Third, the extension of the model to a multifactor setting has been demonstrated. In its multifactor versions (either with discrete or with continuous sets of implied volatilities) the model is capable of reproducing much richer dynamics than one-factor models. The stochasticimplied tree model by Derman and Kani  is the one that is closest in scope and philosophy to this model. Nevertheless, the reader will have realized by now that there are fundamental differences between both approaches, most importantly the no-bubbles restrictions (which are not in Derman and Kani) in this model and the market-based approach (as opposed to Derman and Kani’s ’effective volatility’ approach). Although the analysis in this model is based on European Call options, the methods presented can also be used with the implied volatilities of other options (e.g. options of American type) as underlying factors. Then, the partial derivatives of the options are needed to derive the no-arbitrage drift restrictions (like it was done in section 3.3), but qualitatively the model would not change.
The structural framework is based on the idea that a company defaults as soon as it decreases below a given barrier. Thus, we define the distance to default of each of the assets. We obtain a large marketmodel by passing to the limit with the multi- dimensional model. We derive a Stochastic Partial Differential Equation governing the behaviour of the limit density as a result. This enables us to define the loss of our portfolio of assets evolving through time as a functional of the limit density. The value of credit derivative based on a large basket of assets is a function of the loss. By varying the parameters of the dynamics, we observe which segments of the model are important for pricing and risk management of complex credit instruments such as Single Tranche Collateralized Debt Obligations (STCDOs) and Forward Start CDOs.
It turned that for these data sets the stochasticvolatility parameter r needed to be taken rather close to one, r ≡ 0.9. A qualitative impression of the calibration can be obtained from Figure 2. From the last down to the sixed tenor the relative average price calibration fit is about 5% for both data sets. For the short term tenors (up to the fifth) the calibration errors growth up to about 13%–25% unfortunately, and are therefore not reported. We found out however that the main reason for this bad fit for small maturities is the erratic behavior of the yield curve over this period at the calibration dates (see Table 3). For instance after replacing the actual yield curve with a smoothed one we also got a good fit for small maturities. The overall relative root-mean-square fit we have reached shows to be 0.5%–5%, when the caplet maturity ranges from 0.5 to 20.
The dynamics of all market variables can be expressed in terms of the same, vector–valued Brownian motion and correlation between market variables is obtained via the sum products of the respective vector–valued volatilities. As a consequence, the calibration across multiple sources of risk can be broken down into stages, simplifying the high–dimensional optimisation problems to be solved at each stage. The interest rate market can be calibrated separately using well established procedures. We chose to base our interest rate volatility calibration on the robust and widely used method of ?, but this could easily be replaced by a different method without impacting the remainder of our calibration approach. The model is fitted to an exogenously given correlation structure (typically estimated statistically from historical data) between forward interest rates and commodity prices (cross–correlation) — without impacting the interest rate volatility calibration already obtained — via a modified version of the “orthonormal Procrustes” problem in linear algebra, for which an efficient numerical solution exists.
Calibration of the Heston model is not covered in the literature in such extent that could be comparable to the coverage of the simulation schemes. We take the real data from the market and try to calibrate the model to them. This is not an easy task, espe- cially when one uses rather high number of options. We approached it by combining the local optimizers, which tend to be sensitive to an initial guess and thus we think one should avoid using only those, with the global optimizers. We suggested using the Ge- netic algorithm from the MATLAB’s Global Optimization Toolbox and we developed what we believe is a stable procedure combining the Genetic algorithm providing an initial guess, which is later refined by non-linear least squares method (lsqnonlin). Having the opportunity to run long computations on the departmental server 1 we were able to find benchmark values by creating a grid of the state space for the parameters and find out what the model is capable of. We managed to match the magnitude of the errors by our procedure, but of course in shorter computational time. We believe this is an approach that could be further used to compare for example SABR model to the Heston model. SABR model by Hagan, Kumar, Lesniewski and Woodward [ 13 ] is a newer model of the stochasticvolatility and is very popular these days. Our procedure could be used for comparison of these two while avoiding inadequate results because of the calibration procedure. We also used Adaptive Simulated Annealing and Excel’s Solver and compared the results for two consecutive days. Another conclusion is that slight adjustments to weights used during calibration can play significant role and one can improve the results when experimenting with them.
