these MCRRs have been exceeded by price movements in day t + 1. We roll this process forward and we calculate the MCRRs for 252 days. 2 In Table 18, we present the number of violations of the MCRR estimates generated by the models. We observe, relatively to the GARCH model, that the number of violations (in percentage) never exceeds the 5% nominal value for all series, which may indicate that the model generates ”slight” excessive MCRRs, but the inclusion of realisedvolatility seems, in all cases, to improve the performance of the model. Finally, the best performances are registered by the asymmetric models: the GJR model for the Disney returns, the EGARCH for the American Express and the EGARCH-RV for the Coca-Cola returns, with failure rates closer to the nominal 5% level.
Forecasting daily returns volatility is crucial in finance. Tra- ditionally, volatility is modelled using a time-series of lagged information only, an approach which is in essence atheoretical. Although the relationship of market conditions and volatility has been studied for decades, we still lack a clear theoretical framework to allow us to forecast volatility, despite having many plausible explanatory variables. This setting of a data- rich but theory-poor environment suggests a useful role for powerful model induction methodologies such as Genetic Programming. This study forecasts one-day ahead realisedvolatility (RV) using a GP methodology that incorporates information on market conditions including trading volume, number of transactions, bid-ask spread, average trading dura- tion and implied volatility. The forecasting result from GP is significantly better than that produced by the heterogeneous autoregressive model (HAR), the benchmark model from the traditional finance literature. The error measures and R 2
The relationship has been an important research topic and many studies have been devoted to this topic. The first study is done by Latane and Rendleman (1976). They use closing prices of options and stocks for 24 companies whose options are traded on the Chicago Board Options Exchange (CBOE) and conclude that implied volatility outperforms historical volatility in forecasting future realized volatility. After that, Chiras and Manaster (1978) and Beckers (1981) also obtain the same conclusion based on a broader sample of CBOE stock options. It should be noted that these studies concentrate on static cross-sectional rather than time-series forecasts.
Consider now some statistical/econometric issues around the pricing of op- tions when volatility is stochastic and has short memory. Here the modeling commences with separate diffusion specifications for the asset prices and its volatility such that option prices then depend on several parameters including a volatility risk premium and the correlation between the differentials of the Wiener processes in the separate diffusions. Duan (1995) provides an explicit option valuation framework for a GARCH (1,1) process with short memory. Comte and Renault (1998) and Bollerslev and Mikkelson (1996, 1999) describe methods to price options when volatility has a long memory. Here, they re- place the Wiener process in the volatility equation with fractional Brownian motion. However, their option pricing formula requires independence between the Wiener process in the price equation and the volatility process which is inconsistent with the empirical evidence on stock returns. Furthermore, ap- plication of the theory of option pricing when volatility follows a discrete-time ARCH process with fractional integration requires approximation because the fundamental filter is an infinite order polynomial that must be truncated at some power N. Option prices will therefore be sensitive to the truncation point and large values and long price histories from an assumed stationary process are required.
Our main results can be summarised as follows. Consistent with Sand˚ as’s (2001) results, our formal tests also reject the structural model. The informativeness parameter, Glosten’s α, turns insignificant when based on the updating conditions for an equilibrium LOB. One interpretation is that book replenishment from transaction to transaction is noisy and potentially incomplete. This motivates our alternative approach that does not rely on a parametrised model, and uses the long-term price impact of trades to measure their informativeness. This approach finds empirical support for the main prediction of static LOB models that informativeness matters for book liquidity, which we measure through price concessions of market orders of different sizes. We find that large order price concessions are most sensitive to trade informativeness, as opposed to any of the control variables employed in the multivariate analysis, e.g. realisedvolatility. However, we also find that the bid-ask spread and the average-size order price concessions respond stronger to our proxy of market volatility, i.e. picking-off risk that is unrelated to private information.
