Variance swaps valuation under non-affine GARCH models and their diffusion limits

In this article, we investigate the pricing and convergence of general non-affine non-Gaussian GARCH-based discretely sampled variance swaps. Explicit solutions for fair strike prices under two different sampling schemes are derived using the extended Girsanov principle as the pricing kernel candidate. Following standard assumptions on time-varying GARCH parameters, we show that these quantities converge respectively to fair strikes of discretely and continuously sampled variance swaps that are constructed based on the weak diffusion limit of the underlying GARCH model. An empirical study which relies on a joint estimation using both historical returns and VIX data indicates that an asymmetric heavier tailed distribution is more appropriate for modelling the GARCH innovations. Finally, we provide several numerical exercises to support our theoretical convergence results in which we further investigate the effect of the quadratic variation approximation for the realized variance, as well as the impact of discrete versus continuous-time modelling of asset returns.     

Variance swaps on time-changed Lévy processes

We prove that a multiple of a log contract prices a variance swap, under arbitrary exponential Lévy dynamics, stochastically time-changed by an arbitrary continuous clock having arbitrary correlation with the driving Lévy process, subject to integrability conditions. We solve for the multiplier, which depends only on the Lévy process, not on the clock. In the case of an arbitrary continuous underlying returns process, the multiplier is 2, which recovers the standard no-jump variance swap pricing formula. In the presence of negatively skewed jump risk, however, we prove that the multiplier exceeds 2, which agrees with calibrations of time-changed Lévy processes to equity options data. Moreover, we show that discrete sampling increases variance swap values, under an independence condition; so if the commonly quoted multiple 2 undervalues the continuously sampled variance, then it undervalues even more the discretely sampled variance. Our valuations admit enforcement, in some cases, by hedging strategies which perfectly replicate variance swaps by holding log contracts and trading the underlying.     

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.     

Variation and share-weighted variation swaps on time-changed Lévy processes

For a family of functions G, we define the G-variation, which generalizes power variation; G-variation swaps, which pay the G-variation of the returns on an underlying share price F; and share-weighted G-variation swaps, which pay the integral of F with respect to G-variation. For instance, the case G(x)=x ² reduces these notions to, respectively, quadratic variation, variance swaps, and gamma swaps. We prove that a multiple of a log contract prices a G-variation swap, and a multiple of an FlogF contract prices a share-weighted G-variation swap, under arbitrary exponential Lévy dynamics, stochastically time-changed by an arbitrary continuous clock having arbitrary correlation with the Lévy driver, under integrability conditions. We solve for the multipliers, which depend only on the Lévy process, not on the clock. In the case of quadratic G and continuity of the underlying paths, each valuation multiplier is 2, recovering the standard no-jump variance and gamma-swap pricing results. In the presence of jump risk, however, we show that the valuation multiplier differs from 2, in a way that relates (positively or negatively, depending on the specified G) to the Lévy measure’s skewness. In three directions this work extends Carr–Lee–Wu, which priced only variance swaps. First, we generalize from quadratic variation to G-variation; second, we solve for not only unweighted but also share-weighted payoffs; and third, we apply these tools to analyze and minimize the risk in a family of hedging strategies for G-variation.     

Pricing by hedging and no-arbitrage beyond semimartingales

We show that pricing a big class of relevant options by hedging and no-arbitrage can be extended beyond semimartingale models. To this end we construct a subclass of self-financing portfolios that contains hedges for these options, but does not contain arbitrage opportunities, even if the stock price process is a non-semimartingale of some special type. Moreover, we show that the option prices depend essentially only on a path property of the stock price process, viz. on the quadratic variation. We end the paper by giving no-arbitrage results even with stopping times for our model class.      

Fed funds futures variance futures

We develop a novel contract design, the fed funds futures (FFF) variance futures, which reflects the expected realized basis point variance of an underlying FFF rate. The valuation of short-term FFF variance futures is completely model-independent in a general setting that includes the cases where the underlying FFF rate exhibits jumps and where the realized variance is computed by sampling the FFF rate discretely. The valuation of longer-term FFF variance futures is subject to an approximation error which we quantify and show is negligible. We also provide an illustrative example of the practical valuation and use of the FFF variance futures contract.     

Hedging variance options on continuous semimartingales

We find robust model-free hedges and price bounds for options on the realized variance of [the returns on] an underlying price process. Assuming only that the underlying process is a positive continuous semimartingale, we superreplicate and subreplicate variance options and forward-starting variance options, by dynamically trading the underlying asset and statically holding European options. We thereby derive upper and lower bounds on values of variance options, in terms of Europeans.     

