LSV models with stochastic interest rates and correlated jumps

Abstract

Pricing and hedging exotic options using local stochastic volatility models drew a serious attention within the last decade, and nowadays became almost a standard approach to this problem. In this paper we show how this framework could be extended by adding to the model stochastic interest rates and correlated jumps in all three components. We also propose a new fully implicit modification of the popular Hundsdorfer and Verwer and Modified Craig-Sneyd finite-difference schemes which provides second order approximation in space and time, is unconditionally stable and preserves positivity of the solution, while still has a linear complexity in the number of grid nodes.

Pricing foreign exchange options under stochastic volatility and interest rates using an RBF–FD method

“…The PM method does not include any special consideration to preserve positivity of the solution, as this was done, e.g., in [26]. Therefore, to verify it our results are For the European vanilla Put option 2, the results are brought forward in Table 5.7.…”

“…Having in mind that a majority of the FX options are exotic, nowadays it is a common approach for the FX models to treat the underlying domestic and foreign interest rates (IRs) to be stochastic as well as volatility of the FX rate itself, [23,34,43]. However, because of this complexity, pricing options under such a model requires using numerical methods, in more detail see [26,38,47] and references therein. While Monte Carlo methods are traditionally slow, the FD methods suffer from the curse of dimensionality and the FE approaches requires an extensive triangulization.…”

“…Following a standard no-arbitrage argument, the pricing PDE could be derived for the FX-HHW model and reads, [18,26]:…”

Modelling stochastic skew of FX options using SLV models with stochastic spot/vol correlation and correlated jumps

“…To resolve this problem, in Itkin (2016b) it is proposed to replace the explicit step for the mixed derivatives with the implicit one. It was proved there that the scheme proposed for calculating of the mixed derivative terms guarantees the positivity of the solution, while the whole scheme becomes fully implicit.…”

“…By construction of the HV scheme, the first step in Eq. (32) has to be solved with the accuracy O(∆τ ), again see discussion in Itkin (2016b). With this accuracy, the previous equation can be factorized as V 0 (τ n−1 ) = e ∆τ F n−1 Sv e ∆τ F n−1 Sr e ∆τ F n−1 vr V (τ n−1 ), or using splitting…”