In modelling the IVS one faces two main challenges. First, the data design is degenerated: due to trading conventions, observations of the IVS occur only for a small number of ma- turities such as 1, 2, 3, 6, 9, 12, 18, and 24 months to expiry on issuance. Consequently, implied volatilities appear in a row like pearls strung on a necklace, Figure 1, or, in short: as ‘strings’. This pattern is also visible in the right panel of Figure 1, which plots the data design as seen from top. Options belonging to the same string have a common time to maturity. As time passes, the strings move through the maturity axis towards expiry while changing levels and shape in a random fashion. Second, also in the moneyness dimension, the observation grid does not cover the desired estimation grid at any point in time with the same density. Thus, even when the data sets are huge, for a large number of cases impliedvolatility observations are missing for certain sub-regions of the desired estimation grid. This is particularly virulent when transaction based data is used.
A primary goal in modelling the impliedvolatility surface (IVS) for pricing and hedging aims at reducing complexity. For this purpose one fits the IVS each day and applies a principal component analysis using a functional norm. This approach, however, neglects the degenerated string structure of the impliedvolatility data and may result in a modelling bias. We propose a dynamic semiparametric factor model (DSFM), which approximates the IVS in a finite dimensional function space. The key feature is that we only fit in the local neighborhood of the design points. Our approach is a combination of methods from functional principal component analysis and backfitting techniques for additive models. The model is found to have an approximate 10% better performance than a sticky moneyness model. Finally, based on the DSFM, we devise a generalized vega-hedging strategy for exotic options that are priced in the local volatility framework. The generalized vega-hedging extends the usual approaches employed in the local volatility framework.
one-step approach, is simplicity: we view our method as a simple extension of what practitioners do already in practice and we show it works well. We nevertheless realize that further gains in forecasting the IVS could potentially be obtained with a Kalman Þlter approach. We leave this interesting extension for future research. Second, additional experiments could be useful in terms of specifying the most useful prediction models. For instance, both Harvey and Whaley (1992) and Noh, Engle and Kane (1994) Þnd that there are substantial days-of-the-week eﬀects in ATM impliedvolatility. It might be important to account for these kinds of eﬀects also when modeling the entire surface. Additionally, Noh, Engle and Kane (1994) show that there are considerable advantages in separately modeling the implied surface for call and put options. In this paper we have used data from both calls and puts, but we do not claim that this is an optimal choice. Finally, in our approach we estimate an unrestricted VAR model which does not exploit any non-arbitrage restrictions. Imposing such restrictions in our framework would entail writing a structural model for the IVS, which is beyond the scope of the present paper. We note however that imposing no-arbitrage conditions does not necessarily entail better forecasts. Indeed, our results suggest that our model (which does not exploit non-arbitrage conditions) outperforms Heston and Nandi’s (2000) model, which is arbitrage-free.
Since the price of an option is decisively determined by the market’s assessment of the volatility of the underlying asset over the remaining life of the option, market prices of options implicitly contain information about market participants’ volatility expectations. Consequently, given an option pricing model, these volatility expectations implied by option prices can be extracted. 1 Most conventionally, the Black-Scholes (1973) / Merton (1973) option pricing framework is applied to extract volatilities, and hence impliedvolatility is typically defined as the value of standard deviation of the underlying asset price process that produces the observed market price of an option when substituted into the Black-Scholes option pricing formula. Although the Black- Scholes model assumes constant volatility, this assumption is obviously not made by the market. 2 It is now well known that implied volatilities differ across times to maturity of option contracts, and thereby form a term structure of volatilities (see e.g., Heynen et al., 1994; Xu and Taylor, 1994; Campa and Chang, 1995). 3 Given that impliedvolatility may be regarded as the market expectation of future volatility, differences in implied volatilities across times to maturity should reflect differences in market participants’ perceptions of uncertainty over given future horizons.