Options on realized variance and convex orders

Realized variance option and options on quadratic variation normalized to unit expectation are analysed for the property of monotonicity in maturity for call options at a fixed strike. When this condition holds the risk-neutral densities are said to be increasing in the convex order. For Lévy processes, such prices decrease with maturity. A time series analysis of squared log returns on the S&P 500 index also reveals such a decrease. If options are priced to a slightly increasing level of acceptability, then the resulting risk-neutral densities can be increasing in the convex order. Calibrated stochastic volatility models yield possibilities in both directions. Finally, we consider modeling strategies guaranteeing an increase in convex order for the normalized quadratic variation. These strategies model instantaneous variance as a normalized exponential of a Lévy process. Simulation studies suggest that other transformations may also deliver an increase in the convex order.     

Pricing swaps and options on quadratic variation under stochastic time change models—discrete observations case

We use a forward characteristic function approach to price variance and volatility swaps and options on swaps. The swaps are defined via contingent claims whose payoffs depend on the terminal level of a discretely monitored version of the quadratic variation of some observable reference process. As such a process we consider a class of Levy models with stochastic time change. Our analysis reveals a natural small parameter of the problem which allows a general asymptotic method to be developed in order to obtain a closed-form expression for the fair price of the above products. As examples, we consider the CIR clock change, general affine models of activity rates and the 3/2 power clock change, and give an analytical expression of the swap price. Comparison of the results obtained with a familiar log-contract approach is provided.     

Binomial Option Pricing Biases and Inconsistent Implied Volatilities

We evaluate the binomial option pricing methodology (OPM) by examining simulated portfolio strategies. A key aspect of our study involves sampling from the empirical distribution of observed equity returns. Using a Monte Carlo simulation, we generate equity prices under known volatility and return parameters. We price American–style put options on the equity and evaluate the risk–adjusted performance of various strategies that require writing put options with different maturities and moneyness characteristics. The performance of these strategies is compared to an alternative strategy of investing in the underlying equity. The relative performance of the strategies allows us to identify biases in the binomial OPM leading to the well–known volatility smile . By adjusting option prices so as to rule out dominated option strategies in a mean–variance context, we are able to reduce the pricing errors of the OPM with respect to option prices obtained from the LIFFE. Our results suggest that a simple recalibration of inputs may improve binomial OPM performance.     

A numerical method to price exotic path-dependent options on an underlying described by the Heston stochastic volatility model

We consider the problem of pricing European exotic path-dependent derivatives on an underlying described by the Heston stochastic volatility model. Lipton has found a closed form integral representation of the joint transition probability density function of underlying price and variance in the Heston model. We give a convenient numerical approximation of this formula and we use the obtained approximated transition probability density function to price discrete path-dependent options as discounted expectations. The expected value of the payoff is calculated evaluating an integral with the Monte Carlo method using a variance reduction technique based on a suitable approximation of the transition probability density function of the Heston model. As a test case, we evaluate the price of a discrete arithmetic average Asian option, when the average over n = 12 prices is considered, that is when the integral to evaluate is a 2n = 24 dimensional integral. We show that the method proposed is computationally efficient and gives accurate results.     

Analytical approximations for the critical stock prices of American options: a performance comparison

Many efficient and accurate analytical methods for pricing American options now exist. However, while they can produce accurate option prices, they often do not give accurate critical stock prices. In this paper, we propose two new analytical approximations for American options based on the quadratic approximation. We compare our methods with existing analytical methods including the quadratic approximations in Barone-Adesi and Whaley (J Finance 42:301–320, 1987) and Barone-Adesi and Elliott (Stoch Anal Appl 9(2):115–131, 1991), the lower bound approximation in Broadie and Detemple (Rev Financial Stud 9:1211–1250, 1996), the tangent approximation in Bunch and Johnson (J Finance 55(5):2333–2356, 2000), the Laplace inversion method in Zhu (Int J Theor Appl Finance 9(7):1141–1177, 2006b), and the interpolation method in Li (Working paper, 2008). Both of our methods give much more accurate critical stock prices than all the existing methods above.     

