Quadratic hedging in affine stochastic volatility models

We determine the variance-optimal hedge for a subset of affine processes including a number of popular stochastic volatility models. This framework does not require the asset to be a martingale. We obtain semiexplicit formulas for the optimal hedging strategy and the minimal hedging error by applying general structural results and Laplace transform techniques. The approach is illustrated numerically for a Lévy-driven stochastic volatility model with jumps as in Carr et al. (Math Finance 13:345–382, 2003).      

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.     

Pricing options on discrete realized variance with partially exact and bounded approximations

We derive efficient and accurate analytic approximation formulas for pricing options on discrete realized variance (DRV) under affine stochastic volatility models with jumps using the partially exact and bounded (PEB) approximations. The PEB method is an enhanced extension of the conditioning variable approach commonly used in deriving analytic approximation formulas for pricing discrete Asian style options. By adopting either the conditional normal or gamma distribution approximation based on some asymptotic behaviour of the DRV of the underlying asset price process, we manage to obtain PEB approximation formulas that achieve a high level of numerical accuracy in option values even for short-maturity options on DRV.     

Geometric Asian option pricing in general affine stochastic volatility models with jumps

In this paper, we present some results on Geometric Asian option valuation for affine stochastic volatility models with jumps. We shall provide a general framework into which several different valuation problems based on some average process can be cast, and we shall obtain closed form solutions for some relevant affine model classes.     

Attainability of European path-independent claims in incomplete markets

In this paper we consider the question which path-independent claims are attainable through self-financing trading strategies in an incomplete market. For continuous-time stochastic volatility models we show that only affine payoffs can be replicated. We provide a simple proof for this proposition based on the requirement that, for replication, the stock and the claim must be locally perfectly correlated, and based on the partial differential equation that any path-independent claim has to satisfy. Moreover, we show that this result does not carry over to discrete setups.     

Pricing options under stochastic volatility: a power series approach

In this paper we present a new approach for solving the pricing equations (PDEs) of European call options for very general stochastic volatility models, including the Stein and Stein, the Hull and White, and the Heston models as particular cases. The main idea is to express the price in terms of a power series of the correlation parameter between the processes driving the dynamics of the price and of the volatility. The expansion is done around correlation zero and each term is identified via a probabilistic expression. It is shown that the power series converges with positive radius under some regularity conditions. Besides, we propose (as in Alós in Finance Stoch. 10:353–365, 2006) a further approximation to make the terms of the series easily computable and we estimate the error we commit. Finally we apply our methodology to some well-known financial models.      

Real-world jump-diffusion term structure models

This paper considers interest rate term structure models in a market attracting both continuous and discrete types of uncertainty. The event-driven noise is modelled by a Poisson random measure. Using as numeraire the growth optimal portfolio, interest rate derivatives are priced under the real-world probability measure. In particular, the real-world dynamics of the forward rates are derived and, for specific volatility structures, finite-dimensional Markovian representations are obtained. Furthermore, allowing for a stochastic short rate in a non-Markovian setting, a class of tractable affine term structures is derived where an equivalent risk-neutral probability measure may not exist.     

Implied volatilities,stochastic interest rates,and currency futures options valuation: an empirical investigation

Different models of pricing currency call and put options on futures are empirically tested. Option prices are determined using different models and compared to actual market prices. Option prices are determined using historical as well as implied volatility. The different models tested include both constant and stochastic interest rate models. To determine if the model prices are different from the market prices, regression analysis and paired t-tests are performed. To see which model misprices the least, root mean square errors are determined. It is found that better results are obtained when implied volatility is used. Stochastic interest rate models perform better than constant interest rate models.     

Can interest rate volatility be extracted from the cross section of bond yields?

Most affine models of the term structure with stochastic volatility predict that the variance of the short rate should play a ‘dual role’ in that it should also equal a linear combination of yields. However, we find that estimation of a standard affine three-factor model results in a variance state variable that, while instrumental in explaining the shape of the yield curve, is essentially unrelated to GARCH estimates of the quadratic variation of the spot rate process or to implied variances from options. We then investigate four-factor affine models. Of the models tested, only the model that exhibits ‘unspanned stochastic volatility’ (USV) generates both realistic short rate volatility estimates and a good cross-sectional fit. Our findings suggest that short rate volatility cannot be extracted from the cross-section of bond prices. In particular, short rate volatility and convexity are only weakly correlated.     

