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Understanding of Black-Scholes-Merton Model


New Member
Dear David,

I have been struggling with understanding of the Black-Scholes-Merton model and I have a few confusions that I hope you can kindly enlighten.

1) for d1:
after studying your video, I seem to understand that d1 means "future expected return IN(S/K) divided by future volatility (sigma*square root(t))" and also plus "half of one future volatility(sigma*square root(t))". Assuming it follows a normal distribution, graphically, it represents how far away it sits to the right hand side of the mean (since the "plus" sign means that we are kind of trying to find out how far upward the return can go).
if my understanding is not wrong, why "half" of the future volatility is added, not 0.4, 0.8 or 1? stock return can be higher, right?

2) for d2:
I'm thinking that if my explanation for d1 as above is correct, likewise d2 graphically represents how far away it sits to the left hand side of the mean (since the "minus" sign means that we are kind of trying to find out how far downward the return can go). Then I have the same confusion as for d1: why "half" of the future volatility is added, not 0.4, 0.8 or 1? stock return can be higher, right?

3) in the video-cast titled" Importance of d2 in Black-Scholes to Merton Model in Credit Risk – 10 min screencast" (page (http://www.bionicturtle.com/learn/article/importance_of_d2_in_black_scholes_to_merton_model_in_credit_risk_10_min_scr/ ), the formula you provided for d2 considers In(S/K). I've checked with Hull (page 291)and FRM handbook(5th) (page 144), and found out that Hull's formula is consistent with yours but the FRM formula uses a slight different item, that is "In{S/K*exp(-rt)}". I'm not sure if FRM handbook is making another mistake again.

4) What's more, how can I relate d1 to delta?

Sorry about so many stupid questions. Thank you very much!


David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi Liming,

It is admirable to attempt a direct intuition of the BSM, however, I think it is difficult or maybe impossible
(my own techniques, that you reference, are sort of metaphors I developed in client work...they are not a means to direct intuition)

The BSM is based on a dynamic hedge: Long Delta Shares plus short the cash necessary to exercise the strike in the future;
i.e., delta*Stock - discounted (Strike price)

...so my own preferred method to "access" BSM is to start with minimum value (lower bound) of European call:
lower bound (call) = S - K * exp(-rT); i.e., value if volatility = 0
and "wrap-in" the N(d1) and N(d2) which increase to account for volatility:
N(d1)*S - N(d2)*K * exp(-rT); i.e., must increase b/c N(d1) > N(d2)

so, the key ideas are:
1. N(d1) is delta (first partial derivative)
2. N(d2) is the probability the option will be expire ITM (i.e., prob option will be struck)

my recommendation is always to memorize d2 because d2 = distance to default in Merton and can itself be understood intuitively(!)
N(d2) = N(-DD) = PD in Merton model *except* Merton uses firm growth rate instead of riskless rate.
let me say it another way: Merton PD is not option pricing: d2 can be understood graphically.

So, I think your (1) and (2) are good tries but they cannot really ever get directly to the intuition (let me remind you something Hull says: the BSM does not depend on the expected return of the stock, this throws a wrench in our attempts to intuit); the BSM is a differential equation and ultimate it derives from the PDE based on the dynamic hedge.

(3) Thank you for pointing me to this formula p. 291. This is an atypical expression, IMO, but it is mathematically equivalent to Hull; e.g., it is easily show that the exp(-rt) solves out to become the (r + variance)*T in Hull's

(4) N(d1) is delta for the Euro call option; it literally is solved by taking the first partial derivative of the BSM. Hopefully, this makes sense?
...in fact, mentally, i cheat:
c = N(d1)*S + etc; BSM
assume N(d1) is a contant = a
c = a*S + etc
dc/dS = a = N(d1)
...that is a cheat, the actual first derivative is more complicated (but not too hard) but it helps me keep track

Thanks, David