Recently, the word subsampling has been used in connection with the estimation of quadratic variation of a latent price process subject to market microstructure noise, see Zhang, Mykland, and Aït-Sahalia (2005) and Barndor¤-Nielsen and Shephard (2007). The subsampling scheme in this setting is slightly di¤erent from the usual one and is perhaps better called ‘In…ll Price’subsampling. Zhang, Mykland, and Aït-Sahalia (2005) use this ‘In…ll Price’subsampling to de…ne a bias correction method that achieves consistent estimation. For these type of subsamples, equation (1) holds, so we explore the possibility of their use for inference. We show that ‘In…ll Price’ subsampling does not deliver consistent inference for the realisedvolatility, due to high correlation between subsamples. We recover consistency by modifying the ‘In…ll Price’ subsampling in ‘In…ll Returns’ subsampling. However, this method requires restrictive assumptions on the volatility path to achieve consistency. Finally, we propose ‘Subset Centered In…ll’subsampling, which does achieve the required consis- tency without any smoothness assumptions on the volatility path. It builds on the basic principles of Politis and Romano (1994), but uses a di¤erent centering for each subsample.
SV models can be estimated by quasi maximum likelihood methods but the main emphasis will be on methods for exact maximum likelihood using Monte Carlo importance sampling methods. The performance of the models is evaluated, both within sample and out-of-sample, for daily returns on the Standard & Poor's 100 index. Similar studies have been undertaken with GARCH models where ndings were initially mixed but recent research has indicated that implied volatility provides superior forecasts. We nd that implied volatility outperforms historical returns in-sample but that the latter contains incremental information in the form of stochastic shocks incorporated in the SVX models. The out-of-sample volatility forecasts are evaluated against daily squared returns and intradaily squared returns for forecasting horizons ranging from 1 to 10 days. For the daily squared returns we obtain mixed results, but when we use intradaily squared returns as a measure of realisedvolatility we nd that the SVX + model produces the most accurate
In active markets characterised by a signiﬁcant increase in the proportion of utilitarian traders, liquidity appears to deteriorate, with larger transaction costs paid by participating banks against the background of lower total volume. Spreads also appear to increase when realisedvolatility is higher. We conjecture that, con- sistent with the literature, information asymmetry increases at the turn of the period, which results in wider spreads posted by market participants. Figure 6 ex- hibits the evolution of our liquidity measures over our stylised RMP. Quoted (i.e., EWQS and EWPQS) and eﬀective (i.e., EWES, EWPES, SWES and SWPES) spreads appear to increase towards the end of the RMP. Although those spreads exhibit little variation until the last MRO allotment of the period, transacting on the last days of the period eventually becomes more costly, with the last open day of the period being the most expensive day. Apart from a greater stability of ef- fective spreads during the non-last days, no signiﬁcant change is observed after 10 March 2004 (Figure 7). This analysis extends to traded spreads. However, after the implementation of the changes to the operational framework, traded spreads for buy and sell orders evolve in opposite directions on non-last days, before start- ing a common increase over the last days of the RMP. Over non-last days, traded spreads for buy orders broadly follow an inverted U-shaped path, while traded spreads for sell orders exhibit a regular U-shaped pattern. This picture is not aﬀected by the frequency of FTOs.
The Black-Scholes (BS) formula is widely used by traders because it is easy to use and understand. An important characteristicof the model is the assumption that the volatility of the underlying security is constant. However, practitioners have observed, especially after the crash of 1987, the so called volatility \smile" eect. Namely, options written on the same underlying asset usually trade, in Black-Scholes term, with dierent implied volatilities 3 . Deep-in-the-money
Columbia2000 pdf 1 Andrew Matytsin New York, 29 January 2000 (212) 648 0820 Perturbative Analysis of Volatility Smiles 2 This report represents only the personal opinions of the author and not those o[.]
In the past dozen years ARCH-models have become popular for modelling nancial time series since they are able to account for several empirical features like volatility clustering and leptokurtosis (fat tails) in the distribution of returns. While they dier substantially in their detailed expression, most ARCH-models involve a sequence of uncorrelated innovations whose
princeton dvi ?? Conclusions ? UVM ? New method for quantifying volatility risk in derivative market?making ? Extremal prices are computed using the non?linear BSB equation ? Non?linear PDE quanti?es[.]
SpSV ps Saddlepoint Approximations for Stochastic Volatility Models Michael Studer Dept Mathematik, ETH Z?urich, 8092 Z?urich, CH E mail studerm@math ethz ch It is well known that the saddlepoint appr[.]
