the increase and reduction in uncertainty around a meeting is not reducing with increasing tenors, showing that the impact of information release has a similar impact over all horizons ranging from 1 to 30 years. If, on one side, this evidence witnesses the capacity of the central bank to control the long end of the curve, on the other it indicates the possibility that overreaction to news may generate the puzzling excess volatility which is observed in long term rates.

Estimating Trend Inflation with Moving Average and Stochastic Volatility for Nigeria

This study was carried out to estimate underlying inflation in Nigeria using Unobserved Component (UC) model. Also, different channels were used to identify the source of inflation persistence and volatility. This was estimated using Bayesian analysis in order to examine the role of priors in improving the precision of the estimates. Following the analysis, a number of observations were made. Firstly, in terms of methodological contribution, we found that the observed inflation in Nigeria can be modelled by assuming constant volatility as the evidence shows that stochastic volatility (SV) is not a source of observed inflation persistence. However, the variance of the trend inflation can be modelled using regime switching models, where both the mean and variance can be subjected to regime changes. This is because we were only able to establish an evidence of instability not volatility in the variance of the trend inflation. Secondly, we found that the inclusion of autoregressive component in the UC specification leads to volatility in the variance of the observed inflation. Therefore, unobserved component model with MA is more appropriate in modeling inflation persistence in Nigeria which is consistent with empirical regularities. Furthermore, the study found that the observed core inflation (which is the headline less food inflation as released by Bureau of National Statistics, NBS) is being understated. This finding has significant implication for the current measure of core inflation and its application in policy decisions. Therefore, we recommend that the NBS should remove energy components from the observed core inflation in order to have a consistent permanent inflation component in Nigeria.

Truncated EM numerical method for generalised Ait-Sahalia-type interest rate model with delay

The original Ait-Sahalia model of the spot interest rate proposed by Ait-Sahalia assumes constant volatility. As supported by several empirical studies, volatility is never constant in most financial markets. From application viewpoint, it is important we generalise the Ait-Sahalia model to incorporate volatility as a function of delay in the spot rate. In this paper, we study analytical properties for the true solution of this model and construct several new techniques of the truncated Euler–Maruyama (EM) method to study properties of the numerical solutions under the local Lipschitz condition plus Khasminskii-type condition. Finally, we justify that the truncated EM approximate solution can be used within a Monte Carlo scheme for numerical valuations of some financial instruments such as options and bonds.

Fast mean-reversion asymptotics for large portfolios of stochastic volatility models

We consider an asymptotic SPDE description of a large portfolio model where the underlying asset prices evolve according to certain stochastic volatility models with default upon hitting a lower barrier. The asset prices and their volatilities are correlated through systemic Brownian motions, and the SPDE is obtained on the positive half-space along with a Dirichlet boundary condition. We study the convergence of the loss from the system, which is given in terms of the total mass of a solution to our stochastic initial-boundary value problem, under fast mean-reversion of the volatility. We consider two cases. In the first case, the volatilities are sped up towards a limiting distribution and the system converges only in a weak sense. On the other hand, when only the mean-reversion coefficients of the volatilities are allowed to grow large, we see a stronger form of convergence of the system to its limit. Our results show that in a fast mean-reverting volatility environment, we can accurately estimate the distribution of the loss from a large portfolio by using an approximate constant volatility model which is easier to handle.

Pricing holder-extendable call options with mean-reverting stochastic volatility

Options with extendable features have many applications in finance and these provide the motivation for this study. The pricing of extendable options when the underlying asset follows a geometric Brownian motion with constant volatility has appeared in the literature. In this paper, we consider holder-extendable call options when the underlying asset follows a mean-reverting stochastic volatility. The option price is expressed in integral forms which have known closed-form characteristic functions. We price these options using a fast Fourier transform, a finite difference method and Monte Carlo simulation, and we determine the efficiency and accuracy of the Fourier method in pricing holder-extendable call options for Heston parameters calibrated from the subprime crisis. We show that the fast Fourier transform reduces the computational time required to produce a range of holder-extendable call option prices by at least an order of magnitude. Numerical results also demonstrate that when the Heston correlation is negative, the Black–Scholes model under-prices in-the-money and over-prices out-of-the-money holder-extendable call options compared with the Heston model, which is analogous to the behaviour for vanilla calls.
 
 doi:10.1017/S1446181119000142

Volatility Timing: Pricing Barrier Options on DAX XETRA Index

This paper analyses the impact of different volatility structures on a range of traditional option pricing models for the valuation of call down and out style barrier options. The construction of a Risk-Neutral Probability Term Structure (RNPTS) is one of the main contributions of this research, which changes in parallel with regard to the Volatility Term Structure (VTS) in the main and traditional methods of option pricing. As a complementary study, we propose the valuation of options by assuming a constant or historical volatility. The study implements the GARCH (1,1) model with regard to the continuously compound returns of the DAX XETRA Index traded at daily frequency. Current methodology allows for obtaining accuracy forecasts of the realized market barrier option premiums. The paper highlights not only the importance of selecting the right model for option pricing, but also fitting the most accurate volatility structure.

