# Abstracts

**Discrete Barrier Options. Exact Geometric Solution.**

*Alexander Skabelin (One of the top global investment banks, USA)*

Wednesday June 4, 11:30-12:00 | session 4.5 | Options, Futures | room G

Barrier options are some of the most actively traded exotic derivatives. However, simple analytical solution for option prices exists only in the approximation of continuously monitored barrier. In reality all barriers are monitored discretely but no simple analytical solution for barrier options with discrete monitoring has been found so far even in the standard Black-Scholes framework. We discovered exact analytical solution for discrete barrier options that does not require numerical inversion of complex z-trasforms and has significantly simpler form than anything reported in the literature. The novel approach reveals geometric nature of discrete barrier options for the first time.

**Multivariate Subordination using Generalised Gamma Convolutions with Applications to V.G. Processes and Option Pricing**

*Boris Buchmann (Australian National University, Australia)*

Wednesday June 4, 12:00-12:30 | session 4.5 | Options, Futures | room G

We unify and extend a number of approaches related to constructing multivariate Variance-Gamma (V.G.) models for the pricing of options on multiple assets. An overarching model is derived by subordinating multivariate Brownian motion to a subordinator from the Thorin (1977) class of generalised Gamma convolution subordinators. A special case is the well-known Madan-Seneta V.G. model, but our multivariate generalization is considerably wider, allowing in particular for processes with unbounded variation and a variety of dependencies between the underlying processes. Multivariate classes developed by Pérez-Abreu and Stelzer (2012) and Semeraro (2008) are also submodels.

We draw out some interesting connections in this respect. The new models are shown to be invariant under Esscher transforms, and quite explicit expressions for canonical measures (and transition densities in some cases) are obtained, which permit option pricing using PIDEs or tree based methodologies. We illustrate with best-of and worst-of European and American options on two assets.

**FVA via semi-replication strategies – pre-default costs vs post-default windfalls**

*Christoph Burgard (Barclays, UK)*

Wednesday June 4, 12:30-13:00 | session 4.5 | Options, Futures | room G

We present a unified model for a generalised bilateral CVA with funding and collateral costs derived using semi-replication. The model is generalised in two ways. First it is formulated in terms of general boundary conditions that include, amongst others, bilateral close-outs with one-way and two-way CSAs, extinguishers and set-offs. Second it makes only weak assumptions on the own bonds available to the issuer and how these bonds can be traded, and we prove that any valuation adjustment that includes counterparty risk, funding and collateral is given by the classical (generalised) bilateral CVA plus a funding cost adjustment. This term is shown to be equal to the expected value of the issuer hedge error at own default. This proves that each issuer bond / strategy combination has it unique set of economic valuation adjustment formulas, resulting in non-symmetric prices.

We also show how funding aware close-outs can be incorporated into the framework and discuss the impact of derivative assets and funding liabilities on the balance sheet of the derivative issuer and theoretical and practical effects thereof.

Finally we present three concrete models with different assumptions on issuer bonds and trading strategies and give their specific CVA, DVA, FCA, FVA and collateral adjustments. We show that under certain assumptions set-off close-outs result in vanishing funding cost adjustments and symmetric prices. The models are illustrated by practical examples.