Sergei Levendorskiĭ

Prices and Sensitivities of Barrier and First-Touch Digital Options in Levy-Driven Models

We present a fast and accurate FFT-based method of computing the prices and sensitivities of barrier options and first-touch digital options on stocks whose log-price follows a Levy process. The numerical results obtained via our approach are demonstrated to be in good agreement with the results obtained using other (sometimes fundamentally different) approaches that exist in the literature. However, our method is computationally much faster (often, dozens of times faster). Moreover, our technique has the advantage that its application does not entail a detailed analysis of the underlying Levy process: one only needs an explicit analytic formula for the characteristic exponent of the process. Thus our algorithm is very easy to implement in practice. Finally, our method yields accurate results for a wide range of values of the spot price, including those that are very close to the barrier, regardless of whether the maturity period of the option is long or short.

Valuation of Continuously Monitored Double Barrier Options and Related Securities

In this article we apply Carrs randomization approximation and the operator form of the Wiener-Hopf method to double barrier options in continuous time. Each step in the resulting backward induction algorithm is solved using a simple iterative procedure that reduces the problem of pricing options with two barriers to pricing a sequence of certain contingent claims with first-touch single barrier features. This procedure admits a clear financial interpretation that can be formulated in the language of embedded options.Our approach results in a fast and accurate pricing method that can be used in a rather wide class of Levy-driven models including Variance Gamma processes, Normal Inverse Gaussian processes and KoBoL processes (a.k.a. the CGMY model). At the same time, our work gives new insight into the known explicit formulas obtained by other authors in the setting of the Black-Scholes model. The operator form of the Wiener-Hopf method is generalized for wide classes of processes including the important class of Variance Gamma processes.Our method can be applied to double barrier options with arbitrary bounded terminal payoff functions, which, in particular, allows us to price knock-out double barrier put/call options as well as double-no-touch options.

Efficient Pricing and Reliable Calibration in the Heston Model

We suggest a general scheme for improvement of FT-pricing formulas for European options and give efficient recommendations for the choice of the parameters of the numerical scheme, which allow for very accurate and fast calculations. The efficiency of the method stems from the properties of functions analytical in a strip, which were introduced to finance by Feng and Linetsky (2008). We demonstrate that an indiscriminate choice of parameters of a numerical scheme leads to an inaccurate pricing and calibration. As applications, we consider the Heston model and its generalization. For many parameter sets documented in empirical studies of financial markets, relative accuracy better than 0.01% can be achieved by summation of less than 10-20 terms even in the situations in which the standard approach requires more than 200. In some cases, the one-term formula produces an error of several percent, and the summation of two terms — less than 0.5%. Typically, 10 terms and fewer suffice to achieve the error tolerance of several percent and smaller.

Pricing Discrete Barrier Options and Credit Default Swaps Under Levy Processes

We consider discretely monitored barrier options under Levy models, including single and double barrier options and first-touch digitals, as well as CDS and defaultable bonds. At each step of backward induction, we use piece-wise polynomial interpolation and an efficient version of the Fourier transform technique, which allows for efficient error control. We derive accurate recommendations for the choice of parameters of the numerical scheme, and produce numerical examples showing that oversimplified prescriptions in other methods can result in large errors.

PRICES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS IN LÉVY-DRIVEN MODELS, NEAR BARRIER

We calculate the leading term of asymptotics of the prices of barrier options and first-touch digitals near the barrier for wide classes of Levy processes with exponential jump densities, including the Variance Gamma model, the KoBoL (a.k.a. CGMY) model and Normal Inverse Gaussian processes. In the case of processes of infinite activity and finite variation, with the drift pointing from the barrier, we prove that the price is discontinuous at the boundary. This observation can serve as the basis for a simple robust test of the type of processes observed in real financial markets. In many cases, we calculate the second term of asymptotics as well. By comparing the exact asymptotic results for prices with those of Carrs randomization approximation, we conclude that the latter is very accurate near the barrier. We illustrate this by including numerical results for several types of Levy processes commonly used in option pricing.

American Options in the Heston Model with Stochastic Interest Rate and Its Generalizations

Abstract We consider the Heston model with the stochastic interest rate of Cox–Ingersoll–Ross (CIR) type and more general models with stochastic volatility and interest rates depending on two CIR-factors; the price, volatility and interest rate may correlate. Time-derivative and infinitesimal generator of the process for factors that determine the dynamics of the interest rate and/or volatility are discretized. The result is a sequence of embedded perpetual options arising in the time discretization of a Markov-modulated Lévy model. Options in this sequence are solved using an iteration method based on the Wiener–Hopf factorization. Typical shapes of the early exercise boundary are shown, and good agreement of option prices with prices calculated with the Longstaff–Schwartz method and Medvedev–Scaillet asymptotic method is demonstrated.

Efficient Laplace Inversion, Wiener-Hopf Factorization and Pricing Lookbacks

We construct fast and accurate methods for (a) approximate Laplace inversion, (b) approximate calculation of the Wiener-Hopf factors for wide classes of Levy processes with exponentially decaying Levy densities, and (c) approximate pricing of lookback options. In all cases, we use appropriate conformal change-of-variable techniques, which allow us to apply the simplified trapezoid rule with a small number of terms (the changes of variables in the outer and inner integrals and in the formulas for the Wiener-Hopf factors must be compatible in a certain sense). The efficiency of the method stems from the properties of functions analytic in a strip (these properties were explicitly used in finance by Feng and Linetsky 2008). The same technique is applicable to the calculation of the pdfs of supremum and infimum processes, and to the calculation of the prices and sensitivities of options with lookback and barrier features.

Efficient Pricing Of Barrier Options And Credit Default Swaps In Lévy Models With Stochastic Interest Rate

Recently, advantages of conformal deformations of the contours of integration in pricing formulas for European options have been demonstrated in the context of wide classes of Levy models, the Heston model, and other affine models. Similar deformations were used in one-factor Levy models to price options with barrier and lookback features and credit default swaps (CDSs). In the present paper, we generalize this approach to models, where the dynamics of the assets is modeled as , where X is a Levy process, and the interest rate is stochastic. Assuming that X and r are independent, and , the infinitesimal generator of the pricing semigroup in the model for the short rate, satisfies weak regularity conditions, which hold for popular models of the short rate, we develop a variation of the pricing procedure for Levy models which is almost as fast as in the case of the constant interest rate. Numerical examples show that about 0.15 second suffices to calculate prices of 8 options of same maturity in a two-factor model with the error tolerance and less; in a three-factor model, accuracy of order 0.001–0.005 is achieved in about 0.2 second. Similar results are obtained for quanto CDS, where an additional stochastic factor is the exchange rate. We suggest a class of Levy models with the stochastic interest rate driven by 1–3 factors, which allows for fast calculations. This class can satisfy the current regulatory requirements for banks mandating sufficiently sophisticated credit risk models.

Ultra-Fast Pricing Barrier Options and CDSs

We construct a new approximate method for pricing barrier options and CDSs. In many cases, prices of barrier options and CDS of maturities T ≥ 1 years, at the log-distance 0.1 from the barrier and farther, for eight spots, can be calculated adding up 4–16 fairly simple terms, with relative errors of order 5 ⋅ 10−5 and smaller, in 4–12msc.