the possibility of such an “impossible model.” There, I elaborated the idea that the regimes are variables with no names and no pre-determined level of “stochasticity,” and that freedom was therefore left completely to the procedure of cali- bration and recalibration, in other words, to the future, to determine the level at which the model would operate. When the option vanilla surface was found insufficient for determining the smile dynamics, calibration was attempted against path-dependent options, such as barriers and cli- quets, which are sensitive to the future smile. And when they, in turn, are found insufficient, calibration will be attempted against more com- plex structures still. Every instrument used in calibration is liable to be used in hedging. Needless to say, such an open model can only be tested against real market conditions. This was attempted by my co-workers Pedro Ferreira, Philippe Henrotte and Willy Lorange, in a study that they will soon publish separately. I will close the present article by showing the graphs (taken from their study) of the cumulative profit and loss of a dynamically hedged position spanning a period of one year, when the model is recalibrat- ed everyday to the market prices of various deriv- ative instruments, and the corresponding hedges executed at the corresponding market prices.
Actually, according to Xing, Zhang, and Zhao (2010), the volatility skew contains at least 3 levels of information. These levels are “the likelihood of a negative price jump, the expected magnitude of the price jump, and the premium that compensates investors for both the risk of a jump and the risk that the jump could be large”.
Another interesting extension of the paper would be the incorporation of independent dynamics for options of the same maturities but different strike prices. The problem here is that the no-bubbles restrictions still must be satisfied as maturity approaches. Thus the final value for the implied volatilities would be pre-determined (via the value of the spot volatility and equation (3.13)). Further research will have to show whether there is a sufficiently simple way to ensure the final condition while still allowing richer dynamics within the smile.
considered the case of models with stochastic volatility like the Heston model (see [10, 13, 14]) and the SABR model (see ), or the context of interest rates with the Hull-White model (see ). An alternative modeling is the use of Levy processes proposed for instance in . Recently, Jacquier and Roome  provided an expansion formula of the forward impliedvolatility using calculations based on the forward characteristic function and large deviations techniques. Such an enthusiasm for the stochastic volatility models or more generally for two or more factors models in the literature can be explained by the potential availability of closed formulas using the (semi) explicit computation of the forward characteristic function owing to the tower property for conditional expectations. In the class of models mentioned above, we start with a price process or a joint process price- volatility and deduce more and less the dynamic of the future volatility. To get a better control on the dynamic of the impliedvolatility, Bergomi modelises jointly the dynamics for the forward variance swap and the spot consistently and discuss about the calibration and pricing in [2, 3]. To the best of our knowledge the case of local volatility models is not handled in the literature and the purpose of this work is to provide an accurate and tractable analytical approximation of the forward impliedvolatility in time-dependent local volatility models. ⊲ Formulation of the problem and our contribution. In this work, we consider financial products in a world with no interest rate written w.r.t. a single asset which price at time t is denoted by S t paying no
This study develops a new KOSPI200 impliedvolatility index and examines its infor- mational content and nonlinear dynamics. The construction of this new benchmark for volatility expectations follows the methodology for calculating the new VIX index from S&P500 options. The empirical evidence suggests that the expected level of volatility in the Korean stock market has been steadily falling since the inception of option trading and the onset of the Asian financial crisis. Impliedvolatility is found to reflect useful in- formation on future volatility that is not contained in the history of returns, even after allowing for leverage effects. Markov regime-switching models suggest that nonlinearities in volatility expectations are not likely to be driven solely by the asymmetric impact of news but also by regime-dependencies in the realignment mechanism adjusting for fore- cast errors. The adjustment process is likely to be significant during regimes of lower volatility expectations but financial crises seem to elevate the level of anticipated volatil- ity and impair its adaptive dynamics.
