For example we would know the maximum price so far the minimum the average we

For example we would know the maximum price so far

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. For example we would know the maximum price so far, the minimum, the average, we can answer questions like if the price has yet crossed some barrier, and so on. It follows that IE Q { S t | F u } = e r ( t - u ) S u , (40) which is what we expect: the stock grows on average at rate r under Q . Now define ˆ S t = e - rt S t , the discounted stock price. Another way to write (40) is IE Q { e - rt S t | F u } = e - ru S u , or IE Q { ˆ S t | F u } = ˆ S u . This shows that the discounted stock price process is a martingale under Q . That is we price derivatives as expected discounted payoffs pretending that discounted stock prices are martingales. The no arbitrage or fair price is closely connected to martingales because we price as though there is no identifiable drift to the stock price (after discounting). If there were a drift other than the time-value of money, it would be eliminated by arbitrageurs. The connection is even deeper because absence of arbitrage is equivalent to the existence of a risk-neutral world in which the discounted prices of all traded securities are martingales. (This is sometimes known as the Fundamental Theorem of Asset Pricing). To illustrate this consider the price of any European derivative V t = e - r ( T - t ) IE Q { h ( S T ) | F t } . (We have replaced conditioning by S t with conditioning on F t ; this makes no difference in this case, because of the Markov or efficient markets property of this stock price model). Then the discounted price is ˆ V t = e - rT IE Q { h ( S T ) | F t } . This is a martingale (with respect to the information generated by observing the stock price) because for any u < t IE Q { ˆ V t | F u } = IE Q { e - rT IE Q { h ( S T ) | F t } | F u } = IE Q { e - rT h ( S T ) | F u } = ˆ V u using the iterated expectations property of conditional expectations . This is just that the (weighted) average at time u of the (weighted) averages taken at time t > u of quantities 59
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at time T > t is the (weighted) average of the quantities at T from time u (missing the intermediate averaging at t ). Hence ( ˆ V t ), the discounted price of the derivative is also a martingale (with respect to the information generated by observing the stock price) under Q . The result that prices of traded securities are martingales after discounting in the risk- neutral world allows us to generalize the derivative pricing formula to the price of any (exotic) derivative security (barring early exercise features – we do not consider American options here). These are characterized by a random variable H whose value (payoff) is revealed at time T . Let V t be the price of this derivative at times t T . Then as ( e - rt V t ) is a Q -martingale, IE Q { e - rT V T | F t } = e - rt V t , and since V T = H , we obtain V t = e - r ( T - t ) IE Q { H | F t } . (41) This then tells us that, in the Black-Scholes model, pricing any derivative (except Americans) is just calculation of the discounted expected payoff in the risk-neutral world. Hedging of non-European claims is generally more complicated (in many cases the hedging ratio is just the Delta of the price, but it may not be that simple in other cases), and we do not discuss that here.
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  • Fall '11
  • Dividend, Mathematical finance, Black–Scholes

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