Estimating Stochastic Volatility under the Assumption of Stochastic Volatility of Volatility

We propose novel nonparametric estimators for stochastic volatility and the volatility of volatility. In doing so, we relax the assumption of a constant volatility of volatility and therefore, we allow the volatility of volatility to vary over time. Our methods are exceedingly simple and far simpler than the existing ones. Using intraday prices for the Standard & Poor’s 500 equity index, the estimates revealed strong evidence that both volatility and the volatility of volatility are stochastic. We also proceeded in a Monte Carlo simulation analysis and found that the estimates were reasonably accurate. Such evidence implies that the stochastic volatility models proposed in the literature with constant volatility of volatility may fail to approximate the discrete-time short rate dynamics.

MULTI-ASSET PORTFOLIO OPTIMIZATION WITH STOCHASTIC SHARPE RATIO UNDER DRAWDOWN CONSTRAINT

We consider an investor who seeks to maximize his expected utility of the portfolio, consisting of multiple risky assets and one risk-free asset, derived from the terminal wealth relative to the maximum wealth achieved over a fixed time horizon. This is achieved under a portfolio draw down constraint, in a market with local stochastic volatility. In empirical application, considering two risky assets, the assets have been identified with the help of pairs trading. In the absence of closed form solution of the value function and the optimal strategy, we obtain the approximates of these quantities using coefficient series expansion techniques and finite difference schemes. We utilize the risk tolerance factor function to ease our approximations of this value functions and the strategies. All the parameters were estimated from the triplets and used to illustrate and compare the stochastic volatility with the constant volatility situation, and how an investor can deploy different portfolio plans.

PENENTUAN NILAI KONTRAK OPSI SAHAM TIPE EROPA MENGGUNAKAN MODEL CONSTANT ELASTICITY OF VARIANCE

Investment is a very sensitive matter especially relating to securities commonly known as shares. Shares are not merely as securities or certificates of ownership but as a business area in achieving profits. One alternative factor for investment is option. Stock options are one of the trading tools used to secure stock investments owned by investors. The real value of stock options can be known when the due date. The stock option value formula can be used to find out the value before the due date. The most widely known stock option value is to use the Black-Scholes equation which is obtained from a constant volatility value. Then it was developed because it saw the conditions in the market based on the volatility of the value (not constant). The purpose of this study is to determine the value of stock options in the market based on volatile values ??that change using the Constant Elasticity of Variance model with the limit of European stock purchase options. If the resulting stock option value is greater than the option price in the market, investors are advised to buy the stock option.

Classical Option Pricing and Some Steps Further

This paper considers the asset price p as relations C=pV between the value C and the volume V of the executed transactions and studies the consequences of this definition for the option pricing equations. We show that the classical BSM model implicitly assumes that value C and volume V of transactions follow identical Brownian processes. Violation of this identity leads to 2-dimensional BSM-like equation with two constant volatilities. We show that agents expectations those approve execution of transactions can further increase the dimension of the BSM model. We study the case when agents expectations may depend on the option price data and show that such assumption can lead to the nonlinear BSM-like equations. We reconsider the Heston stochastic volatility model for the price determined by the value and the volume and derive 3-dimensional BSM-like model with stochastic value volatility and constant volume volatility. Variety of the BSM-like equations states the problem of reasonable balance between the accuracy and the complexity of the option pricing equations.

Importance Sampling for Monte Carlo Simulation to Evaluate Collar Options under Stochastic Volatility Model

The collar option is one kind of exotic options which is useful when institutional investors wish to lock in the profit they already have on the underlying asset. Under the constant volatility assumption, the pricing problem of collar options can be solved in the classical Black Scholes framework. However the smile-shaped pattern of the Black Scholes implied volatilities which extracted from options has provided evidence against the constant volatility assumption, so stochastic volatility model is introduced. Because of the complexity of the stochastic volatility model, a closed-form expression for the price of collar options may not be available. In such case, a suitable numerical method is needed for option pricing under stochastic volatility. Since the dimensions of state variable are usually more than two after the introduction of another volatility diffusion process, the classical finite difference method seems inefficient in the stochastic volatility scenario. For its easy and flexible computation, Monte Carlo method is suitable for evaluating option under stochastic volatility. This paper presents a variance reduction method for Monte Carlo computation to estimate collar option under stochastic volatility model. The approximated price of the collar option under fast mean reverting stochastic volatility model is derived from the partial differential equation by singular perturbation technique. The importance sampling method based on the approximation price is used to reduce the variance of the Monte Carlo simulation. Numerical experiments are carried out under the context of different mean reverting rate. Numerical experiment results demonstrate that the importance sampling Monte Carlo simulation achieves better variance reduction efficiency than the basic Monte Carlo simulation.