Statistical properties of demand fluctuation in the financial market

Numerical integration methods for stochastic volatility models in financial markets are discussed. We concentrate on two classes of stochastic volatility models where the volatility is either directly given by a mean-reverting CEV process or as a transformed Ornstein–Uhlenbeck process. For the latter, we introduce a new model based on a simple hyperbolic transformation. Various numerical methods for integrating mean-reverting CEV processes are analysed and compared with respect to positivity preservation and efficiency. Moreover, we develop a simple and robust integration scheme for the two-dimensional system using the strong convergence behaviour as an indicator for the approximation quality. This method, which we refer to as the IJK (137) scheme, is applicable to all types of stochastic volatility models and can be employed as a drop-in replacement for the standard log-Euler procedure.     

Maximum likelihood estimation of stochastic volatility models

We develop and implement a method for maximum likelihood estimation in closed-form of stochastic volatility models. Using Monte Carlo simulations, we compare a full likelihood procedure, where an option price is inverted into the unobservable volatility state, to an approximate likelihood procedure where the volatility state is replaced by proxies based on the implied volatility of a short-dated at-the-money option. The approximation results in a small loss of accuracy relative to the standard errors due to sampling noise. We apply this method to market prices of index options for several stochastic volatility models, and compare the characteristics of the estimated models. The evidence for a general CEV model, which nests both the affine Heston model and a GARCH model, suggests that the elasticity of variance of volatility lies between that assumed by the two nested models.     

Options on realized variance by transform methods: a non-affine stochastic volatility model

In this paper we study the pricing and hedging of options on realized variance in the 3/2 non-affine stochastic volatility model by developing efficient transform-based pricing methods. This non-affine model gives prices of options on realized variance that allow upward-sloping implied volatility of variance smiles. Heston's model [Rev. Financial Stud., 1993, 6, 327–343], the benchmark affine stochastic volatility model, leads to downward-sloping volatility of variance smiles—in disagreement with variance markets in practice. Using control variates, we propose a robust method to express the Laplace transform of the variance call function in terms of the Laplace transform of the realized variance. The proposed method works in any model where the Laplace transform of realized variance is available in closed form. Additionally, we apply a new numerical Laplace inversion algorithm that gives fast and accurate prices for options on realized variance, simultaneously at a sequence of variance strikes. The method is also used to derive hedge ratios for options on variance with respect to variance swaps.     

Double-jump diffusion model for VIX: evidence from VVIX

This paper studies the continuous-time dynamics of VIX with stochastic volatility and jumps in VIX and volatility. Built on the general parametric affine model with stochastic volatility and jumps in the logarithm of VIX, we derive a linear relationship between the stochastic volatility factor and the VVIX index. We detect the existence of a co-jump of VIX and VVIX and put forward a double-jump stochastic volatility model for VIX through its joint property with VVIX. Using the VVIX index as a proxy for stochastic volatility, we use the MCMC method to estimate the dynamics of VIX. Comparing nested models of VIX, we show that the jump in VIX and the volatility factor are statistically significant. The jump intensity is also stochastic. We analyse the impact of the jump factor on VIX dynamics.     

American stochastic volatility call option pricing: A lattice based approach

This study presents a new method of pricing options on assets with stochastic volatility that is lattice based, and can easily accommodate early exercise for American options. Unlike traditional lattice methods, recombination is not a problem in the new model, and it is easily adapted to alternative volatility processes. Approximations are developed for European C.E.V. calls and American stochastic volatility calls. The application of the pricing model to exchange traded calls is also illustrated using a sample of market prices. Modifying the model to price American puts is straightforward, and the approach can easily be extended to other non-recombining lattices.     

How Persistent is Stock Return Volatility? An Answer with Markov Regime Switching Stochastic Volatility Models

Abstract:  We propose generalised stochastic volatility models with Markov regime changing state equations (SVMRS) to investigate the important properties of volatility in stock returns, specifically high persistence and smoothness. The model suggests that volatility is far less persistent and smooth than the conventional GARCH or stochastic volatility. Persistent short regimes are more likely to occur when volatility is low, while far less persistence is likely to be observed in high volatility regimes. Comparison with different classes of volatility supports the SVMRS as an appropriate proxy volatility measure. Our results indicate that volatility could be far more difficult to estimate and forecast than is generally believed.     

