In this thesis, we study European option pricing and hedging problem under two incomplete market conditions: (1) with liquidity shocks, and (2) with regime-switching transaction costs. We consider market illiquidity as the inability to trade timely and transaction costs as proportional to the trading (buy/sell) amounts. Working within a Markovian regime-switching setting, we deem the market to have two regimes - one a regular trading regime and the other an illiquid no-trading regime, though in the transaction costs model, we generalize the two regimes to be both regular trading regimes but with different fundamental parameters. The pricing problem is solved by extending the Black-Scholes model in each regime with an extra accommodation for regime switching possibilities. This thesis consists of four chapters.