Abstracts
Tuesday June 3, 14:00-14:30 | session 2.7 | Stochastic Volatility | room I
Using positive supOU (superposition of Ornstein-Uhlenbeck type) processes to describe the volatility, we introduce a stochastic volatility model for financial data which is capable of modelling long range dependence effects and other important stylized features of financial data.
The finiteness of moments and the second order structure of the volatility, the log returns, as well as their “squares'” are discussed in detail. Moreover, we give a concrete example in which long memory effects occur.
Thereafter we give conditions for the discounted stock price to be a martingale, compute the characteristic function and show that calculating option prices in this model is well possible using a Fourier approach.
Finally we show how the model can be estimated from historical data and how it can be calibrated to observed option prices. We conclude by presenting some concrete data examples and by commenting on multivariate extensions.
Tuesday June 3, 14:30-15:00 | session 2.7 | Stochastic Volatility | room I
In our work we compare a large variety (not fifty, but a large number) of simulation schemes for the SABR model due to Hagan et al. [2002]. All schemes are inspired by the recent works of Islah [2009] and Chen, Oosterlee and Van der Weide [2011]. Islah has shown that, conditional on the integrated variance, an asset in the SABR model follows, after a suitable transformation, a squared Bessel process. In Chen et al. this result has been utilised to arrive at a low-bias simulation scheme for the SABR model.
To be precise, Chen et al. approximate the dynamics of the squared Bessel process by drawing from the exact distribution when the process is close to its absorbing boundary at zero, and drawing from a quadratic Gaussian process (matched to the approximations of the first two moments) elsewhere. For the integrated variance process, approximate first two moments are derived and a lognormal distribution is fitted to approximate the distribution.
Our first set of schemes improves on the work of Chen et al. in a number of ways. First of all, we demonstrate that matching the distribution of the integrated variance is of lesser importance when we use a short-stepped approach such as Chen et al. do, so that simpler approximations, as in Andersen [2008], suffice. In longer-stepped simulations the exact distribution does matter, and we can use conditioning techniques from Asian option pricing to improve on the lognormal approximation.
Second, we derive the exact first two moments of the asset, conditional on the integrated variance, and use a moment-matching scheme with a mixture distribution (a mixture of zero to reflect the absorbing boundary condition, and a distribution resulting in positive values) to fit its distribution.
We also study the recent work of Makarov and Glew [2010] on the simulation of squared Bessel processes with or without absorption at zero. They show that their distribution is that of a randomised Gamma distribution, and devise several schemes for their simulation.
Finally, we also consider a variety of much simpler schemes, such as a simple Euler scheme based on the work of Lord et al. [2010] and an approximation based on Sankaran’s [1963] approximation of the non-central chi-squared distribution.
All schemes are compared in numerical examples for both the CEV model, of which we know an analytical solution, and for the more general SABR case.
Tuesday June 3, 15:00-15:30 | session 2.7 | Stochastic Volatility | room I
We present in our work a continuous time Capital Asset Pricing Model where the volatilities of the market index and the stock are both stochastic. Using a singular perturbation technique, we provide approximations for the prices of european options on both the stock and the index. These approximations are functions of the model parameters. We show then that existing estimators of the parameter beta, proposed in the recent literature, are biased in our setting because they are all based on the assumption that the idiosyncratic volatility of the stock is constant. We provide then an unbiased estimator of the parameter beta using only implied volatility data. This estimator is a forward measure of the parameter beta in the sense that it represents the information contained in derivatives prices concerning the forward realization of this parameter, we test then its capacity of prediction of the forward beta and we draw a conclusion concerning its predictive power.
Tuesday June 3, 15:30-16:00 | session 2.7 | Stochastic Volatility | room I
Commodity futures and their derivatives have become key players in the portfolios of many corporations, especially those in the energy sector. Since plain vanilla options may not be sufficient to address hedging strategies, complex contracts, such as exotic path dependent derivatives become very relevant. This implies that, the dynamics of the future prices must be known for a time interval and not only at the options' maturity. We address this problem by making use of Dupire's local volatility model. Since, for options on commodity futures, when we change maturity, we also change the underlying future, further assumptions ought to be made. Futures with different maturities on the same commodity present very high correlations. This suggests that, in this context of local volatility, we can use only one Brownian motion in order to describe the whole term structure.
In this case, future prices are described by diffusions with a local volatility structure. Thus, we address the correspondent local volatility surface calibration problem by Tikhonov regularization. We review some important result and establish new ones. We also present an online approach, which allows us to incorporate additional information when calibrating local volatility surfaces. We can use, in each procedure, option prices corresponding to different levels of the underlying asset price. We shall see that, if such dependence has some regularity, the convergence results under this online approach imply on the same results under the standard framework. We also analyze the discrete counterpart of the continuous results. This is done in order to substantiate and justify the our numerical experiments.
As in equity markets, the vanilla option prices available are American. However, due to the presence of the unobservable convenience yield in commodity future prices, American call options are more expensive than the European ones. This implies that, in order to perform numerical tests under the context of this article, we have to extract European prices from the American ones.
Due to the intrinsic nonlinearity of of American options pricing problem, an approach similar to Dupire's equation is not available. Thus, the corresponding local volatility surface calibration procedure should be performed for each maturity and strike leading to a computationally intensive problem.