In other words the no arbitrage price of the call option at time t T is found

In other words the no arbitrage price of the call

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In other words, the no-arbitrage price of the call option at time t < T is found by taking the risk-neutral discounted expectation of the payoff, conditioned on knowledge of the stock price S t at that time. The pricing function V ( t, S t ) for any European with payoff h ( S T ) is simply given by V ( t, S t ) = IE Q { e - r ( T - t ) h ( S T ) | S t } . 57
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13.6 Filtration It is convenient to have a notation for “what is known” at each time. Let ( F t ) be a filtration , so that F t represents the information known at time t . Typically, it might be the information we acquire by observing the Brownian motion W Q up till time t . In this case, this is equivalent to what we know by observing the stock S up till time t . Without giving a precise definition of the sets F t , which represent information, it is clear that knowing F t is to know much more than just the value of S t , since F t contains path information like the maximum or minimum of W Q up to time t and so on. 13.7 Markov Process We say that ( S t ) is a Markov process if for any N < and times 0 t < t 1 < t 2 < · · · < t N < and any (nice) function g IE Q { g ( S t 1 , S t 2 , · · · , S t N ) | F t } = IE Q { g ( S t 1 , S t 2 , · · · , S t N ) | S t } . This means that the conditional expectation of any function of S at a finite number of times in the future given the past up to time t is equal to the conditional expectation given only the value S t . Another way of saying this is that a Markov process has the property that its future is independent of its past given its present. Or, the statistics of its future evolution depends only on its current state and not how it got there. For any stochastic process S defined by an SDE of the form dS t = b ( S t ) dt + a ( S t ) dW Q t , for some functions a and b , one can see that S is a Markov process because its change depends only on where it is and the Brownian increment dW t , which is independent of the past. Therefore the geometric Brownian motion (39) is a Markov process. As a consequence, we can write the price V ( t, S t ) for any European with payoff h ( S T ) as V ( t, S t ) = IE Q { e - r ( T - t ) h ( S T ) | S t } = IE Q { e - r ( T - t ) h ( S T ) | F t } . 13.8 Connection with Martingales Let ( X t ) be a stochastic process. Then X is a martingale (with respect to the filtration ( F t ) t 0 ) if IE {| X t |} < for all t > 0, and it satisfies the martingale property IE { X t | F s } = X s , for all t > s 0 . In other words IE { X t - X s | F s } = 0, so a martingale is not expected to go anywhere (on average) between times s and t . Returning to the Black-Scholes model, in the risk-neutral world the stock price at time t is related to its time zero value by S t = S 0 e ( r - 1 2 σ 2 ) t + σW Q t , 58
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at any time t , where W Q t ∼ N (0 , t ). More generally, given S u for any time u < t , we can write S t = S u e ( r - 1 2 σ 2 )( t - u )+ σ ( W Q t - W Q u ) , where W Q t - W Q u ∼ N (0 , t - u ). Let ( F t ) be the filtration generated by observing the Brownian motion W Q (or equiv- alently, the stock price path). So, “given F t ” or “conditioned on F t ” means we know the path the stock has taken up till time t . This contains much more information than knowing just the value S t . For example we would know the maximum price so far, the minimum, the
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  • Fall '11
  • COULON
  • Dividend, Mathematical finance, Black–Scholes

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