Optimal Entry and Exit Strategy under Uncertainty with Stochastic Volatility

Motivated by previous papers with conventional models of Geometric Brownian Motion (Hereafter GBM) or Mean-Reverting (Hereafter MR), we discuss the classical investment timing problem in this paper by assuming the output price follows Heston-GBM process. That is, constant volatility in the classical GBM or MR framework is replaced by stochastic volatility in Heston-GBM model. We first derive the asymptotic solution for the investment timing problem. Then the impacts of stochastic volatility on trigger prices and the range of inaction are demonstrated by numerical simulation. Lastly, we examine the analytical properties of trigger prices and the range of inaction quantitatively as well as qualitatively.

Option pricing with bivariate risk-neutral density via copula and heteroscedastic model: A Bayesian approach

Multivariate options are adequate tools for multi-asset risk management. The pricing models derived from the pioneer Black and Scholes method under the multivariate case consider that the asset-object prices follow a Brownian geometric motion. However, the construction of such methods imposes some unrealistic constraints on the process of fair option calculation, such as constant volatility over the maturity time and linear correlation between the assets. Therefore, this paper aims to price and analyze the fair price behavior of the call-on-max (bivariate) option considering marginal heteroscedastic models with dependence structure modeled via copulas. Concerning inference, we adopt a Bayesian perspective and computationally intensive methods based on Monte Carlo simulations via Markov Chain (MCMC). A simulation study examines the bias, and the root mean squared errors of the posterior means for the parameters. Real stocks prices of Brazilian banks illustrate the approach. For the proposed method is verified the effects of strike and dependence structure on the fair price of the option. The results show that the prices obtained by our heteroscedastic model approach and copulas differ substantially from the prices obtained by the model derived from Black and Scholes. Empirical results are presented to argue the advantages of our strategy.

Banks’ Risk Dynamics and Distance to Default

Abstract We adapt structural models of default risk to take into account the special nature of bank assets. The usual assumption of lognormally distributed asset values is not appropriate for banks. Typical bank assets are risky debt claims with concave payoffs. Because of the payoff nonlinearity, bank asset volatility rises following negative shocks to borrower asset values. As a result, standard structural models with constant asset volatility can severely understate banks’ default risk in good times when asset values are high. Additionally, bank equity return volatility is much more sensitive to negative shocks to asset values than in standard structural models.

Time varying structural VARs with sign restrictions: The case of Taiwan

AbstractI apply a Bayesian approach to a time‐varying structural vector autoregression model with stochastic volatility (TVP‐SVAR‐SV) to study the time‐varying nature of the Taiwanese economy. In particular, the structural parameters are identified via the sign information in a three‐variable VAR system. The estimated results show that TVP‐SVAR‐SV model has the best fit to the data, compared to the time‐varying parameters VAR model with constant volatility and a classical VAR model with constant parameters and volatilities. Moreover, I find the time‐varying contemporaneous relationship between the output growth and inflation rates, particularly significant before the year 2000. Lastly, the impulse responses and the volatilities of all the variables are found to be time‐varying.

Extended Black and Scholes model under bankruptcy risk

In the Black-Scholes system, the underlying asset price model follows a geometric Brownian motion with constant volatility and no occurrence of bankruptcy. These two characteristics contradict real financial observations. In order to agree with the nonconstant feature of the volatility and take bankruptcy risk into consideration, we modify the Black and Scholes model and propose a new model based on the efficient market hypothesis. First, we present some probability properties of the bankruptcy risk by our model and demonstrate the statistical inference for the unknown parameters. We also study the European option pricing problem and propose its statistical computation method. In addition, via a real data analysis, we show that our model captures the trend of the stock prices much better than geometric Brownian motion and clarify that the bankruptcy is a crucial factor to be considered in the Black-Scholes system.

PRICING HOLDER-EXTENDABLE CALL OPTIONS WITH MEAN-REVERTING STOCHASTIC VOLATILITY

Options with extendable features have many applications in finance and these provide the motivation for this study. The pricing of extendable options when the underlying asset follows a geometric Brownian motion with constant volatility has appeared in the literature. In this paper, we consider holder-extendable call options when the underlying asset follows a mean-reverting stochastic volatility. The option price is expressed in integral forms which have known closed-form characteristic functions. We price these options using a fast Fourier transform, a finite difference method and Monte Carlo simulation, and we determine the efficiency and accuracy of the Fourier method in pricing holder-extendable call options for Heston parameters calibrated from the subprime crisis. We show that the fast Fourier transform reduces the computational time required to produce a range of holder-extendable call option prices by at least an order of magnitude. Numerical results also demonstrate that when the Heston correlation is negative, the Black–Scholes model under-prices in-the-money and over-prices out-of-the-money holder-extendable call options compared with the Heston model, which is analogous to the behaviour for vanilla calls.