Sample Question #235 (finance – option pricing)

The Black-Scholes option pricing formula involves two terms,

*N*(*d*_{1}) and*N*(*d*_{2}). What are the interpretations of these two terms?[Hint: when an interviewer asks for an interpretation, he or she is asking you to explain, in plain English, what the term means in the context of the framework]

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ANSWER

This question is an excellent example of the kind of unpleasant levels of details an interview might go into when asking about something "everybody" is supposed to know well.

First of all, you should now that the N() function here is the normal cumulative distribution function (cdf), so it stands for a probability, i.e., N(d) = Pr(x<=d) where x is a normally distributed variable. [Bonus question: what’s x here?]

For a call option, N(d2) is simply the probabiliy that the at-expiration stock price will be greater than the strike price, i.e., Pr(P_T > strike). N(d1) is more complicated. It turns out it does not represent a real probability, but rather more of a discount factor in arriving at the expected value of the underlying stock when P_T > strike.

[Bonus question: what are the interpretations of d1 and d2 themselves?]