stochastic.226

# stochastic.226 - CHAPTER 22 Summary of Arbitrage Pricing...

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Unformatted text preview: CHAPTER 22. Summary of Arbitrage Pricing Theory 225 k f P If we introduce a probability measure I under which Sk is a martingale, then martingale, regardless of the portfolio used. Indeed, Xk k will also be a     f f IE Xk+1 F k = IE Xk + k Sk+1 , Sk F k k+1 k k+1 k Xk +  IE  Sk+1 F  , Sk : f = k k Suppose we want to have must have X2 = V , where V k+1 | is some z k =0 k F 2-measurable random variable. Then we 1 X = X1 = IE  X2 F  = IE  V F  ; f f 1 1 1+r 1 1 2   X1  V 2 f f = IE : X0 = X0 = IE 0 To ﬁnd the risk-neutral probability measure 1 2 f IP under which Skk is a martingale, we denote f IP f! k = H g, q = IP f!k = T g, and compute ~ f   f ~ ~ IE Sk+1 F k = pu Sk + qd Sk k+1 k+1 k+1 1 pu + q d Sk : = 1+r ~ ~ k We need to choose p and q so that ~ ~ p= ~ pu + qd = 1 + r; ~ ~ p + q = 1: ~ ~ The solution of these equations is r ~ p = 1 + , , d ; q = u , 1 + r : ~ u d u,d 22.2 Setting up the continuous model Now the stock price S t; 0  t  T , is a continuous function of t. We would like to hedge along every possible path of S t, but that is impossible. Using the binomial model as a guide, we choose 0 and try to hedge along every path S t for which the quadratic variation of log S t accumulates at rate 2 per unit time. These are the paths with volatility 2. To generate these paths, we use Brownian motion, rather than coin-tossing. To introduce Brownian motion, we need a probability measure. However, the only thing about this probability measure which ultimately matters is the set of paths to which it assigns probability zero. ...
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