economic meaning of d1 in Black Scholes

cdsharm75 · July 7th, 2024, 3:17 am

Hello!
Searched this forum and found some questions relating to mine....but not the exact same, hence posting this.
While I follow the derivation of the BS formula and the terms d1 and d2, I was trying to understand if there's an economic meaning of these terms. Most of the literature I've seen discusses Nd1 and Nd2 (Neilson etc.) but not d1 and d2 themselves.

As I starting point, I tried to understand the reason for subtracting the sigma term from the risk free rate in the dynamics of the underlying,

dS/S = (r  - 0.5*sigma^2)*dt + sigma*dW

and learnt that we adjust the risk free rate by the vol term to get to the "geometric mean", which shows the correct aggregation for log returns (hopefully correct). Moving on ahead with the derivations, I reached the BS formula and thought some intuition could be built around d1 and d2. No luck so far:

In the d1 formula we end up adding the vol term back to the risk free rate:

What is the economic/finance/stat/math interpretation of doing this?
In addition, what is the economic intuition of adding a moneyness measure (the lnS/X) to the (r + 0.5sigma^2) term and then scaling it by the vol?

Perhaps it is pointless to look for intuition at this level of detail.....I was just curious to see if someone could provide any direction.
Thanks!

Marsden · July 7th, 2024, 11:53 am

I don't know if there's an "economic meaning" to d1 or d2, but mathematically they might be thought of as reflecting the "tilt" of the exponential nature of the assumed stock price distribution: d1 is bigger because it represents the "fatter" right-hand side of the distribution, and d2 is smaller because it represents just the non-exponential normal probability that the exercise price is exceeded.

And the balance between the two is just whatever makes risk neutral pricing work.

Not sure if that will clarify anything for you.

cdsharm75 · July 7th, 2024, 1:59 pm

@Marsden - thank you! It IS helpful ; the perspective on "tilt" as you put it.....and good to know that other folks may not see an economic meaning behind them.

katastrofa · July 7th, 2024, 9:46 pm

So do they measure what you call "moneyiness"?

cdsharm75 · July 8th, 2024, 6:43 pm

@katastofa - thanks for the response......I'd say d1 does include it.....(the lnS/X indicates the moneyness).....but beyond that, I'm not able to think of any economic/finance meaning. The math intuition, it's helpful the way@Marsden describes it.

economic meaning of d1 in Black Scholes

economic meaning of d1 in Black Scholes

Re: economic meaning of d1 in Black Scholes

Re: economic meaning of d1 in Black Scholes

Re: economic meaning of d1 in Black Scholes

Re: economic meaning of d1 in Black Scholes