Stock prediction is something investors have been trying to achieve since the market first began. Many people believe its impossible to predict the market and while it may be true that it is impossible to predict the market all the time, it might still be possible to predict the shifts in stock value more than half the time. In investing if it's possible to correctly predict stock value even just sixty percent of the time, it's possible to make huge amounts of money by buying and selling shares. In order to achieve this goal investors have created complex algorithms and built costly super computers to analyze data at rapid speed in order to gain an edge on other investors and hedge funds. Even with the expensive equipment and complex equations that extend pages long, it seems as though no one has been able to make a failproof stock prediction model and if it does exist it's unlikely that it would be shared by its creator as it could provide immense
In Al` os, Le´ on and Vives (2007b), the volatility process σ is assumed to be a square-integrable stochastic process with right-continuous trajectories, bounded below by a positive constant and adapted to the filtration generated by W. Here we will assume the same hypothesis, but less restrictively, only that σ is adapted to the bigger filtration generated by W and Z. So, in this paper, we allow the volatility to have non-predictable jump times as advocated by Bakshi, Cao and Chen (1997) and Duffie, Pan and Singleton (2000), among others.
The impliedvolatility index VIX was created in order to estimate, looking forward, the expected realized volatility (RV). CBOE introduced the original VIX (now VXO) in 1986. It was based on an inverted Black-Scholes formula, where S&P 100 near-term, at-the-money options were used to calculate a weighted average of volatilities. However, the Black-Scholes formula assumes that the volatility in the stock returns equation is either a constant, or at least does not have a stochastic component, while in reality it was already understood that volatility itself is stochastic in nature. A number of well-studied models of stochasticvolatility have emerged, such as Heston (HM) (Dragulescu & Yakovenko, 2002; Heston, 1993) and multiplicative (MM) (Ma & Serota, 2014; Nelson, 1990). Consequently, a need arose for an impliedvolatility index, which would not only be based on stochasticvolatility but would also be agnostic to a particular model of the latter (Bollerslev, Mathew & Zhou, 2004; Zhou & Chesnes, 2003).
Furthermore, the SSA-HW combination allows a compromise between model parsimony and forecast accuracy. In brief, the principle of parsimony suggests that one must opt for the model with the smallest number of parameters (simplest model) such that an adequate representation of the actual data is provided (Chatfield, 1996). When combining forecasts, studies indicate that forecasting accuracy can only be improved if forecasts are combined from two adequate parsimonious forecasting models (McLeod, 1993). Parsimony also allows better predictions and generalizations of new data as it helps to distinguish the signal from the noise (Busemeyer et al., 2015). This is in addition to the preference for parsimony as an approach for avoiding over-parameterization when modelling data for forecasting (Booth and Tickle, 2008) and it is a recommended criterion for differentiating between forecasting models (Harvey, 1990). However, the best compromise between model parsimony and forecast accuracy is likely to consider whether the forecasts from the parsimonious model are significantly more accurate than a forecast from a competing model, provided the models in question are not affected by over or under fitting.
Financial literature handled this empirical evidence of not constant impliedvolatility with two broad classes of methods. The first could be labelled “deterministic volatility methods”; in general it refers to the use of a pricing model in which the parameter of constant volatility is replaced by a deterministic volatility function: different examples of this type of models are the approach of Shimko (1993), the implied binomial tree or lattice approach developed by Derman and Kani (1994) and Rubinstein (1994), the non-parametric kernel regression approach of Ait- Sahalia and Lo (1998). The second class of methods could be labelled “two factors models”; besides the risk of the market price of underlying asset, the valuation models price additional non-traded sources of risk, such as the volatility of volatility or market price jumps or even both. One of the first examples belonging to this general class was the stochasticvolatilitymodel of Hull and White (1987); more recent advances are, among others, the stochasticvolatilitymodel of Heston (1993), the random jump model of Bates (1996) and the multifactor model of Bates (2000).
Andersen and Bollerslev (1998) and Barndorff-Nielsen and Shephard (2001) propose using the sum of the squared intra-daily returns as a proxy measure for the corresponding daily volatility. This measure provides a consistent estimator of the latent volatility under an ideal market condition. A few pa- pers have utilized the realized volatility in the estimation of the SV models, such as Takahashi, Omori and Watanabe (2009), Xu and Li (2010) and etc. Furthermore, recently empirical research has indicated that the financial asset returns and volatilities exhibit asymmetric behavior in different regimes (e.g. bear/bull markets). Li and Lam (1995) have detected the significantly asymmetric movements of the conditional mean structure corresponding to the rise and fall of the previous-day market. In addition, Liu and Maheu (2008) have found strong empirical evidence of asymmetry in the volatility regimes. To accommodate these asymmetric effects, So, Li and Lam (2002) extend the standard SV model into a threshold framework, in which the la- tent volatility dynamic is determined by the sign of the lagged return. They also detected the significant asymmetric behavior in the variance persistence based on their sample data.
