Computational Finance – The Martingale Measure and Pricing of Derivatives

At t = 0, two instruments are available. Any amount (even fractional) of either instrument may be sold or puschased at the specified market price i.e., arbitrary short or long positions are allowed. A risk free asset or bond, B, and a stock, S. At t = 0 (the first period), the bond is worth B(0), and, the stock is worth S(0) = 100. At t = T (the second period), the economy can be in one of two states. In both states, the bond is valued at B(T) and hence is risk free. In the first state the stock is valued at S(T ) = 100 and in the second state, the stock is valued at S(T ) = 150. Suppose that Pup is the probability that the market goes up. We have thus completely specified the market dynamics for our simple economy. In this simplified economy, it is clear that one can guarantee an amount B(T ) at t = T by investing B(0) in the bond at t = 0. The risk free discount factor is defined by