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 **stochastic** **implied** tree **model** by Derman and Kani [6] 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.

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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 **market** **model** 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.

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It turned that for these data sets the **stochastic** **volatility** 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.

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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.

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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 **stochastic** **volatility** 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.

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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

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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.

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The **implied** **volatility** 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 **stochastic** **volatility** 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 **implied** **volatility** index, which would not only be based on **stochastic** **volatility** but would also be agnostic to a particular **model** of the latter (Bollerslev, Mathew & Zhou, 2004; Zhou & Chesnes, 2003).

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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.

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Financial literature handled this empirical evidence of not constant **implied** **volatility** 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 **stochastic** **volatility** **model** of Hull and White (1987); more recent advances are, among others, the **stochastic** **volatility** **model** of Heston (1993), the random jump **model** of Bates (1996) and the multifactor **model** of Bates (2000).

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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.

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The Jacobi **model** is a highly tractable and versatile **stochastic** **volatility** **model**. It contains the Heston **stochastic** **volatility** **model** 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 **implied** **volatility** 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.

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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 **implied** **model** volatilities display exactly the opposite pattern from that of call options 5 .

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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**.

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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 **stochastic** **volatility** **model** in the fi eld of fi nancial leverage in the equity index.

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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.

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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 **implied** **volatility** surface. This potential issue has been pointed out by Andersen and Piterbarg [5] 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 [7] 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 **market** **model**. Gogala and Kennedy [29] 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 **market** **model** and expect that it is similar to the **stochastic** **volatility** 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 **market** **model** 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 **market** **implied** correlations. A numerical investigation of the calibration performance is also given.

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eyness categories and reports the number of contracts, the average option price, the average Black- Scholes **implied** **volatility** (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%.

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The empirical literature about the forecasting performance of corridor **implied** **volatility** 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 **implied** **volatility** are more useful for **volatility** forecasting than broad corridor measures, which tend to **model**-free **implied** **volatility** 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 **implied** **volatility** mainly an empirical question.

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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” **implied** **volatility**, 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.

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