The Jacobi model is a highly tractable and versatile stochasticvolatilitymodel. It contains the Heston stochasticvolatilitymodel as a limit case. The moments of the finite dimensional distributions of the log prices can be calculated explicitly thanks to the polynomial property of the model. As a result, the series approximation techniques based on the Gram–Charlier A expansions of the joint distributions of finite sequences of log returns allow us to efficiently compute prices of options whose payoff depends on the underlying asset price at finitely many time points. Compared to the Heston model, the Jacobi model offers additional flexibility to fit a large range of Black–Scholes impliedvolatility surfaces. Our numerical analysis shows that the series approximations of European call, put and digital option prices in the Jacobi model are computationally comparable to the widely used Fourier transform techniques for option pricing in the Heston model. The truncated series of prices, whose computations do not require any numerical integration, can be implemented efficiently and reliably up to orders that guarantee accurate approximations as shown by our numerical analysis. The pricing of forward start options, which does not involve any numerical integration, is significantly simpler and faster than the iterative numerical integration method used in the Heston model. The minimal and maximal volatility parameters are universal bounds for Black–Scholes implied volatilities and provide additional stability to the model. In particular, Black–Scholes implied volatilities of forward start options in the Jacobi model do not experience the explosions observed in the Heston model. Furthermore, our density approximation technique in the Jacobi model circumvents some limitations of the Fourier transform techniques in affine models and allows us to price discretely monitored Asian options.
we are moving from ITM to OTM call options and from OTM to ITM put options. Starting with the call options, we can note that the model displays some degree of skewness reflecting the fact that, due to mean reversion, prices do not fluctuate freely but are pulled back toward their mean. In the base and high cases, where the risk-neutral mean is equal and above the current spot price respectively, mean reversion is good for ITM call options since it pulls prices above the strike price and thus increases the likelihood that the options will end ITM; same is also true for the OTM options, however, since these options have lower probability of ending ITM, the impact of mean reversion is less beneficial compared to ITM options. Overall, this results in the skew evidenced in Figure 1. Turning next to the case where the equilibrium level is below the current spot price (i.e. the low case), we can see that volatility increases as strike price increases. This suggests that prices are now pulled toward lower levels due to mean reversion, and thus ITM calls are cheaper compared to calls calculated using the BSM model. OTM options on the other hand, are less affected by this since they have already a low probability of ending ITM, as the previous results showed. A similar pattern is observed in the case of put options, which are shown in Figure 2, where the impliedmodel volatilities display exactly the opposite pattern from that of call options 5 .
paper, which is called the volatility of the under- lying share and written in a specific context for the sake of pricing a specific derivative! Implying the value of volatility from the options market might be, as a matter of fact, the last thing to do! One usually looks for a firm and stable empirical reality against which to perform a measurement and solve a given inverse problem. For instance, one selects a solid material and exerts tension upon it in order to infer Young’s modulus. As changing and fleeting and revisable and unwar- ranted as elasticity theory may be to the empiri- cally minded, at least the material we are leaning against to invert our equation will not change! If option prices were listed in some immutable reg- ister, then it might be a good idea indeed to infer volatility from that unexpected source of infor- mation. But we are talking here of inferring volatility from the option market! Instead of clos- ing the mystery of the “first” market (that of the underlying share) and the mystery of its volatility, we are now introducing a much bigger mystery right into the heart of our problem and opening a much wider meaning for the word “market” than was initially intended by our model.
By comparison to this paper there are several authors who considered the leverage eff ect in commodities market, including Kristoufek (2012). He investigates the relationship between the returns and volatility of energy commodities futures, including Brent, WTI crude oil and natural gas. Li et al. (2016) suggest signifi cant inverse leverage eff ects for the corn commodity using GARCH regime‑dependent models. The paper of Assa et al. (2016) deals with estimating the parameters of the CEV process, which are focused on estimating the leverage eff ect. He uses the maximum likelihood estimation for calibrating the CEV model. The Linetsky and Mendoza (2009) paper surveys the application of the CEV model in the fi elds of credit derivatives and bancruptcy. Chen (2015) shows that the CEV model exceeds other stochastic models in the case of derivatives. Carr (2011) proposes the application of the CEV stochasticvolatilitymodel in the fi eld of fi nancial leverage in the equity index.