Referee articles for: Econometrica, Journal of Finance, Journal of Financial Economics, Review of Financial Studies, Journal of Financial and Quantitative Analysis, Management Science, Journal of Business and Economics Statistics, Journal of Econometrics, Journal of Financial Econometrics, Econometric Theory, Journal of the American Statistical Association, Journal of Banking and Finance, Financial Analyst Journal, Review of Finance, Journal of Economic Dynamics and Control, Journal of Empirical Finance, European Journal of Finance, Journal of Derivatives, Financial Review, Energy Economics, American Journal of Agricultural
Among others, David and Veronesi (2002) and Guidolin and Timmerman (2003) have developed asset pricing models that explain theoretically why impliedvolatility may change in a predictable fashion. The main idea is that investors’ uncertainty about the economic fundamentals (e.g., dividends) affects impliedvolatility. This uncertainty evolves over time. In the case where it is persistent, the models induce predictable patterns in impliedvolatility. The empirical evidence on the predictability of impliedvolatility is mixed. Dumas et al. (1998) and Gonçalves and Guidolin (2006) have investigated whether the dynamics of the S&P 500 implied surface can be predicted over different time periods. The first study finds that the specifications under scrutiny are unstable over time for the purposes of option pricing and hedging. The second finds a statistically predictable pattern. However, for given trading strategies, this pattern cannot be exploited in an economically significant way since no abnormal profits can be obtained as transaction costs increase. There is also some literature that has explored whether the evolution of short-term at-the- money impliedvolatility, rather than the entire impliedvolatility surface, can be forecasted over time in various markets. Harvey and Whaley (1992), Guo (2000) and Brooks and
Regarding the predictability of IV itself, we run the CPA test for the squared forecast error. Table 4 reports the p-values for testing the null hypothesis of equal predictive ability between the row and column models.Rejection of the null hypothesis is indicated by the superscripts + and −. A positive (negative) sign indicates that the row (column) model is outperformed by the column (row) model. There are 9 cases (out of 18) in which we reject the null hypothesis that the random walk and the ARMA-type models perform equally well. In these cases, the + sign denotes the ARMA models performs better than the random walk. This suggests that there is a predictable pattern in the dynamics of impliedvolatility indices, which is in line with the results of Konstantinidi et al. (2008). When the predictive ability of the random walk is tested against the asymmetric ARMA models, the latter performs significantly better than the random walk at the 1% level. The importance of asymmetry found in-sample carries over to the out-of-sample analysis. When the model under consideration is an ARMA model that takes into account the contemporaneous asymmetric effect - ARMAX, ARIMAX, ARFIMAX models - always outperforms not only the random walk, but also the symmetric ARMA models. In these cases, the null hypothesis of equal predictive ability is always rejected at the 1% level.
Daouk and Guo (2004), Wagner and Szimayer (2004), and Dotsis et al. (2007) have studied the dynamics of impliedvolatility indices for the purposes of pricing impliedvolatility derivatives. However, the question whether the dynamics of impliedvolatility indices can be predicted has received little attention. To the best of our knowledge, Aboura (2003), Ahoniemi (2006), and Fernandes et al. (2007) are the only related studies. All three studies differ in the time period they consider, focus on a limited number of indices and forecasting models, and provide only point forecasts. They all find that the evolution of impliedvolatility indices is statistically predictable. Only the second paper examines the economic significance of the obtained forecasts and finds that performing a trading strategy with the S&P 500 options cannot attain abnormal profits. Our research approach is more general; a range of European and U.S. impliedvolatility indices is employed over a common time period, point and interval forecasts are formed by a number of alternative model specifications, and both their statistical and economic significance is assessed.
A drawback even of the most sophisticated models is the failure to correctly describe the dynamics of the IVS. This can be inferred from frequent recalibration of the model and has been best understood in the context of LV-models . Consequently, studying the IVS as an additional market fac- tor has become a vital stand of research. The main focus is on a low dimensional approximation of the IVS based on principal components analysis (PCA). The PCA is applied both to the term structure of the IVS ( or ) and strike dimension (eg. ). The common PCA for several matu- rity groups is studied in  and the functional PCA was dis- cussed in  and .
market have led to the development of a considerable literature on alternative option pricing models, in which the dynamics of the underlying asset is considered to be a nonlinear diffusion, a jump-diffusion process or a latent volatility model. These models attempt to explain the various empirical deviations from the Black–Scholes model by introducing additional degrees of freedom in the model such as a local volatility function, a stochastic diffusion coefficient, jump intensities, jump amplitudes etc. However, these additional parameters describe the infinitesimal stochastic evolution of the underlying asset while the market usually quotes options directly in terms of their market-implied volatilities which are global quantities. In order to see whether the model reproduces empirical observations, one has to relate these two representations: the infinitesimal description via a stochastic differential equation on one hand, and the market description via implied volatilities on the other hand.
