Notes : Black - Scholes - Merton Model ( IEOR 4707 ,
نویسنده
چکیده
denote an increment of the BM (with ds > 0). We also use N(μ, σ2) to denote a normal distribution with mean μ and variance σ2. Recall some of the key properties of BM: (i) B0 = 0; (ii) independent increments, i.e., dBs and dBt are independent, for any s + ds ≤ t; (iii) stationary increments, i.e., dBs follows a normal distribution N(0, ds). Note this last distribution depends only on the length of the increment, not on when it starts (hence, “shift invariant”, or stationary). Also note that the variance is proportional to the length of the increment. The BM is a Markov process, with a continuous trajectory over time. We state without formal proof the following result:
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