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 and hedging exotic options using local stochastic volatility (LSV) models drew a serious attention within the last decade, and nowadays became almost a standard approach to this problem. For the detailed introduction into the LSV among multiple available references we mention a recent comprehensive literature overview in Homescu (2014) Despite LSV has a lot of attractive features allowing simultaneous pricing and calibration of both vanilla and exotic options, it was observed that in many situations, e.g., for short maturities, jumps in both the spot price and the instantaneous variance need to be taken into account to get a better replication of the market data on equity or FX derivatives. This approach was pioneered by Bates (1996) who extended the Heston model by introducing jumps with finite activity into the spot price (a jump-diffusion model). Then Lipton (2002) further extended this approach by considering local stochastic volatility to be incorporated into the jump-diffusion model (for the extension to an arbitrary Lévy model, see, e.g., Pagliarani and Pascucci (2012)). Later Sepp (2011a,b) investigated exponential and discrete jumps in both the underlying spot price S and the instantaneous variance v, and concluded that infrequent negative jumps in the latter are necessary to fit the market 1 arXiv:1511.01460v3 [q-fin.CP]

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 Table 5.4: The RE ǫ obtained in the experiment 3 as a function of the number of nodes in various directions, and the elapsed time.…”

“…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]:…”