We present an empirical investigation of long memory dynamics in the fac- tors of ImpliedVolatility Strings. The factor loadings series are obtained by applying a Dynamic Semiparametric Factor Model (DSF M ) for impliedvolatility strings on the German DAX index market. Long range depen- dence in the factor loadings series is tested using the rescaled variance V /S and the semiparametric LobRob tests. We estimated the degree of long mem- ory based on the log-periodogram GP H regression estimator and the GSP estimator based on the Whittle approximate maximum likelihood estimate. Results are indicative of long-range dependence in the factor loading series in levels and absolute returns. The factors can be interpreted in terms of a long term, middle long term and short term impact on the dynamics of IV S. The first factor loading, z1 is highly persistent and influences all op- tions similarly, irrespective of maturity. The impact of the second factor loading, z2 gradually diminishes for longer maturities and the third factor governs large volatility changes in relatively short maturities. Such depen- dence or persistence has importance implications for short-term trading and long range investment strategies. As a consequence, hedging strategies of a long position should take into consideration the long-memory effects in a short position in a call option. This would certainly provide more secure protection against negative effects of long-range persistence in volatility. On the other hand, better results could be obtained for models that price and hedge derivative securities when there is prior information on long-memory volatility in terms of expectation on the potential level of volatility and the rate at which volatility changes.
Market smirks are usually managed by using the local volatility model of Derman and Kani (1994), Dupire (1994) and Rubinstein (1994) in the financial industry. Ncube (1996) fits the impliedvolatility surface of FTSE 100 index options with parameter models. Du- mas, Fleming and Whaley (1998) examines the predictive and hedging performance of option-pricing models under deterministic (local) volatility by using S&P 500 options from June 1988 through December 1993. They find it is no better than an ad hoc procedure that merely smooths the impliedvolatility across strike prices and times to maturity. Ski- adopoulos, Hodges and Clewlow (1999) analyzes the dynamics of the impliedvolatility smile of S&P 500 futures options by applying principal components analysis. Pe˜na, Rubio and Serna (1999) studies the determinants of the impliedvolatility function, such as transaction costs, time to expiration, market uncertainty and relative market momentum. They use the Spanish IBEX-35 index options from January 1994 to April 1996. Hagan et al. (2002) finds that the dynamics of the market smile (or smirk) predicted by the local volatility model is opposite of the observed market behavior. They propose a stochastic volatility constant-
In existing literature, Cheng and Madhavan (2009) analyze the return dynamics of LETFs in a discrete-time model. They illustrate that, due to daily re-balancing, the LETF value depends on the realized volatility of the underlying index. This path-dependence can lead to value erosion over time. In a related study, Avellaneda and Zhang (2010) discuss the performance and potential tracking errors of LETFs and demonstrate the path-dependent characteristics under both discrete- time and continuous-time frameworks. In the context of constant proportion trading, Haugh (2011) derives similar price dynamics of LETFs under a jump-diffusion model and examines the tracking errors of LETFs. On the other hand, there is relatively little literature on pricing LETF options. In their theses, Russell (2009) analyzes LETF implied volatilities with the CEV and other models, and Zhang (2010) provides numerical results for pricing LETF options assuming the underlying follows the Heston model. However, they do not compare LETF option prices with different leverage ratios. Most recently, Ahn et al. (2012) propose an approximation to compute LETF option prices with Heston stochastic volatility and jumps for the underlying.
! Often, following an M&A deal we can see share prices stagnate following an initial jump, while the deal gets ratified by shareholders etc (i.e. low vol long term, following an initial high vol period). If an investor sold a long- dated call option or variance swap against the other position, they could lock in a higher implied vol and a higher option premium long dated, while still benefiting in the short-term from their long call option/variance swap.