Interpretation of N(d1) and N(d2)

[sridhar]

David:

Is there is a straightforward "English language" interpretation for the BSM-related probabilities N(d1) and N(d2) -- for example, since N(d1) seems to be "linked" to S (current stock price) and N(d2) to the PV of the strike price, is there a simple way to understand the meaning of N(d1) and N(d2) -- I am wondering if N(d1) means the probability of the stock price at option expiration exceeding the strike price and N(d2) -- the prob. of the stock price at expiration being below the strike price.

To be honest, I am just graspig at straws here...But from the BSM equation -- S * N(d1) and PV(K) * N(d2) seems to smell (respectively) like the expected value of S and expected value of PV(K) -- but I am struggling to connect the dots.

Since the call option payoff is max (0, S-t - K) [S-t being stock at time T the time of expiration]

I see the symmetry between this and the call option price in the BSM equation...I will stop rambling now.

Summary: What is the English interpretation of N(d1) and N(d2)? Since the delta of a long call option is N(d1), there must be some generic interpretation....I must have missed it.

--sridhar

 

[David Harper CFA FRM]

Hi sridhar,

Option pricing was my specialty in practice (for many years, educating clients, etc) and I think you have come about as near as you can to an intuition based on direct examination. First, IMO, it may be unrealistic to expect an English language intuition for N(d1) and N(d2) based on direct examination. (Did you consider these dots may not be directly connect-able?!) B-S comes from differential equation and I'm not totally sure we have the right to expect an intuitive view on the solution to a differential equation. That said, we can get close enough for the important points where, I think, the important idea is to see that the formula is based on a hedge (long delta shares and short the bond/cash).

I've said often on the forum here that I think you should start with N(d2) because d2 is similar to distance to default (DD) in the Merton. d2 in Black-Scholes is DD in the Merton model (de Servigny Chapter 3) with two exceptions:

1. A riskless rate (d2 in BS) is replaced by the asset's growth rate (d2 in Merton for Credit), and
2. The strike price (X or K in BS) is replaced by default point [e.g., ST debt + 50% of LT debt] in the Merton).

The FRM candidate especially does well to see that:

1. N(d2) is not option pricing, it is mechanical and, if you walk through it, intuitive Here is 10 min screencast where i walk through N(d2). It is useful b/c it employs three or more FRM building block ideas!.
2. N(d2) is equal to the probability the Stock Price (Future Firm Asset value) will breach the Strike Price (Default point) in the future. Under assumptions, mainly: lognormal asset prices.

Given that, we can say N(2) is NOT exactly the probability the Euro call option will be struck, but rather (and this is a profound distinction leading to the core of option pricing) N(2) is the probability the option will be struck in the risk-neutral world where the expected return on the stock is the riskless rate (r); i.e., the actual probability the option will be struck is greater than N(d2). Put another way, the risk neutral valuation idea applies to the option value but not to N(d2). My suggestion here is, if you seek intuition, examine N(2) in the context of a Merton model because you will see it is mechanical (not option pricing theory) and then you can switch over to N(d2) in the Black-Scholes to see how it is the prob the call option is struck in the riskneutral but not the real world.

And, yes you are totally correct, N(d1) is the delta. So, ultimately where we are with an intuition is the following:

Stock*delta - Discounted Strike*(prob of striking at expiration in the RISKNEUTRAL world).

For myself, the above is satisfying intuition (i.e., I don't try to take it further; short of an encounter with the differential equation) because the Black-Scholes, we might remind ourselves, is a DYNAMIC HEDGE. It can solve for the call by the no-arbitrage idea that our hedge has a riskless payoff:

We are LONG delta N(d1) shares of stock and SHORT N(d2) bond/cash;

My point here is, IMO, rather than seeking further intuition of N(d1) and N(d2), I might focus on the self-financing hedge that requires the call option to be worth this hedge. Then, yes, N(d1) is the delta, and we can say more about that. And N(d2) is, like Merton, 1 - PD but with the important difference around the riskless rate and a risky growth rate. Hope that helps!

David