Lech A. Grzelak

On Cross-Currency Models with Stochastic Volatility and Correlated Interest Rates

We construct multi-currency models with stochastic volatility and correlated stochastic interest rates with a full matrix of correlations. We frst deal with a foreign exchange (FX) model of Heston-type, in which the domestic and foreign interest rates are generated by the short-rate process of Hull-White [HW96]. We then extend the framework by modeling the interest rate by a stochastic volatility displaced-diffusion Libor Market Model [AA02], which can model an interest rate smile. We provide semi-closed form approximations which lead to effcient calibration of the multi-currency models. Finally, we add a correlated stock to the framework and discuss the construction, model calibration and pricing of equity- FX-interest rate hybrid payoffs.

An Equity-Interest Rate Hybrid Model with Stochastic Volatility and the Interest Rate Smile

We define an equity-interest rate hybrid model in which the equity part is driven by the Heston stochastic volatility [Hes93], and the interest rate (IR) is generated by the displaced-diffusion stochastic volatility Libor Market Model [AA02]. We assume a non-zero correlation between the main processes. By an appropriate change of measure the dimension of the corresponding pricing PDE can be greatly reduced. We place by a number of approximations the model in the class of affine processes [DPS00], for which we then provide the corresponding forward characteristic function. We discuss in detail the accuracy of the approximations and the efficient calibration. Finally, by experiments, we show the effect of the correlations and interest rate smile/skew on typical equity-interest rate hybrid product prices. For a whole strip of strikes this approximate hybrid model can be evaluated for equity plain vanilla options in just milliseconds.

The Affine Heston Model with Correlated Gaussian Interest Rates for Pricing Hybrid Derivatives

In this article we define a multi-factor equity–interest rate hybrid model with non-zero correlation between the stock and interest rate. The equity part is modeled by the Heston model and we use a Gaussian multi-factor short-rate process. By construction, the model fits in the framework of affine diffusion processes, allowing fast calibration to plain vanilla options. We also provide an efficient Monte Carlo simulation scheme.

Determination of the Corrosion Rate of a MIC Influenced Pipeline Using Four Consecutive Pig Runs

In-line inspections have been part of the verification of pipeline integrity since the late seventies in N.V. Nederlandse Gasunie. The discovery of external microbial corrosion (MIC) in 1999 in one of the high pressure pipelines changed the inspection policy from inspection of a randomly selected pipeline once every 5 year to an inspection program for the whole high pressure grid (approximately 5.600 km) to be completed in 10–12 years. One of the MIC influenced lines is used to qualify MFL-pig from different suppliers. In the time period 1999–2004 4 different MFL qualification runs have taken place, resulting in 18 excavations. Rather simple calculation of corrosion rates after every pig run indicated values of 0.2–0.3 mm/yr as an average value for the MIC corrosion. After the fourth pig run (5 years after the first) Gasunie decided to determine the corrosion rate also in a more sophisticated way. In coorperation with the mathematical department of the Technical University of Delft statistical analyses were performed. The paper describes how the data from pig runs and excavations was used to extract a suitable subset of corrosiondefects for which the corrosion rate was determined. It was decided to use only these defects that were indicated by all suppliers as external corrosion thereby leaving out defects that were indicated as “mill defect but possibly corrosion”. A second criterion was that a defect has to be reported by at least three of the four suppliers. These criteria resulted in a subset of 52 defects to be analyzed. All of the reported defect depths were corrected for the bias that was determined from the excavation results. The corrected values were then used to calculate the corrosion rate using three approaches: calculation of the corrosion rate for every defect in two ways and calculation of the corrosion rate for the pool of defects assuming a corrosion growth that is linear in time. The average corrosion rate for the defects was in the range of 0.12 to 0.24 mm/yr. Dividing the data set into deep and shallow defects showed that the average corrosion rates for both subsets, 0.23 mm/yr and 0.25 mm/yr, are not significantly different. This underpins one of the assumptions that the corrosion rate is constant. Further discussion within Gasunie is envisaged to determine how these values can be used in the calculation of re-inspection intervals for this line. And at the same time another discussion will focus on whether these values can be used on other lines.Copyright

Calibration and Monte Carlo Pricing of the SABR-Hull-White Model for Long-Maturity Equity Derivatives

We model the joint dynamics of stock and interest rate by a hybrid SABR-Hull-White model, in which the asset price dynamics are modeled by the SABR model and the interest rate dynamics by the Hull-White short-rate model. We propose a projection formula, mapping the SABR-HW model parameters onto the parameters of the nearest SABR model. A time-dependent parameter extension of this SABR-HW model is adopted to make the calibration of the model consistent across maturity times. The calibration procedure is then finalized by employing the weighted Monte Carlo technique. The Monte Carlo weights are not uniform and chosen to replicate the calibration market instruments.

