Saturday, July 2, 2016

black scholes - Understanding $N(d_1)$ and how to use the stock itself as the numeraire?


Assume the stock price follows a geometric Brownian motion Then in Black-Scholes pricing model, $N(d_2)$ is the risk-neutral probability that the option expires in-the-money. However, it is said that $N(d_1)$ is also the probability that the option expires in-the-money under the measure that uses the stock itself as the numeraire.


I understand that risk-neutral measure uses discounted stock price $\frac{S}{B}$ as the numeraire, but how do you use the stock itself as the numeraire?




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