Derivative Securities , Fall 2010

The dynamic replication strategy of Black and Scholes is important enough that it is worth repeating from last week. Recall the setup. From day k − 1 to day k, the stock (risky asset price) either goes up Sk−1 → Sk = uSk or goes down Sk = dSk−1 (recall that we actually did not necessarily need u > 1 or d < 1, but it is convenient to think of u as up and d as down.) The replicating portfolio is a dynamically rebalanced combination of stock and cash. At time t0 = 0 the value is f0(S0), which is known today. At time tk, the value will be fk(Sk), which is not known today. More precisely, the numbers fk(s) are known today for all possible values of Sk, but we do not know Sk. At the expiration time, the value will be fn(Sn) = V (Sn). No matter which value Sn takes, the value of the portfolio at time tn = T will be exactly the payout of the option. The replicator will be able to satisfy the option holder by liquidating the portfolio. Repeating from last week, there also is an arbitrage argument. If the option is not selling for f0(S0), the arbitrager can buy or sell the option and make a guaranteed profit by replicating the option and keeping the price difference. We review in more detail the rebalancing step on day k. The replicator ended day tk−1 with ∆k−1 units of stock and Mk−1 “units” of cash (bond). The value of the stock position was Xk−1 = ∆k−1Sk−1. The total value of the portfolio was Xk−1 +Mk−1. Let us assume that this was equal to the planned value fk−1: ∆k−1Sk−1 + Mk−1 = fk−1(Sk−1) .