Stochastic Volatility Option Pricing Research Articles

Empirical performance of stochastic volatility option pricing models

This paper examines the empirical performance of four stochastic volatility option pricing models: Heston, Heston[Formula: see text], Bates and Heston–Hull–White. To compare these models, we use individual stock options data from January 1996 to August 2014. The comparison is made with respect to pricing and hedging performance, implied volatility surface and risk-neutral return distribution characteristics, as well as performance across industries and time. We find that the Heston model outperforms the other models in terms of in-sample pricing, whereas Heston[Formula: see text] model outperforms the other models in terms of out-of-sample hedging. This suggests that taking jumps or stochastic interest rates into account does not improve the model performance after accounting for stochastic volatility. We also find that the model performance deteriorates during the crises as well as when the implied volatility surface is steep in the maturity or strike dimension.

Numerical aspects of integration in semi-closed option pricing formulas for stochastic volatility jump diffusion models

ABSTRACTIn mathematical finance, a process of calibrating stochastic volatility (SV) option pricing models to real market data involves a numerical calculation of integrals that depend on several model parameters. This optimization task consists of large number of integral evaluations with high precision and low computational time requirements. However, for some model parameters, many numerical quadrature algorithms fail to meet these requirements. We can observe an enormous increase in function evaluations, serious precision problems and a significant increase of computational time. In this paper, we numerically analyse these problems and show that they are especially caused by inaccurately evaluated integrands. We propose a fast regime switching algorithm that tells if it is sufficient to evaluate the integrand in standard double arithmetic or if a higher precision arithmetic has to be used. We compare and recommend numerical quadratures for typical SV models and different parameter values, especially for problematic cases.

Pricing of stochastic volatility stock index option based on Feynman path integral

<sec>Under the background that stock index options urgently need launching in China, the research on stock option pricing model has important theoretical and practical significance. In the traditional B-S-M model it is assumed that the volatility remains unchanged, which differs tremendously from the market’s reality. When the market fluctuates drastically, it is difficult to realize the risk management function of stock index options. Although in the Heston model, as one of the traditional stochastic volatility option pricing models, the correlation risk between the volatility and underlying price is taken into consideration, its pricing accuracy is still to be improved. From the quantum finance perspective, in this paper we use the Feynman path integral method to explore a more practical stock index option pricing model.</sec><sec>In this paper, we construct a Feynman path integral pricing model of stock index options with stochastic volatility by taking Hang Seng index option as the research object and Heston model as the control group. It is found that the Feynman path integral pricing model is significantly superior to the Heston model either at different strike prices on the same expiration date or at different expiration dates for the same strike price. The stock index option pricing model constructed in this paper can give the numerical solution of Feynman's pricing kernel, and directly realizes the forecast of stock index option price. The pricing accuracy is significantly improved compared with the pricing accuracy given by the Heston model through using the characteristic function method.</sec><sec>The remarkable advantages of Feynman path integral stock index option pricing model are as follows. Firstly, the path integral has advantages in solving multivariate problems: the Feynman pricing kernel represents all the information about pricing and can be easily expanded from one-dimensional to multidimensional case, so the change of closing price of stock index and volatility of underlying index can be taken into account simultaneously. Secondly, based on the relationship between the Feynman path generation principle and the law of large number, the mean values of pricing kernel obtained by MATLAB not only optimizes the calculation process, but also significantly improves the pricing accuracy. Feynman path integral is the main method in quantum finance, and the research in this paper will provide reference for its further application in the pricing of financial derivatives.</sec>

Are there gains from using information over the surface of implied volatilities?

We investigate the out‐of‐sample predictability of implied volatility using the information over the implied volatility surface. We show that implied volatility surface is useful for the out‐of‐sample forecast of implied volatility up to 1 week ahead. Trading strategies based on the predictability of implied volatility could generate significant risk‐adjusted gains after controlling for transaction costs. Significant results also depend on the way of modeling implied volatility surface. We then calibrate a two‐factor stochastic volatility option pricing model to implied volatility data. Results show that implied volatility is better explained by both long‐ and short‐term variance factors.

