# “Understanding N(d1) and N(d2): Risk-Adjusted Probabilities in the Black-Scholes Model”

By Lars Tyge Nielsen

Finance 14 (1993), 95-106

### Abstract

This paper uses risk-adjusted lognormal probabilities to derive the Black- Scholes formula and explain the factors N(d1) and N(d2). It also shows how the one-period and multi-period binomial option pricing formulas can be restated so that they involve analogues of N(d1) and N(d2) which have the same interpretation as in the Black-Scholes model.

## 5 thoughts on ““Understanding N(d1) and N(d2): Risk-Adjusted Probabilities in the Black-Scholes Model””

1. Benjamin Roscoe

Understanding N(d1) and N(d2):
Black-Scholes Model 1
The expected future
payoff is

EC1T = −XP{ ST > X}.
where P is the risk-adjusted probability, and so the value is
−e−rτ XP{ST > X}.

This implies that e−rτ is equal to 1.

I believe this is incorrect. Please tell me what I am missing.
What I am actually looking for is the probability the option will be exercised before it expires. I have some literature that states that probability is equal to delta, one of the Greeks associated with the Black Scholes model. Can you tell me if delta is indeed that probability?
Yours truly,
Benjamin Roscoe PhD

2. LTN Post author

The expected future payoff (to the part which consists in paying the exercise price) is $$-XQ(S_T > X)$$ as you say (where I use $$Q$$ for the risk adjusted probability). Multiply by the discount factor $$e^{-rT}$$ and you get the present value. The discount factor is not one. The present value is different from the expected future payoff.

The (risk adjusted) probability that the option will be in the money and will be exercised when it expires is $$-XQ(S_T > X) = N(d_2)$$. The delta of the option is $$N(d_2)$$, which is different from $$N(d_1)$$. I tried to explain this in the paper.

3. db

This is an excellent paper. I believe the proof in section 8 “Computing the Call Value” could be further simplified, but this paper was written in 1992. Has there been any further literature on simplification and intuitive approaches to Black Scholes since then?

4. db

p.6 “The present value of unconditionally receiving the stock at time T is obviously equal S, the current stock value.” Is this obviously true? Suppose the stock is one side of a currency pair, e.g. pounds sterling, priced in dollars. The principle of sufficient reason suggests that there should be no more bias in the expected return in dollars when holding pounds, as when holding dollars, returned price in pounds.

So the lognormal distribution should not have a bias, and so (assuming £/\$ exchange rate= 1) the present value (in dollars) of unconditionally receiving 1 pound sterling should be greater than 1, and similarly for the present value (in sterling) of unconditionally receiving 1 dollar. Both will be greater than 1.