Lars Tyge Nielsen provides an interpretation of N(d1) and N(d2) and an explanation behind the difference between N(d1) and N(d2) under the Black Scholes Model. He does this by considering the value of European call option on a stock which pays no dividends prior to the expiry date of the option as given by the following formula:
C= SN(d1) – Xe-rtN(d2)
Where C is value of the European call option
S is the current value of the stock price
X is the exercise price
t is the remaining time to expiry
r is the risk free rate of interest
N(.) is the cumulative standard normal distribution, the risk adjusted probabilities.
To understand this process the payoff of the call option is divided into 2 components, the payment of exercise price and the receipt of stock. Both of these are dependent on the option being exercised which will only occur if the option is in the money or alternatively when the stock price on the exercise date, ST, rises above the exercise price, X, i.e.
The expected value and present value of each component is then calculated:
Payment of Exercise Price and N(d2)
The expected value of payment of the exercise price is exercise price times probability of stock price exceeding exercise price (probability of exercise):
The present value of the expected value of payment of the exercise price is determined by discounting this expected value using the risk free interest rate over the time remaining to expiry of
-X* P(ST>X)* e-rt
Hence comparing this with the second portion of the call option value equation above,
-X* P(ST>X)* e-rt = – Xe-rtN(d2),
we see that N(d2) = P(ST>X), i.e. N(d2) is the risk adjusted probability that the option will be exercised.
Receipt of stock and N(d1)
Explanation of N(d1) is a bit more complex. We begin with the expected value of the contingent receipt of stock.
The expected value of the receipt of stock is contingent on exercise of the option. It is therefore the product of the conditional expected value of the receipt of ST given that
exercise has occurred times the probability of exercise:
Statistically this is written as
This equation can also be written as follows:
Note that the first term in this equation is a conditional expectation. It is the expected value of ST given that we are now only considering those future values of ST
which exceed the exercise price X. If this constraint were not added we would have E(ST)= Sert the unconditional expectation of ST. Given the
conditionality therefore E(ST|ST>X) will always be greater than E(ST). We may understand this concept through the following simple probability
A six sided fair die is rolled. The probability of rolling 4 is 1/6. Now suppose we have additional information that tells us that the number rolled is greater than 3. In this instance what is the
probability of a 4 having being rolled given that the number rolled is greater than 3. The conditional probability works out to 1/3 > 1/6. The conditional probability is higher because we are
only considering those outcomes that exceed three in our calculation.
In a similar fashion E(ST|ST>X) > E(ST) because the expected value will only consider those stock prices which exceeds the exercise price in
the calculation of the expectation. Hence E(ST|ST>X)* N(d2) > E(ST)* N(d2)=
Let us now consider the present value of this expected value by discounting it with the risk free rate over the time remaining to option expiry. We have:
E(ST|ST>X)* N(d2)* e-rt > E(ST)* e-rt*N(d2)= S*N(d2)
Comparing the left hand side of this inequality with first portion of the Black Scholes equation for the call option, SN(d1), we have:
SN(d1)= E(ST|ST>X)* N(d2)* e-rt > E(ST)* e-rt*N(d2) = S*N(d2)
In other words N(d1) ensures that the discounted expected value of the contingent stock price received on exercise will be greater than this current value of stock.
Difference between N(d1) and N(d2)
SN(d1) > SN(d2) ? N(d1) > N(d2)
As mentioned above N(d2) is simply the risk adjusted probability that the option will be exercise. It’s linkage to X suggests that it only depends on when the event
On the other hand N(d1) will always be greater than N(d2) because in linking it with the contingent receipt of stock in the Black Scholes equation, N(d1) must not only account for the probability of exercise as given by N(d2) but must also account for the fact that exercise or rather receipt of stock on exercise is dependent on the conditional future values that the stock price takes on the expiry date. In particular this means stock prices being greater than the exercise price are taken as a given condition when calculating the expected future value of stock on the expiry date.
If you are interested in putting this intuition to work within an Excel spreadsheet to build a deeper understanding of the framework, take a look at our newly released video lecture series from our youtube channel.
The video is the fourth video in a series of lectures that build up the framework presented above and translate it into a discrete Excel model. See “An intuitive derivation of N(d2) using the Black Scholes model” a post the covers an alternate derivation of N(d2) and includes the four video collection in one easy to access location for free.