Critical Point on Stochastic Volatility Option Pricing

Critical Point on Stochastic Volatility Option Pricing

Herding and Stochastic Volatility

In this paper we develop a one-factor non-affine stochastic volatility option pricing model where the dynamics of the underlying is endogenously determined from micro-foundations. The interaction and herding of the agents trading the underlying asset induce an amplification of the volatility of the asset over the volatility of the fundamentals. Although the model is non-affine, a closed form option pricing formula can still be derived by using a Gauss-Hermite series expansion methodology. The model is calibrated using S&P 500 index options for the period 1996-2013. When its results are compared to some benchmark models we find that the new non-affine one-factor model outperforms the affine one-factor Heston model and it is competitive, especially out-of-sample, with the affine two-factor double Heston model.

Rare Shock, Two-Factor Stochastic Volatility and Currency Option Pricing

In this paper, we develop an option valuation model where the dynamics of the spot foreign exchange rate is governed by a two-factor Markov-modulated jump-diffusion process. The short-term fluctuation of stochastic volatility is driven by a Cox–Ingersoll–Ross (CIR) process and the long-term variation of stochastic volatility is driven by a continuous-time Markov chain which can be interpreted as economy states. Rare events are governed by a compound Poisson process with log-normal jump amplitude and stochastic jump intensity is modulated by a common continuous-time Markov chain. Since the market is incomplete under regime-switching assumptions, we determine a risk-neutral martingale measure via the Esscher transform and then give a pricing formula of currency options. Numerical results are presented for investigating the impact of the long-term volatility and the annual jump intensity on option prices.

Realizing smiles: Options pricing with realized volatility

We develop a discrete-time stochastic volatility option pricing model exploiting the information contained in the Realized Volatility (RV), which is used as a proxy of the unobservable log-return volatility. We model the RV dynamics by a simple and effective long-memory process, whose parameters can be easily estimated using historical data. Assuming an exponentially affine stochastic discount factor, we obtain a fully analytic change of measure. An empirical analysis of Standard and Poor's 500 index options illustrates that our model outperforms competing time-varying and stochastic volatility option pricing models.

Estimating and testing non‐affine option pricing models with a large unbalanced panel of options

Summary In this paper, we consider joint estimation of objective and risk-neutral parameters for stochastic volatility option pricing models using both stock and option prices. A common strategy simplifies the task by limiting the analysis to just one option per date. We first discuss its drawbacks on the basis of model interpretation, estimation results and pricing exercises. We then turn the attention to a more flexible approach, that successfully exploits the wealth of information contained in large heterogeneous panels of options, and we apply it to actual S&P 500 index and index call options data. Our approach breaks the stochastic singularity between contemporaneous option prices by assuming that every observation is affected by measurement error, essentially recasting the problem as a non-linear filtering one. The resulting likelihood function is evaluated using a Monte Carlo Importance Sampling (MC-IS) strategy, combined with a Particle Filter algorithm. The results provide useful intuitions on the directions that should be followed to extend the model, in particular by allowing jumps or regime switching in the volatility process.

The performance of popular stochastic volatility option pricing models during the subprime crisis

Using daily options prices on the Eurostoxx 50 stock index over the whole year 2008, we compare the performance of three popular stochastic volatility models (Heston, 1993; Bates, 1996; Heston and Nandi, 2000), in addition to the traditional Black-Scholes model and a proprietary trading desk model. We show that the most consistent in-sample and out-of-sample statistical performance is obtained for the internal model. However, the Bates model seems to be better suited to short term (out-of-the-money) options while the Heston model seems to perform better for medium or long term options. In terms of hedging performance, the Heston and Nandi model exhibits the best average, albeit most volatile, result and the Heston model outperforms the Black and Scholes model in terms of hedging errors, mainly for option contracts that mature in-the-money.

Price and Volatility Dynamics Implied by the VIX Term Structure

A particle-lter based estimation method is developed for the stochastic volatility model with/without jumps and applied on the S&P 500 index value and the VIX term structure jointly. The model encompasses all mean-reverting stochastic volatility option pricing models with a constant elasticity of variance, and can allow for price jumps. Our contention is that using the VIX term structure in estimation can help reach a more reliable conclusion on the nature of the risk-neutral volatility dynamic. Our empirical

Realizing Smiles: Options Pricing with Realized Volatility

We develop a discrete-time stochastic volatility option pricing model, which exploits the information contained in high-frequency data. The Realized Volatility (RV) is used as a proxy of the unobservable log-returns volatility. We model its dynamics by a simple but effective (pseudo) long memory process, the Heterogeneous Auto-Regressive Gamma with Leverage (HARGL) process. Both the discrete-time specification and the use of the RV allow us to easily estimate the model using observed historical data. Assuming a standard, exponentially affine stochastic discount factor, we obtain a fully analytic change of measure. An extensive empirical analysis of SP and (ii) the specification of our model, which, on the one hand, is able to accurately reproduce the volatility persistence and, on the other hand, provides the necessary smoothing of the noise present in the RV dynamics.

