No short sale constraints, i.e., we can take any long and short positions in all traded assets. There is no bid-ask spread, i.e., we sell and we buy at the same price. There is no counterparty risk.
Financial Engineering with Stochastic Calculus I 27 / 30 The binomial model One-period binomial model: Solution Simplest example: One-period binomial model We assume that prices for the money market account B t and stock S t are quoted at times t = 0 (today) and t = 1 (option maturity). Suppose B 0 = B 1 = 1 , S 0 = 100 , and S 1 is a random variable taking values S 1 = 115 or S 1 = 90 each with probability p = 1 2 . Consider a call with strike K = 100 . Value at time 0 ? Naive response: Expectation of the payoff ( S 1 - 100) + ? Gives E ( S 1 - 100) + = 1 2 (115 - 100) + + 1 2 (90 - 100) + = 7 . 5 Turns out to be too expensive : A suitable investment strategy can generate the option payoff out of less capital! Indeed, suppose at time 0 we invest into η shares of bond and Δ shares of stock, and require that at time 1 η + Δ · 115 = (115 - 100) + , (2.1) η + Δ · 90 = (90 - 100) + . (2.2) We find Δ = 0 . 6 , η = - 54 . This requires η + Δ · 100 = 6 initial capital and does the same job as the call.
Financial Engineering with Stochastic Calculus I 28 / 30 The binomial model One-period binomial model: Solution The initial capital C 0 = 6 is the fair value at time 0 of the option in the above example. Any price C 0 6 = 6 for the option would introduce an arbitrage opportunity into the market: If C 0 > 6 , the seller could make a guaranteed profit of C 0 - 6 by selling the option and simultaneously employing the above investment strategy. If C 0 < 6 , the buyer could make a guaranteed profit of 6 - C 0 by buying the option and simultaneously employing the reverse strategy ( Δ = - 0 . 6 , η = 54 ). Such arbitrage opportunities (possibility of a riskless profit without net investment of capital) are unrealistic in most markets. Absence of arbitrage is a fundamental concept in financial market models and in the theory of derivative pricing.
- Fall '09