Pricing and Hedging Options in Discrete Time with Liquidity Risk
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Different derivative securities, including European options, are very popular and widely used in forms of exchange-traded instruments or over-the-counter products. For practical purposes the European options are often priced using analytic solution to the Black-Scholes formula. Hedging, according to the Black-Scholes model, is accomplished via the construction of dynamically rebalanced replicating portfolio. However, the model makes several critical assumptions. I extend the Black-Scholes model by relaxing the assumption of no trading costs and considering the market liquidity risk for the underlying asset. Liquidity risk is understood as the effect of the trade size on the price of the underlying asset. I use stochastic supply curve to model liquidity risk. The problem is to hedge a European option in the presence of the market liquidity risk for an underlying asset. One hedges with the underlying, as the option price depends on the price of the underlying asset. The underlying asset has market liquidity risk; thus, studying the impact of market liquidity risk is important for devising more effective and efficient option hedging algorithms. The main contributions of the thesis arise from the investigation of mathematical techniques for hedging and pricing of European options in discrete time with liquidity risk. First, I study delta hedging in Chapter 3. I show L2 convergence of the replicating trading strategy payoff to the option payoff. In other words, the optimal strategy minimizes the mean squared replication error. I also show that for European put and call options with varying trading times the recommendation is to trade closer to expiry as the spot price of the underlying asset deviates from the strike price. Then I apply the local risk-minimizing hedge in Chapter 4. This time the optimal strategy minimizes the conditional mean squared hedging error. I prove the existence of the local risk-minimizing trading strategy and characterize its structure.