On the Role of Behavioral Finance in the Pricing of Financial Derivatives: The Case of the S&P 500

The object of this study was to investigate some implications of the tenets of behavioral finance on the pricing of financial derivatives. In particular, based on the work by Wolff, et al (2009) we have investigated how prospect theory (Kahneman and Tversky, 1979) can be intregrated into the Black and Scholes (1973) option pricing framework. We have then used the resulting “behavioral version” of the Black-Scholes equation to price market quoted options. As an empirical test we have calibrated three-month market-quoted call options on the Standard & Poor’s 500 index (SPX) at the Chicago Board of Options Exchange (CBOE) during the period January to December 2007. As a comparison, we have also calibrated the Heston (1993) stochastic volatility option pricing model to the same contracts. Our results show that during the period of study the market option prices are captured better by the behavioral version of the Black-Scholes equation than by the Heston stochastic volatility model. Further work is required to investigate if this is the case for other option types and under different market conditions.

Jump and volatility risk premiums implied by VIX

An estimation method is developed for extracting the latent stochastic volatility from VIX, a volatility index for the S&P 500 index return produced by the Chicago Board Options Exchange (CBOE) using the so-called model-free volatility construction. Our model specification encompasses all mean-reverting stochastic volatility option pricing models with a constant-elasticity of variance and those allowing for price jumps under stochastic volatility. Our approach is made possible by linking the latent volatility to the VIX index via a new theoretical relationship under the risk-neutral measure. Because option prices are not directly used in estimation, we can avoid the computational burden associated with option valuation for stochastic volatility/jump option pricing models. Our empirical findings are: (1) incorporating a jump risk factor is critically important; (2) the jump and volatility risks are priced; (3) the popular square-root stochastic volatility process is a poor model specification irrespective of allowing for price jumps or not. Our simulation study shows that statistical inference is reliable and not materially affected by the approximation used in the VIX index construction.

Regime switching stochastic volatility option pricing

Stochastic volatility option pricing has become popular in financial mathematics due to its theoretical and empirical consistencies. However, stochastic volatility models generally suffer from analytical and calibration intractability, except for regime switching stochastic volatility. However, regime switching models neither model volatility nor price options over short time periods accurately, hence they are of limited use. This paper proposes a general method of pricing regime switching options over short time periods. We achieve this by relating Fouque's perturbation based option pricing method to regime switching models. We conduct numerical experiments to validate our method using empirical S&P 500 index option prices and compare our results to Black-Scholes and Fouque's standard option pricing method. We demonstrate our pricing method which provides lower relative error compared to the Fouque's standard option pricing method and Black-Scholes pricing. In addition, this paper can be seen as an extension to Fouque's standard option pricing method.

The lead‐lag relationship between stock index options and the stock index market

PurposeThe purpose of this paper is to examine the price discovery role of the Korea Composite Stock Price Index 200 (KOSPI 200) stock index options market in contrast to other developed options markets.Design/methodology/approachThe price discovery roles of the stock and options markets using the error‐correction model derived from the co‐integration relationship are examined. Various analyses are conducted. First, Heston's stochastic volatility option pricing model is employed to confirm its usefulness, and compare the results with the Black and Scholes (BS) model. Second, whether the out of the money (OTM) options purchased by individual investors have a stronger price discovery role than options with other moneyness is examined. Finally, whether options have a stronger price discovery role in bullish or bearish markets than in normal markets is tested.FindingsIt is found that stock index prices lead implied index prices estimated from option prices using both BS and Heston models. In regards to the OTM options, the lead‐effect of real stock index to implied index prices holds. Also it is shown that there is a weak rise in the lead effect of the options to the stock index, but the lead effect of stock index market rules over that of the options market.Originality/valueThe paper examines the price discovery role of the KOSPI 200 stock index options market in contrast to other developed options markets and the results indicate that the consensus on the Korean financial markets may be incorrect.

