The objective of this thesis is to provide a simulations-free approximation to the price of multivariate derivatives and for the calculation of risk measures like Value at Risk (VaR). The first chapters are dedicated to the pricing of multivariate derivatives. In particular we focus on multivariate derivatives under switching regime Markov models. We consider the cases of two and three states of the switching regime Markov model, and derive analytic expressions for the first and second order moments of the occupation times of the continuous-time Markov process. Then we use these expressions to provide approximations for the derivative prices based on Taylor expansions. We compare our closed form approximations with Monte Carlo simulations. In the last chapter we also provide a simulations-free approximation for the VaR under a switching regime model with two states. We compare these VaR estimations with those obtained using Monte Carlo.
Taylor expansions on Lefschetz thimbles
