Numerical Simulation of Exchange Option with Finite Liquidity:
Controlled Variate Model
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In this paper we develop numerical pricing methodologies for European style
Exchange Options written on a pair of correlated assets, in a market with
finite liquidity. In contrast to the standard multi-asset Black-Scholes
framework, trading in our market model has a direct impact on the asset's
price. The price impact is incorporated into the dynamics of the first asset
through a specific trading strategy, as in large trader liquidity model.
Two-dimensional Milstein scheme is implemented to simulate the pair of assets
prices. The option value is numerically estimated by Monte Carlo with the
Margrabe option as controlled variate. Time complexity of these numerical
schemes are included. Finally, we provide a deep learning framework to
implement this model effectively in a production environment.