The fair value of guaranteed annuity options

We discuss the fair valuation of Guaranteed Annuity Options, i.e. options providing the right to convert deferred survival benefits into annuities at fixed conversion rates. The use of doubly stochastic stopping times and of affine processes provides great computational and analytical tractability, while enabling to set up a very general valuation framework. For example, the valuation of options on traditional, unit-linked or indexed annuities is encompassed. Moreover, security and reference fund prices may feature stochastic volatility or discontinuous dynamics. The longevity risk is also taken into account, by letting the evolution of mortality present stochastic dynamics subject not only to random fluctuations but also to systematic deviations.     

A new closed-form solution as an extension of the Black–Scholes formula allowing smile curve plotting

In this paper, as a generalization of the Black–Scholes (BS) model, we elaborate a new closed-form solution for a uni-dimensional European option pricing model called the J-model. This closed-form solution is based on a new stochastic process, called the J-process, which is an extension of the Wiener process satisfying the martingale property. The J-process is based on a new statistical law called the J-law, which is an extension of the normal law. The J-law relies on four parameters in its general form. It has interesting asymmetry and tail properties, allowing it to fit the reality of financial markets with good accuracy, which is not the case for the normal law. Despite the use of one state variable, we find results similar to those of Heston dealing with the bi-dimensional stochastic volatility problem for pricing European calls. Inverting the BS formula, we plot the smile curve related to this closed-form solution. The J-model can also serve to determine the implied volatility by inverting the J-formula and can be used to price other kinds of options such as American options.     

A new approach for option pricing under stochastic volatility

We develop a new approach for pricing European-style contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance swap market, we directly assume continuous dynamics for the time T variance swap rate. The initial value of this variance swap rate can either be directly observed, or inferred from option prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption the stationary volatility ratio hypothesis (SVRH). The instantaneous volatility of the futures follows an unspecified stochastic process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this, we show that the payoff to a path-independent contingent claim can be perfectly replicated by dynamic trading in futures contracts and variance swaps of the same maturity. As a result, the contingent claim is uniquely valued relative to its underlying’s futures price and the assumed observable variance swap rate. In contrast to standard models of stochastic volatility, our approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility. As a consequence of our SVRH, the partial differential equation (PDE) governing the arbitrage-free value of the contingent claim just depends on two state variables rather than the usual three. We then focus on the consistency of our SVRH with the standard assumption that the risk-neutral process for the instantaneous variance is a diffusion whose coefficients are independent of the variance swap maturity. We show that the combination of this maturity independent diffusion hypothesis (MIDH) and our SVRH implies a very special form of the risk-neutral diffusion process for the instantaneous variance. Fortunately, this process is tractable, well-behaved, and enjoys empirical support. Finally, we show that our model can also be used to robustly price and hedge volatility derivatives.     

The term structure of S&P 100 model-free volatilities

We develop an improved method to obtain the model-free volatility more accurately despite the limitations of a finite number of options and large strike price intervals. Our method computes the model-free volatility from European-style S&P 100 index options over a horizon of up to 450 days, the first time that this has been attempted, as far as we are aware. With the estimated daily term structure over the long horizon, we find that (i) changes in model-free volatilities are asymmetrically more positively impacted by a decrease in the index level than negatively impacted by an increase in the index level; (ii) the negative relationship between the daily change in model-free volatility and the daily change in index level is stronger in the near term than in the far term; and (iii) the slope of the term structure is positively associated with the index level, having a tendency to display a negative slope during bear markets and a positive slope during bull markets. These significant results have important implications for pricing and hedging index derivatives and portfolios.     

Calibration and advanced simulation schemes for the Wishart stochastic volatility model

In this article, we deal with calibration and Monte Carlo simulation of the Wishart stochastic volatility model. Despite the analytical tractability of the considered model, being of affine type, the implementation of Wishart-based stochastic volatility models poses non-trivial challenges from a numerical point of view. The goal of this article is to overcome these problems providing efficient numerical schemes for Monte Carlo simulations. Moreover, a fast and accurate calibration procedure is proposed.