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.

Download pdf: Understanding N(d1) and N(d2): Risk-Adjusted Probabilities in the Black-Scholes Model

SiloHere’s VBA and an Excel spreadsheet to price a European option and calculate the greeks: http://investexcel.net/2888/black-scholes-greeks-vba/

Benjamin RoscoeUnderstanding N(d1) and N(d2):

Risk-Adjusted Probabilities in the

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

LTNPost authorThe 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.