Pricing discretely monitored Asian options under Lévy processes

We present methodologies to price discretely monitored Asian options when the underlying evolves according to a generic Lévy process. For geometric Asian options we provide closed-form solutions in terms of the Fourier transform and we study in particular these formulas in the Lévy-stable case. For arithmetic Asian options we solve the valuation problem by recursive integration and derive a recursive theoretical formula for the moments to check the accuracy of the results. We compare the implementation of our method to Monte Carlo simulation implemented with control variates and using different parametric Lévy processes. We also discuss model risk issues.     

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.     

Efficient and accurate quadratic approximation methods for pricing Asian strike options

This study is on valuing Asian strike options and presents efficient and accurate quadratic approximation methods that work extremely well, both with regard to the volatility of a wide range of underlying assets, and longer average time windows. We demonstrate that most of the well-known quadratic approximation methods used in the literature for pricing Asian strike options are special cases of our model, with the numerical results demonstrating that our method significantly outperforms the other quadratic approximation methods examined here. Using our method for the calculation of hundreds of Asian strike options, the pricing errors (in terms of the root mean square errors) are reasonably small. Compared with the Monte Carlo benchmark method, our method is shown to be rapid and accurate. We further extend our method to the valuing of quanto forward-starting Asian strike options, with the pricing accuracy of these options being largely the same as the pricing of plain vanilla Asian strike options.     

A New Computational Scheme for Computing Greeks by the Asymptotic Expansion Approach

We developed a new scheme for computing “Greeks” of derivatives by an asymptotic expansion approach. In particular, we derived analytical approximation formulae for Deltas and Vegas of plain vanilla and average European call options under general Markovian processes of underlying asset prices. Moreover, we introduced a new variance reduction method of Monte Carlo simulations based on the asymptotic expansion scheme. Finally, several numerical examples under CEV processes confirmed the validity of our method.     

Pricing a class of exotic commodity options in a multi-factor jump-diffusion model

In a recent paper, Crosby introduced a multi-factor jump-diffusion model which would allow futures (or forward) commodity prices to be modelled in a way which captured empirically observed features of the commodity and commodity options markets. However, the model focused on modelling a single individual underlying commodity. In this paper, we investigate an extension of this model which would allow the prices of multiple commodities to be modelled simultaneously in a simple but realistic fashion. We then price a class of simple exotic options whose payoff depends on the difference (or ratio) between the prices of two different commodities (for example, spread options), or between the prices of two different (i.e. with different tenors) futures contracts on the same underlying commodity, or between the prices of a single futures contract as observed at two different calendar times (for example, forward start or cliquet options). We show that it is possible, using a Fourier transform-based algorithm, to derive a single unifying form for the prices of all these aforementioned exotic options and some of their generalizations. Although we focus on pricing options within the model of Crosby, most of our results would be applicable to other models where the relevant ‘extended’ characteristic function is available in analytical form.     

Empirical risk aversion functions-estimates and assessment of their reliability

In this paper, we examine investor's risk preferences implied by option prices. In order to derive these preferences, we specify the functional form of a pricing kernel and then shift its parameters until realized returns are best explained by the subjective probability density function, which consists of the ratio of the risk-neutral probability density function and the pricing kernel. We examine, alternatively, pricing kernels of power, exponential, and higher order polynomial forms. Using S&P 500 index options, we find surprising evidence of risk neutrality, instead of risk aversion, in both the power and exponential cases. When extending the underlying assumption on the specification of the pricing kernel to one of higher order polynomial functions, we obtain functions exhibiting ‘monotonically decreasing’ relative risk aversion (DRRA) and anomalous ‘inverted U-shaped’ relative risk aversion. We find, however, that only the DRRA function is robust to variation in sample characteristics, and is statistically significant. Finally, we also find that most of our empirical results are consistent, even when taking into account market imperfections such as illiquidity.     

Power and Bipower Variation with Stochastic Volatility and Jumps

This article shows that realized power variation and its extension,realized bipower variation, which we introduce here, are somewhatrobust to rare jumps. We demonstrate that in special cases,realized bipower variation estimates integrated variance instochastic volatility models, thus providing a model-free andconsistent alternative to realized variance. Its robustnessproperty means that if we have a stochastic volatility plusinfrequent jumps process, then the difference between realizedvariance and realized bipower variation estimates the quadraticvariation of the jump component. This seems to be the firstmethod that can separate quadratic variation into its continuousand jump components. Various extensions are given, togetherwith proofs of special cases of these results. Detailed mathematicalresults are reported in Barndorff-Nielsen and Shephard (2003a).