The bivariate Ito’s lemma is used to derive the fundamental partial differential equations. The derivation fol- lows the no-arbitrage argument, similar to that of the BS model. However, as opposed to the BS model, two de- rivative assets are required here to make the resulting portfolio risk-free, as there are two sources of randomness. The two derivative assets are on the same underlying, but differ in strike price and maturity. Because of the second Brownian motion, it is not possible to find a closed-form solution for the European options. Here, the SVH model has an advantage over other models in its category that it has a semi-closed form (or quasi-closed- form) solution available for the plain vanilla European options. This, in turn, makes it feasible to calibrate the model to market prices.
the input prices of the set of digital caplets which will be fed into the model. The separation of a driver and marginals provides more flexibility for a Markov-functional model. But this may also cause an issue. A mismatch of a driving process and marginals could potentially lead to nontransparent dynamics of forward LIBORs and could result in an unstable evolution of the impliedvolatility surface. This potential issue has been pointed out by Andersen and Piterbarg  who argue that a non-parametric formulation of the marginal distribution for LI- BORs may result in unrealistic evolution of the volatility smile through time. On the other hand, Bennett and Kennedy  showed that a LIBOR Markov-functional model with a Gaussian driver together with the Black’s formula for (digital) caplets is numerically similar to the one-factor separable LIBOR marketmodel. Gogala and Kennedy  extended the above link to a more general local-volatility case. Based on this link, the authors propose an approach for choosing an appropriate combination of a driving process and (digital) caplet prices, and such a combination leads to desirable dynamics of future implied volatilities. In Chapter 4 we con- sider a separable SABR-LIBOR marketmodel and expect that it is similar to the stochasticvolatility Markov-functional model with a SABR driver together with a SABR marginals. Based on this link the intuition behind the SDEs of the separable SABR-LIBOR marketmodel can be applied to the corresponding stochastic volatil- ity LIBOR Markov-functional model. This gives us an appropriate combination of the driver and marginals. Based on this link we develop a calibration routine to feed in SABR marginals by calibrating to market prices of caplets or swaptions. Moreover the parameters for the SABR driving process can be chosen by calibrat- ing to the marketimplied correlations. A numerical investigation of the calibration performance is also given.
eyness categories and reports the number of contracts, the average option price, the average Black- Scholes impliedvolatility (IV), and the average bid-ask spread in dollars. Moneyness is defined as the implied index futures price, F , divided by the option strike price X. The implied-volatility row shows that deep out-of-the-money puts, those with F/X > 1.06, are relatively expensive. The implied-volatility for those options is 25.73%, compared with 19.50% for at-the-money options. The data thus display the well-known smirk pattern across moneyness. The middle panel sorts the data by maturity reported in calendar days. The IV row shows that the term structure of volatility is roughly flat, on average, during the period, ranging from 20.69% to 21.87%. The bottom panel sorts the data by the volatility index (VIX) level. Obviously, option prices and IVs are increasing in VIX, and dollar spreads are increasing in VIX as well. More importantly, most of our data are from days with VIX levels between 15% and 35%.
The empirical literature about the forecasting performance of corridor impliedvolatility is very little and mixed: most studies are based on closing prices and investigate only symmetric corridor measures. Andersen and Bondarenko (2007), by using options on the S&P500 futures market, find that narrow corridor measures, closely related to Black Scholes impliedvolatility are more useful for volatility forecasting than broad corridor measures, which tend to model-free impliedvolatility as the corridor widens. A similar finding is obtained in Muzzioli (2010b) who finds that the best forecast for the Italian index options market is the one which operates a 50% cut of the risk neutral distribution. On the other hand, Tsiaras (2009), by using options on the 30 components of the DJIA index, concludes that CIV forecasts are increasingly better as long as the corridor width enlarges. To sum up, the empirical evidence suggests that there is no golden choice for the barriers levels’, which will probably change depending on the underlying asset risk neutral distribution. The latter feature renders the forecasting performance of corridor impliedvolatility mainly an empirical question.
The present investigation is very important for the understanding of the role of the different ingredients of the smile function and can be seen as a preliminary exercise in order to choose different weights for each volatility input in a volatility index. The VDAX-New, the new volatility index of the German equity market, is based on an approximation of the so-called “model free” impliedvolatility, proposed by Britten-Jones and Neuberger (2000), and is derived by using the most liquid at the money and out of the money call and put options. The VDAX- New has replaced the old VDAX, that was computed by using only at the money options (pairs of calls and puts with the four strikes below and above the at the money point). The present investigation suggests some directions in order to improve the information content of the VDAX-New: overall put options are more informative than call options, ATMP are preferred to ATMC, OTMP predict future realised volatility better than both ATMC and OTMC. How these rules can be embedded in the index and the empirical comparison between the suggested modifications and the existent VDAX-New is left for future research.