HIGH ORDER SPLITTING METHODS FOR FORWARD PDEs AND PIDEs

Abstract

This paper is dedicated to the construction of high-order (in both space and time) finite-difference schemes for both forward and backward PDEs and PIDEs, such that option prices obtained by solving both the forward and backward equations are consistent. This approach is partly inspired by Andreasen & Huge (2011) who reported a pair of consistent finite-difference schemes of first-order approximation in time for an uncorrelated local stochastic volatility model. We extend their approach by constructing schemes that are second-order in both space and time and that apply to models with jumps and discrete dividends. Taking correlation into account in our approach is also not an issue.

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

“…Forward PIDEs can be derived using techniques proposed in Cont and Bentata (2012); Andersen and Andreasen (2000); Carr (2006). Alternatively, we can exploit the approach which is discussed in Itkin (2015) and for the financial literature is originated by Lipton (2001Lipton ( , 2002, see a literature survey in Itkin (2015). Briefly, the idea is as follow.…”

“…This problem is solved in Itkin (2015) where a splitting finite-difference scheme for the 2D forward equation is constructed which is fully consistent in the above-mentioned sense with the backward counterpart. For the latter, two popular scheme are chosen, namely: Hundsdorfer and Verwer (HV) and a modified Craig-Sneyd scheme, In't Hout and Welfert (2007).…”

“…For the latter, two popular scheme are chosen, namely: Hundsdorfer and Verwer (HV) and a modified Craig-Sneyd scheme, In't Hout and Welfert (2007). Also a consistent forward scheme is proposed in Itkin (2015) for solving the corresponding PIDEs.…”

“…which is required at the first and third steps of Eq.(26). For doing that in Itkin (2015) two forward finite-difference schemes are proposed for the 2D case. The first scheme, which is consistent with the HV scheme is…”

“…The forward analog for another popular backward finite-difference schemea modified Craig-Sneyd (MCS) scheme, see In't Hout and Foulon (2010), can also be found in Itkin (2015).…”