This examples involves a larger set of input options. We consider the Telmex (TMX) Advanced Depository Receipts and options on this security traded in the NYSE in the months following the December 1994 Peso devaluation. During this period, the market exhibited very large implied volatilities in comparison with 1994 levels. Variations in im- plied volatilities on a given trading date were also large, according to both strike levels and maturities. Overall, the volatility term-structure was inverted, decreasing as the maturity increased. A complex volatility structure such as this one is well-suited for applying the optimality algorithm.
In many markets prior to the 1987 stock market crash, there appeared to be a symmetry around the zero moneyness, where out-of-the-money and in-the-money options traded at higher implied volatilities than the implied volatilities for at-the- money options. This dependence of implied volatility on the strike, for a given maturity became known as the smile effect, although the exact structure of volatility varied across markets and even within a particular market from day to day. However, since the 1987 stock market crash the smile has changed shape in many markets, particularly for traded options on stock indexes, where the function has gone from a smile shape to more of a ‘sneer’. The idea of the volatility ‘smile’ had its genesis in the early papers documenting the systematic pricing biases of the Black and Scholes (1973) option pricing model. Black (1975) suggests that the non-stationary behaviour of volatility would lead the Black-Scholes model to overprice or underprice options. Other authors have confirmed the existence of systematic biases in the model. 6
a 1-factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the local volatility function from a finite set of observation data. Assuming that the underlying indeed follows a 1-factor model, it is emphasized that accurately approximating the local volatility function prescribing the 1- factor model is crucial in hedging even simple European options, and pricing exotic options. A spline functional approach is used: the local volatility function is represented by a spline whose values at chosen knots are de- termined by solving a constrained nonlinear optimization problem. The optimization formulation is amenable to various option evaluation methods; a partial differential equation implementation is discussed. Using a synthetic European call option example, we illustrate the capability of the proposed method in reconstructing the unknown local volatility function. Accuracy of pricing and hedging is also illustrated. Moreover, it is demonstrated that, using different implied volatilities for options with different strikes/maturities can produce erroneous hedge fac- tors if the underlying follows a 1 -factor model. In addition, real market European call option data on the S&P 500 stock index is used to compute the local volatility function; stability of the approach is demonstrated.
On the Martingale Problem for Jumping Di?usions M A H Dempster Centre for Financial Research Judge Institute of Management Studies University of Cambridge, Cambridge, England mahd2@cam ac uk www cfr j[.]
Over the last 20 years the classical Black-Scholes model has proven to be a very eective tool for the valuation and the risk-management of derivative securities, and even today most of the trading activity on markets for equity and currency options is based on this model. Nonetheless, in recent years a number of empirical observations have been compiled that are dicult to reconcile both with the assumptions the model imposes on the price process of the underlying asset and with the predictions the model makes on the behaviour of option prices. To mention only a few of these issues that currently mark many debates in derivative asset analysis, most time series of asset returns are said to exhibit \excess kurtosis" and \fat tails" and on options markets we encounter \smile" or \skew" patterns of implied volatility.
A wide range of models that deal with systematic volatility have been developed since the seminal proposed by Engle (1982) 11 . Since then, the vast majority of volatility work has often focused on series that where the trajectory of the series cannot be predicted from its past. Financial and stock prices behave in this way. Simply focusing on the variability of the differenced series is sufficient in this case. However, for many other series (such as agricultural prices) this may not really be appropriate, as there is evidence that these series are cyclical, sometimes with, or without, trends that require modelling within a flexible and unified framework. Deaton and Laroque (1992), citing earlier papers, note that many commodity prices also behave in a manner that is similar to stock prices (the so called random walk model). However, they also present evidence that is inconsistent with this hypothesis. They note that within the random walk model, all shocks are permanent, and that this is implausible with regard to agricultural commodities (i.e. weather shocks would generally be considered transitory). In view of the mixed evidence about the behaviour of agricultural prices, we would emphasis the importance of adopting a framework that can allow the series to have either trends or cycles or a combination of both. Importantly, there may be alterations in the variances that drive both these components. Therefore, the approach adopted within this report allows for