We discuss the Heston model [Rev. Financ. Stud., 6 (1993), pp. 327-343] with stochastic interest rates driven by Hull-White (HW) [J. Derivatives, 4 (1996), pp. 26-36] or Cox-Ingersoll-Ross (CIR) [Econometrica, 53 (1985), pp. 385-407] processes. Two projection techniques to derive affine approximations of the original hybrid models are presented. In these approximations we can prescribe a nonzero correlation structure between all underlying processes. The affine approximate models admit pricing basic derivative products by Fourier techniques [P. P. Carr and D. B. Madan, J. Comput. Finance, 2 (1999), pp. 61-73, F. Fang and C. W. Oosterlee, SIAM J. Sci. Comput., 31 (2008), pp. 826-848] and can therefore be used for fast calibration of the hybrid model.

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On the Heston Model with Stochastic Interest Rates

Spread options are a fundamental class of derivative contracts written on multiple assets and are widely traded in a range of financial markets. There is a long history of approximation methods for computing such products, but as yet there is no preferred approach that is accurate, efficient, and flexible enough to apply in general asset models. The present paper introduces a new formula for general spread option pricing based on Fourier analysis of the payoff function. Our detailed investigation, including a flexible and general error analysis, proves the effectiveness of a fast Fourier transform implementation of this formula for the computation of spread option prices. It is found to be easy to implement, stable, efficient, and applicable in a wide variety of asset pricing models.

https://epubs.siam.org/doi/pdf/10.1137/090750421

A Fourier Transform Method for Spread Option Pricing

Given a multi-dimensional Ito process whose drift and diffusion terms are adapted processes, we construct a weak solution to a stochastic differential equation that matches the distribution of the Ito process at each fixed time. Moreover, we show how to match the distributions at each fixed time of functionals of the Ito process, including the running maximum and running average of one of the components of the process. A consequence of this result is that a wide variety of exotic derivative securities have the same prices when the underlying asset price is modeled by the original Ito process or the mimicking process that solves the stochastic differential equation.

Mimicking an It\^{o} process by a solution of a stochastic differential equation

A formula is derived for the ‘effective’ skew in a stochastic volatility model with a time‐dependent local volatility function. The formula relates the total amount of skew generated by the model over a given time period to the time‐dependent slope of the instantaneous local volatility function. A new ‘effective’ volatility approximation is also derived. The utility of the formulas is demonstrated by building a forward Libor model that can be calibrated to swaption smiles that vary across the swaption grid.

Stochastic Volatility Model with Time‐dependent Skew

The introduction of Central Clearing Counterparties (CCPs) in most derivative transactions will dramatically change the landscape of derivatives pricing, hedging and risk management, and, according to the TABB Group, will lead to an overall liquidity impact of about USD 2 trillions. In this paper, we develop for the first time a comprehensive approach for pricing under CCP clearing, including variation and initial margins, gap credit risk and collateralization, showing concrete examples for interest rate swaps. This framework stems from our 2011 framework on credit, collateral and funding costs in Pallavicini et al. (Pallavicini, A., D. Perini and D. Brigo, 2011, Funding Valuation Adjustment: FVA consistent with CVA, DVA, WWR, Collateral, Netting and Re-hypothecation, arxiv.org, ssrn.com). Mathematically, the inclusion of asymmetric borrowing and lending rates in the hedge of a claim, and a replacement closeout at default, lead to nonlinearities showing up in claim dependent pricing measures, aggregation dependent prices, nonlinear Partial Differential Equations (PDEs) and Backward Stochastic Differential Equations (BSDEs). This still holds in presence of CCPs and CSA. We introduce a modeling approach that allows us to enforce rigorous separation of the interconnected nonlinear risks into different valuation adjustments where the key pricing nonlinearities are confined to a funding costs component that is analyzed through numerical schemes for BSDEs. We present a numerical case study for Interest Rate Swaps that highlights the relative size of the different valuation adjustments and the quantitative role of initial and variation margins, of liquidity bases, of credit risk, of the margin period of risk and of wrong-way risk correlations.

Nonlinear consistent valuation of CCP cleared or CSA bilateral trades with initial margins under credit, funding and wrong-way risks

In the current low-interest-rate environment, extending option models to negative rates has become an important issue. This paper describes one such extension of the widely used SABR model. We stress that our solution is more natural and attractive than the shifted SABR.An exact formula is derived for the option prices in the case of zero correlation between the rate and its volatility. For nonzero correlation, a mapping procedure onto a mimicking zero-correlation model is applied. Analytical results for the suggested free-boundary SABR model are compared with Monte Carlo simulations.

The Free Boundary SABR: Natural Extension to Negative Rates

In this paper, we present advanced analytical formulas for SABR model option pricing. The first technical result consists of a new exact formula for the zero correlation case. This closed form is a simple 2D integration of elementary functions, particularly attractive for numerical implementation. The second result is an effective approximation of the general correlation case. We use a map to the zero correlation case having a nice behavior on strike edges. The map formulas are easily implemented and do not contain any numerical integration.These formulas are important in volatility surface construction and CMS products replication because they provide correct behavior for far strikes and reduced approximation error. The latter is also helpful for dynamic SABR models.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID2141691_code340600.pdf?abstractid=2026350&mirid=1&type=2

Advanced Analytics for the SABR Model

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.

https://papers.ssrn.com/sol3/Delivery.cfm?abstractid=937860

Markovian Projection onto a Displaced Diffusion: Generic Formulas with Applications

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.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1106223_code340600.pdf?abstractid=1106223&rulid=286840&mirid=1

Markovian Projection to a Displaced Volatility Heston Model

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.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1352606_code340600.pdf?abstractid=1352606&rulid=5209342&mirid=4

Alexandre Antonov

Papers

The Free Boundary SABR: Natural Extension to Negative Rates

Advanced Analytics for the SABR Model

Markovian Projection onto a Displaced Diffusion: Generic Formulas with Applications

Markovian Projection to a Displaced Volatility Heston Model

Analytical Formulas for Pricing CMS Products in the LIBOR Market Model with the Stochastic Volatility