ABSTRACT
This chapter considers an economy with two securities: a single tradable risky asset, namely, a stock, and a money market (bank) account or a bond. All appropriately discounted tradable assets are martingales under an equivalent martingale (risk-neutral) measure. In particular, by using a self-financing replication strategy, one arrives at the risk-neutral pricing formula that expresses the current price of any attainable European-style derivative, including contracts with path-dependent payoffs, as a conditional expectation (under the risk-neutral or equivalent martingale measure) of the discounted payoff. The chapter focuses on risk-neutral pricing and hedging of options. It considers a derivation of the risk-neutral pricing formula for a European option with arbitrary piecewise linear payoff, assuming the standard geometric Brownian motion model for the stock. The chapter discusses the application of risk-neutral pricing to path-dependent European options whose payoff is dependent on the underlying stock price history over the lifetime of the contract.
