In order to reconstruct the cube, a set of parameters ( β, σ, α, ρ ) needs to be mapped to each unobserved couple ( T, t ). The previous section explained how to calibrate the parameters of the shifted SABR model for each pair ( T, t ), where we have at least one observation of swaptions prices.

As explained in Johnson and Nonas ( 2009), in addition to the no call spread and butterfly spread arbitrage, the interpolated swaption prices must also respect an additional arbitrage condition, named “in-plane triangular”. In order to interpolate with respect to maturity and tenor, it is possible to use functional interpolation directly on swaptions prices, or in a more elaborate manner, to use functional interpolation on the parameters of stochastic models. When cap market data are not available, parameters of the SABR model are taken as the closest SABR parameters that are calibrated with quotes. The validity of both approaches decreases when the tenor of the considered swaptions increases. The second, discussed in Skantzos and Garston ( 2019), consists of mapping underlying caplet volatilities to the swaption volatility by making an assumption in relating the moneyness of the swaption to the moneyness of the underlying caplets. The first approach consists of using caplet SABR parameters as a basis for the calibration of the swaption SABR parameters (see, e.g., Hagan and Konikov ( 2004)). When there is only one quoted strike observation, for a given tenor and maturity, three methods are usually employed in the industry to interpolate the swaption smile. However, this construction method requires several observed quotes at different strikes for each considered pair of tenors and maturities. The most common method for interpolating in an arbitrage-free way between quoted strikes, as well as extrapolating beyond, is to use the SABR model. A distinction needs to be made between functional interpolation, which is for example polynomial interpolation, and stochastic interpolation, where stochastic models such as SABR and CEV models are used. In the literature, several ways of interpolating the missing prices or volatilities of the swaption cube have been introduced. Using this approach, they achieved to produce arbitrage-free interpolated prices, and local volatility surfaces, with their uncertainties. A Gaussian process regression methodology, constrained on the no-arbitrage conditions for equity options, has been introduced in Chataigner et al. For these volatility surfaces, absence of arbitrage is guaranteed if there is no call spread, butterfly spread and calendar spread arbitrage opportunities. A number of machine learning methods for learning pricing functions have emerged (see, e.g., Ruf and Wang ( 2019) for an in-depth literature review). Finally, the GP approach is able to quantify in- and out-of-sample uncertainty through Hamiltonian Monte Carlo simulations, allowing for the computation of model risk Additional Valuation Adjustment (AVA).įor equity options, the volatility surface, indexed by the maturity and the strike of the derivative, is a topic that has been thoroughly studied. In addition, we show that SABR calibration is better when using the GP cube output as new observations (in terms of predictive error and absence of arbitrage). The constrained GP approach provides better prediction results compared to the SABR approach. In addition, we compare the performance of the GP approach with the SABR model, which is applied to a data set of payer and receiver out-of-the-money (OTM) swaptions. We empirically demonstrate that considering a finite-dimensional weaker form of this condition is enough for the GP to generate swaption cubes with a negligible proportion of violation points, even for a small training set.

The cube is free from static arbitrage along the tenor and maturity directions if swaption prices satisfy an infinite set of in-plane triangular inequalities. Based on the possibly noisy observations of swaption prices, the proposed ‘constrained’ GP regression approach is proven to be arbitrage-free along the strike direction (butterfly and call-spread arbitrages are precluded on the entire 3D input domain). In this paper, we introduce a 3D finite dimensional Gaussian process (GP) regression approach for learning arbitrage-free swaption cubes.

Four dimensional cube