Inferring Risk-Averse Probability Distributions from Option Prices Using Implied Binomial Trees: Additional Theory and Extensions

Arnold, Crack and Schwartz (ACS) (2010) generalize the Rubinstein (1994) risk-neutral implied binomial tree (R-IBT) model by introducing a risk premium. Their new risk-averse implied binomial tree model (RA-IBT) has both probabilistic and pricing applications. They use the RA-IBT model to estimate the pricing kernel (i.e., marginal rate of substitution) and implied relative risk aversion for a representative agent. They also use the RA-IBT to explore the differences between risk-neutral and risk-averse moments of returns. They also discuss practical applications of the RA-IBT model to Value at Risk and stochastic volatility option pricing models. This paper presents additional theoretical details not contained in ACS. We present a deeper discussion on the assumptions required for the risk-averse trees, we discuss details for extending ACS through the use of general utility functions to generate discount rates in the RA-IBT, and we present further theoretical details on the propagation of risk-averse probabilities through an RA-IBT. We also present an alternate CAPM-driven derivation of the certainty equivalent risk-adjusted discounting formula that is derived using no-arbitrage principles in ACS and an alternate direct estimation routine for the RA-IBT that is similar to Rubinstein’s “one-two-three” technique.

Implementing Risk-Averse Implied Binomial Trees: Additional Theory, Empirics, and Extensions

Arnold, Crack and Schwartz (2010) generalize the Rubinstein (1994) risk-neutral implied binomial tree (R-IBT) model by introducing a risk premium. Their new risk-averse implied binomial tree model (RA-IBT) has both probabilistic and pricing applications. They use the RA-IBT model to estimate the pricing kernel (i.e., marginal rate of substitution) and implied relative risk aversion for a representative agent. This paper presents additional theoretical details on the use of assumed utility functions to generate discount rates in the RA-IBT and theoretical details on the propagation of risk-averse probabilities through an RA-IBT (and how this process differs from the propagation of probabilities through a Rubinstein R-IBT). We also present both no-arbitrage and CAPM-driven derivations of the certainty equivalent risk-adjusted discounting formula that is used in Arnold, Crack and Schwartz (2010) and a direct estimation routine for the RA-IBT that is similar to Rubinstein’s “one-two-three” technique. This paper also presents additional empirical applications of the model, including a comparison of risk-neutral and risk-averse implied distributions, and applications of the RA-IBT to financial options trading, time series return forecasting, and a previously infeasible corporate finance real option valuation problem. We also use the RA-IBT to explore the differences between risk-neutral and risk-averse moments of returns. We also discuss practical applications of the RA-IBT model to Value at Risk and stochastic volatility option pricing models.

Using GMM to flatten the option volatility smile

By using an over-identified generalized method of moments (GMM) estimation procedure with careful consideration for data biases existing in the previous literature, parameters are estimated for a stochastic volatility jump diffusion option pricing (SVJ) model. The estimated parameters indicate a statistically significant highly negative infrequent jump process in the underlying security return distribution consistent with market crashes. When comparing to a stochastic volatility (SV) option pricing model, the SVJ is more robust but not always the superior model. The robustness of the models is further gauged by evaluating performance up to a year beyond the estimation data.

PRICING S&P 500 INDEX OPTIONS UNDER STOCHASTIC VOLATILITY WITH THE INDIRECT INFERENCE METHOD

This paper studies the price of SP in the second step, the risk premium, λ, the spot variance, vt, and the correlation coefficient between the asset return and its volatility, ρ, are estimated by a nonlinear least-squares method that minimizes the sum of the squares of the error between the cross-sectional option price and the corresponding model price. The model performance is assessed by directly comparing the computed option model price with the market price. We find that both the Black–Scholes model and the Heston model overprice the out-of-the-money options and underprice the in-the-money options, but the degree of the bias is different. The Heston model significantly outperforms the Black–Scholes model in almost all moneyness-maturity groups. On average, the Heston model can reduce pricing errors by about 25%. However, pricing bias still exists in the Heston model. In particular, the Heston model always overprices short-term options, indicating that some other factors, such as the random jump, may also be needed to explain the option price.