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Abstract:
We develop a systematic approach to the reduction of dimensionality of smile-enabled models by projecting them onto a displaced version of the two-dimensional Heston process. The projection is the key for deriving efficient, analytical approximations to European option prices in such models. This is a further development of the method of Markovian projection previously used for projecting on the displaced-diffusion process (with skew but without smile). The method is derived in a generic form and has a wide range of suitable applications. Examples for spread and basket options are given.

Markovian projection, stochastic volatility, Shifted Heston model, Gyongy lemma, index options, Heston basket options, Heston spread options

Abstract:
We develop a systematic approach to Markovian projection onto an effective displaced diffusion, and work out a set of computationally efficient formulas valid for a large class of non-Markovian underlying processes. The generic derivation is followed by applications, including the calculation of FX options in cross-currency models and swaption pricing in LIBOR Market Models, where we are able to recover in an unambiguous way many known analytical approximations and derive several new ones.

Markovian projection, displaced diffusion, Cross Currency Libor Market Models with Skew, FX volatility skew, LMM swaption formula

Abstract:
We revisit the cross-currency LIBOR Market Model armed with the technique of Markovian projection. We derive an efficient approximation for FX options and show how the FX skew can be modeled consistently with the interest rate skew in a common multifactor model.

Cross Currency Libor Market Models, FX volatility skew, Markovian projection

Abstract:
In this paper, we develop a series of approximations for a fast analytical pricing of European constant maturity swap (CMS) products, such as CMS swaps, CMS caps/floors, and CMS spread options, for the LIBOR Market Model (LMM) with stochastic volatility. The derived formulas can also be used for model calibration to the market, including European swaptions and CMS products.

The first technical achievement of this work is related to the optimal calculation of the measure change. For single-rate CMS products, we have used the standard linear regression of the measure change, with optimally calculated coefficients. For the CMS spread options, where the linear procedure does not work, we propose a new effective 'non-linear' measure change technique. The fit quality of the new results is confirmed numerically using Monte Carlo simulations.

The second technical advance of the article is a theoretical derivation of the generalized spread option price via two-dimensional Laplace transform presented in a closed form in terms of the complex Gamma-functions.

LMM, stochstic volatility, CMS swaps, CMS caps, CMS spread option, Markovian Projection

Abstract:
Markovian Projection is an optimal approximation of a complex underlying process with a simpler one, keeping essential properties of the initial process. The Heston process, as the Markovian Projection target, is an example.

In this article, we generalize the results of Markovian Projection onto a Heston model to a wider class of approximating models, a Heston model with displaced volatility. As an important application, we derive an effective approximation for FX/EQ options for the Heston model, coupled with correlated Gaussian interest rates.

The main technical result is an option evaluation for correlated Heston/Lognormal processes. Unlike the case of exactly solvable (affine) zero correlation or its uncorrelated displacement generalization,considered by Andreasen, non-trivial correlations destroy affine structure and exact solvability.

Using the powerful technique of Markovian Projection onto a Heston model with displaced volatility, we produce an effective approximation and present its numerical confirmation.

Markovian projection, stochastic volatility, Heston model, Gyongy lemma, Heston/Hull-White correlated hybrid, FX-options approximation

Abstract:
This paper presents a discrete framework on event time grid, for a cross-currency term structure modelling. The discrete model is generic, in the sense that it can link together any single currency model to form a multi-factor cross currency model, provided that it is known (analytically or numerically) in a rolling-spot measure. As an example we present a construction of a Markov-Functional model in this framework. For the discrete cross-currency extension, transitional probabilities between adjacent time slices for the FX process can be explicitly calculated if the respective underlying interest rate models have known transitional probabilities for their states. A calibration of the discrete cross-currency model including FX volatility is also discussed in the paper. Namely, we present an analytical FX-rate volatility approximation and a construction of its efficient numerical solution.

Markov-Functional model, Cross-Currency model, rolling spot measure

Abstract:
We develop a technique of parameter averaging and Markovian projection on a quadratic volatility model based on a term-by-term matching of the asymptotic expansions of option prices in volatilities.

In doing so, we revisit the procedure of asymptotic expansion and show that the use of the product formula for iterated Ito integrals leads to a considerable simplification in comparison with the approach currently prevalent in the literature.

Results are applied to the classic problem of LIBOR Market Model (LMM) swaption pricing. We confirm numerically that the retention of the quadratic term gives a marked improvement over the standard approximation based on the projection on a displaced diffusion.

asymptotic expansion, Markovian projection, skew averaging, quadratic volatility model, LIBOR Market Model, swaption, Wiener chaos