On a one time-step Monte Carlo simulation approach of the SABR model

In this work, we propose a one time-step Monte Carlo method for the SABR model. We base our approach on an accurate approximation of the cumulative distribution function of the time-integrated variance (conditional on the SABR volatility), using Fourier techniques and a copula. Resulting is a fast simulation algorithm which can be employed to price European options under the SABR dynamics. Our approach can thus be seen as an alternative to Hagans analytic formula for short maturities that may be employed for model calibration purposes.

On an efficient multiple time step Monte Carlo simulation of the SABR model

In this paper, we will present a multiple time step Monte Carlo simulation technique for pricing options under the Stochastic Alpha Beta Rho model. The proposed method is an extension of the one time step Monte Carlo method that we proposed in an accompanying paper Leitao et al. [Appl. Math. Comput. 2017, 293, 461–479], for pricing European options in the context of the model calibration. A highly efficient method results, with many very interesting and nontrivial components, like Fourier inversion for the sum of log-normals, stochastic collocation, Gumbel copula, correlation approximation, that are not yet seen in combination within a Monte Carlo simulation. The present multiple time step Monte Carlo method is especially useful for long-term options and for exotic options.

From Arbitrage to Arbitrage-Free Implied Volatilities

We propose a method for determining an arbitrage-free density implied by Hagan’s formula. Our technique is based on the stochastic collocation method. The principle is to determine a few collocation points on the implied survival distribution function (SDF) and project them on a polynomial of an arbitrage-free variable for which we choose the Gaussian variable. In this way we have equality in probability at the collocation points while the generated density is arbitrage-free. Analytic European option prices are available and the implied volatilities stay very close to those initially obtained by Hagan’s formula. The proposed method is very fast and straightforward to implement as it only involves 1D Lagrange interpolation and inversion of a linear system of equations. The technique is generic and may be applied to other variants or other models that generate arbitrage.

Incorporating an Interest Rate Smile in an Equity Local Volatility Model

The focus of this paper is on finding a connection between the interest rate and equity asset classes. We propose an equity interest rate hybrid model which preserves market observable smiles: the equity from plain vanilla products via a local volatility framework and the interest rate from caps and swaptions via the Stochastic Volatility Libor Market Model. We define a multi-factor short-rate process implied from the Libor Market Model via an arbitrage-free interpolation and combine it with the local volatility equity model for stochastic interest rates. We show that the interest rate smile has a significant impact on the equity local volatility. The model developed is intuitive and straightforward, enabling consistent pricing of related hybrid products. Moreover, it preserves the non-arbitrage Heath, Jarrow, Morton conditions.

A Highly Efficient Numerical Method for the SABR Model

In Leitao et al. (Appl Math Comput 293:461–479, 2017), we have presented a one time-step Monte Carlo simulation of the SABR model (Hagan et al. Wilmott Mag 84–108, 2002). The technique is based on an efficient simulation of SABR’s time-integrated variance process. We base our approach on the derivation of the cumulative distribution function of the integrated variance by means of Fourier techniques and the use of a copula to approximate the conditional distribution (integrated variance conditional on the SABR volatility process). Resulting is a fast simulation algorithm which can be employed to price European options under the SABR dynamics. This converts our approach into an alternative to Hagan analytic formula for short maturities, where some known issues of the implied volatility expression for small strike values are overcome. A generalization of this technique to the multiple time-step case has been presented in Leitao et al. (On an efficient multiple time-step Monte Carlo simulation of the SABR model 2016, submitted for publication. Available at SSRN: http://ssrn.com/abstract=2764908).